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INTRODUCTIONTO

PARTIAL DIFFERENTIAL EQUATIONS

2WA90 COURSE NOTES, 1ST EDITION

Luc Florack

c©May 27, 2019, Eindhoven University of Technology

INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

Code: 2WA90

Author: Luc Florack

Version: May 27, 2019

CONTENTS

Preliminaries v

I PDE Systems: Concepts & Motivations 1

1 Introduction 3

1.1 Partial Differential Equations and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Ill-Posed versus Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Generalised Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Classification of Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Existence and Uniqueness 11

2.1 Theorem of Cauchy-Kowalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Calculus of Variations 13

3.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Basic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

i

CONTENTS CONTENTS

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Distribution Theory 21

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Distributions Formalised . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Reparametrisations of the Dirac Point Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Fourier Transformation 31

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 The Fourier Transform on S (Rn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 The Fourier Transform on S ′(Rn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 The Fourier Transform on L2(Rn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5 Fourier Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Complex Analysis 39

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.2 C-Differentiability and Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 39

II PDE Systems: Techniques & Examples 43

7 The Fourier Method 45

7.1 Basic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8 The Method of Characteristics 49

8.1 Basic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9 Separation of Variables 53

9.1 Basic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

ii

CONTENTS CONTENTS

9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10 Fundamental Solutions & Green’s Functions 57

10.1 Basic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

10.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.2.1 Via Classical Homogeneous Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.2.2 Via Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

10.2.3 Via Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

III PDE Systems: Common Cases 63

11 First Order Systems 65

11.1 The Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

11.2 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12 Second Order Hyperbolic Systems 69

12.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

13 Second Order Parabolic Systems 81

13.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

13.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

14 Second Order Elliptic Systems 85

14.1 The Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14.2 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14.3 The Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

iii

CONTENTS CONTENTS

iv

PRELIMINARIES

Numbers

• N = 1, 2, 3, . . .: The set of natural numbers. If 0 is included we write N0.

• Z = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .: The set of integer numbers.

• Q = p/q | p, q ∈ Z, q 6= 0: The set of rational numbers.

• R: The set of real numbers, i.e. the closure Q containing all limit points of Q.

• C = a+ bi | a, b ∈ R: The set of complex numbers: i2 = −1.

• K: Shorthand for either R or C, depending on context.

Functions

It is tacitly understood that functions are K-valued. If relevant, the context should reveal whether K = R orK = C. We abbreviate x = (x1, . . . , xn) ∈ Rn and

∫Ωf(x)dx =

∫Ωf(x1, . . . , xn) dx1 . . . dxn for Ω ⊂ Rn.

The essential supremum ess supx∈Ω f(x) of a function f is the smallest M ≥ 0 s.t. f(x) ≤ M for almost allx ∈ Ω.

• C(Ω): The set of continuous functions with domain Ω.

• Ck(Ω), k ∈ N0: The set of k-fold continuously differentiable functions on Ω, with C0(Ω) = C(Ω).

• C∞(Ω): The set of smooth, i.e. infinitely differentiable functions with domain Ω.

• Cω(Ω): The set of analytical functions with domain Ω.

• Lp(Ω), p ≥ 1: The set of functions for which∫

Ω|f(x)|p dx exists. Notation: ‖f‖p =

(∫Ω|f(x)|p dx

)1/p.

• L∞(Ω): The set of functions f for which ess supx∈Ω |f(x)| exists. Notation: ‖f‖∞ = ess supx∈Ω |f(x)|.

• S (Rn): The set of smooth functions of rapid decay on Rn.

v

Preliminaries Preliminaries

• S ′(Rn): The topological dual of S (Rn), a.k.a. the set of tempered distributions on Rn.

• D(Ω): The set of smooth bump functions with compact support in Ω ⊂ Rn.

• D ′(Ω): The topological dual of D(Ω), a.k.a. the set of distributions on Ω ⊂ Rn.

Some frequently used standard Cω(R)-functions:

sinx =eix − e−ix

2i, cosx =

eix + e−ix

2, sinhx =

ex − e−x

2, coshx =

ex + e−x

2(x ∈ R) .

Useful Inequalities

Hölder Inequality. Let 1≤p, q≤∞ such that 1/p+1/q=1, f ∈Lp(Ω) , g∈Lq(Ω), then ‖fg‖1 ≤ ‖f‖p ‖g‖q .

Young Inequality. Let 1≤p, q, r≤∞ with 1/p+1/q=1+1/r, f ∈Lp(Ω), φ∈Lq(Ω), then ‖f∗φ‖r ≤ ‖f‖p‖φ‖q .

Multi-Indices

An n-dimensional multi-index α is an n-tuple of integers (α1, . . . , αn) ∈ Zn. Its order or norm is defined as|α| = α1 + . . . + αn. If α = (α1, . . . , αn) ∈ Zn, β = (β1, . . . , βn) ∈ Zn are multi-indices of dimension n,x = (x1, . . . , xn) ∈ Rn, and f : Ω→ K : x 7→ f(x) is a suitably defined function, then

α+ βdef= (α1 + β1, . . . , αn + βn)

∇αfdef=

∂α1+...+αnf

∂xα11 . . . ∂xαnn

xαdef= xα1

1 . . . xαnn

α!def= α1! . . . αn!(

αβ

)def=

(α1

β1

). . .

(αnβn

)∑α

def=

∑α1

. . .∑αn

vi


Abbreviations

a.e. almost everywhereb.c. boundary conditionh.o.t. higher order term(s)i.c. initial conditioniff if and only ifl.h.s. left hand sideo.d.e. ordinary differential equationp.d.e. partial differential equationp.d.o. partial differential operatorr.h.s. right hand sides.t. such that

Oriented Region and its Boundary

An oriented region Ω (blue) with boundary ∂Ω (black) and outward normal n (arrow). Closure of Ω: Ω = Ω∪∂Ω.Complement of Ω (white background): Ωc = U\Ω, in which U is the “universe” of all admissible sets, in thisexample U = R2.

Stokes’ Theorem

Theorem (Stokes’ Theorem). Let Ω ⊂ Rn be an oriented region with boundary ∂Ω and ω a sufficiently smoothdifferential form. Then ∫

Ω

dω =

∫∂Ω

ω .

At this level of abstraction, Stokes’ Theorem requires knowledge of differential forms and exterior derivatives,cf. Spivak [5]. For our purposes the following special cases in standard vector calculus notation suffice.

Corollary. Cf. Stokes’ Theorem. Let dσ denote the (n−1)-dimensional elementary surface measure of ∂Ω andn the unit outward normal. For sufficiently smooth f, g : Ω→ K and v, w : Ω→ Kn, we have

•∫

Ω

divw dx =

∫∂Ω

w · n dσ.

•∫

Ω

(fdiv v + v · ∇f) dx =

∫∂Ω

f v · n dσ.

vii


•∫

Ω

(f∆g − g∆f) dx =

∫∂Ω

(f∇g − g∇f) · n dσ.

Here div v =∑ni=1 ∂ivi denotes the divergence of v = (v1, . . . , vn). Note that the last two identities follow from

the first by taking w=fv, respectively by taking w=f∇g, and subsequently interchanging roles f ↔ g.

Common Coordinate Reparametrisations

Polar coordinates for (x, y) ∈ R2, with (r, φ) ∈ R+ × [0, 2π):x(r, φ) = r cosφ ,y(r, φ) = r sinφ .

Jacobian determinant:

det∂(x, y)

∂(r, φ)= r .

Cylindrical coordinates for (x, y, z) ∈ R3, with (r, φ, ζ) ∈ R+ × [0, 2π)× R: x(r, φ, ζ) = r cosφ ,y(r, φ, ζ) = r sinφ ,z(r, φ, ζ) = ζ .


det∂(x, y, z)

∂(r, φ, ζ)= r .

Spherical coordinates (ISO convention) for (x, y, z) ∈ R3, with (r, θ, φ) ∈ R+ × [0, π]× [0, 2π): x(r, φ, θ) = r sin θ cosφ ,y(r, φ, θ) = r sin θ sinφ ,z(r, φ, θ) = r cos θ .


det∂(x, y, z)

∂(r, θ, φ)= r2 sin θ .

Dirac function (minuscule/majuscule letters refer to free, respectively fixed point coordinates, l.h.s. and r.h.s.correspond to Cartesian, respectively polar/cylindrical/spherical representations):

δ(x−X)δ(y − Y ) =δ(r −R)δ(φ− Φ)

r,

δ(x−X)δ(y − Y )δ(z − Z) =δ(r −R)δ(φ− Φ)δ(ζ − Z)

r,

δ(x−X)δ(y − Y )δ(z − Z) =δ(r −R)δ(φ− Φ)δ(θ −Θ)

r2 sin θ.

Note: (X,Y ), respectively (X,Y, Z), do not coincide with the coordinate origin.

viii


Unit Balls and Spheres

Let Bn = x ∈ Rn | ‖x‖<1 be the unit ball in Rn, and Sn−1 = ∂Bn = x ∈ Rn | ‖x‖= 1 the correspondingunit n-sphere, then the volume, respectively surface area of Bn and Sn−1 are given by

voln Bn =

√πn

Γ(n2 + 1),

voln−1 Sn−1 =2√πn

Γ(n2 ).

Recursions:voln+1 Bn+1 =

voln Sn

n+ 1with vol0 S0 = 2,

voln+1 Sn+1 = 2π voln Bn with vol0 B0 = 1.

ix


x

IPDE SYSTEMS:

CONCEPTS &MOTIVATIONS

1

1INTRODUCTION

1.1. Partial Differential Equations and Boundary Conditions

Recall the multi-index convention on page vi. A linear partial differential equation (p.d.e.) for a K-valued functionu : Ω→ K with domain Ω ⊂ Rn is an equation of the form

Lu = f on Ω, (1.1)

in which f : Ω→ K is a given function, and L is a linear partial differential operator (p.d.o.):

L =∑|α|≤N

cα∇α . (1.2)

The coefficients cα ∈ K may depend on the independent variable x ∈ Ω. If there exists a coefficient cα 6= 0 forsome α with |α| = N , then the p.d.e. is said to be of orderN . The function f is referred to as the inhomogeneousterm. The p.d.e. is called inhomogeneous if f 6= 0, otherwise it is called homogeneous.

A p.d.e. is often furnished with boundary conditions (b.c.’s):

Bu = g on ∂Ω, (1.3)

in which B is typically a linear p.d.o. and g : ∂Ω→ K a given function on the boundary ∂Ω of Ω:

B =∑|α|≤M

bα∇α . (1.4)

The coefficients bα ∈ K may vary along the boundary ∂Ω. If there is a distinguished entry in the n-tuplex = (x1, . . . , xn) ∈ Rn that is interpreted as time (or some other evolution parameter), then boundary conditionsspecified for fixed initial time are usually referred to as initial conditions (i.c.’s).

The system Eqs. (1.1–1.4) is referred to as a boundary value (respectively initial value) problem, or simply as a(p.d.e.) “system”.

3

1.2. EXAMPLES 1. INTRODUCTION

The following generalisations occur in practice:

• The p.d.e. system may be nonlinear, i.e. the coefficients cα or bα may depend on the dependent variable u.

• The boundary may be partitioned into parts, each with its own boundary condition.

• Instead of a scalar-valued function u : Ω→ K we may have a multi-valued function u : Ω→ Kk.

• The boundary ∂Ω may not be a priori fixed (“free boundary”).

• We may want to admit “non-classical” solutions that are not in CN (Ω), cf. Section 1.5.

1.2. Examples

Certain p.d.e.’s are frequently encountered in physics. Below are some of the most prevalent ones.

Notation 1.2.1. The following notational convention is in effect throughout these lecture notes:

• The Laplacian operator in n spatial dimensions is defined as ∆ =

n∑i=1

∂2

∂x2i

.

• The d’Alembertian operator in n+1 spatiotemporal dimensions is defined as 2 =1

c2∂2

∂t2−∆.

In relativistic theories the universal constant c > 0 represents the speed of light. By convention one often setsc = 1. This means that if one chooses to measure time intervals in the S.I. unit of one second, then the length unitfor spatial distances is a light-second, i.e. 2.997 924 580 × 108 m in S.I. units. If relevant, the spatial dimensionn ∈ N should be clear from the context. The d’Alembertian involves a distinguished time coordinate t.

Caveat. One needs to remain cautious about the above notation for ∆, for several reasons:

• It is tacitly understood that the r.h.s. in the above notation pertains to Cartesian coordinates. A change ofcoordinates will generally result in different expressions. Moreover, on curved manifolds, such as the unitsphere, or general relativistic spacetime, Cartesian coordinates do not exist.

• A rigid motion, i.e. a coordinate transformation of the form x = Ry + a, with R ∈ SO(n) an n × northogonal matrix and a ∈ Rn a constant Euclidean vector, preserves the r.h.s. form of the Laplacian andd’Alembertian.

• A Lorentz transformation x = Ly + a with L ∈ SO(n, 1) and a ∈ Rn ×R, preserves the r.h.s. form of thed’Alembertian.

• In special relativity theory one often encounters the imaginary time coordinate x0 = ict. With this (po-tentially deceptive) convention, the d’Alembertian looks like an (n+1)-dimensional Laplacian, whereas aLorentz transformation resembles an ordinary rotation.

Example 1.2.1. Laplace equation: ∆u = 0.

Example 1.2.2. Poisson equation: ∆u = f , i.e. the inhomogeneous Laplace equation, with data term f 6= 0.

Example 1.2.3. Helmholtz equation: (∆ + ν2ω2)u = 0. In relativistic theories ω represents angular temporalfrequency for a propagating wave in a medium with index of refraction ν = 1/c, i.e. inverse propagation speed.

4

1. INTRODUCTION 1.3. ISSUES

Example 1.2.4. Wave equation: 2u = 0.

Example 1.2.5. Heat or diffusion equation: (∂t − α∆)u = 0. By convention one often sets α = 1, if constant.In heat diffusion problems α>0 represents thermal diffusivity.

Example 1.2.6. Schrödinger equation: (i~∂t +~2

2m∆− V )u = 0. This p.d.e. governs the evolution of the wave

function for a particle with mass m and potential energy V in non-relativistic quantum mechanics. In particularwe have V = 0 for a free particle, and V (x) = 1

2mω2‖x‖2 for an isotropic harmonic oscillator with angular

frequency ω. The universal constant ~ = 1.054571800 × 10−34 J s is called Planck’s constant. By conventionone often sets ~ = 1 instead of using S.I. units.

Example 1.2.7. Klein-Gordon equation:(2 + µ2

)u = 0. This is a relativistic counterpart of the free Schrödinger

equation, with mass parameter µ = mc/~, i.e. in natural units µ = m.

Example 1.2.8. Biharmonic equation: ∆2u = 0. This p.d.e. is encountered in elasticity theory.

Example 1.2.9. Advection or transport equation: (∂t + v · ∇)u = 0, in which v · ∇ =∑ni=1 vi∂i. This p.d.e.

plays a role in fluid dynamics, in which case the coefficients vi represent flow velocity components. Note thatif you consider the trajectory of a point particle comoving with the flow, with position x = x(t) at time t, thenv = x = dx/dt. Using this observation together with the chain rule it follows that along the particle’s trajectory,∂t + v · ∇ acts as a total derivative d/dt on any function depending on (x, t) = (x(t), t).

All of the above p.d.e.’s are linear. An example of a nonlinear p.d.e. is the following.

Example 1.2.10. Korteweg de Vries equation: ut+6uux−uxxx = 0. In this example, and henceforth, subscriptst and x denote partial derivatives w.r.t. t and x, respectively. The Korteweg de Vries equation governs thepropagation of water waves.

The physics of the underlying problems often suggest “natural” boundary conditions.

1.3. Issues

In p.d.e. theory various problems need attention, such as

• the genesis of a p.d.e. system (cf. Chapter 3),

• the mathematical concept of a solution (cf. Section 1.5 and Chapter 4),

• existence of a solution (cf. Chapter 2),

• uniqueness of the solution (cf. Chapter 2),

• regularity of the solution (cf. Chapter 2),

• bounds on the solution (pointwise or with respect to some norm),

• well-posedness of the system (cf. Section 1.4),

• construction of an exact solution,

• numerical approximation of a solution,

• other properties of the solution.

5

1.4. ILL-POSED VERSUS WELL-POSED PROBLEMS 1. INTRODUCTION

Not all of these issues will be addressed in-depth. It is clear though that many of these problems are related. Forinstance, it makes no sense to try and approximate the solution if no solution exists, or if the solution does notdepend continuously on the data (i.e. if the problem is ill-posed).

The situation for p.d.e.’s is more complicated than for ordinary differential equations (o.d.e.’s). For instance,whereas anN -th order o.d.e. typically has anN -dimensional solution space, a p.d.e. is, in general, not guaranteedto have a solution at all. If it does, the solution space tends to be infinite-dimensional. Moreover, uniqueness andwell-posedness are issues more intricate for p.d.e.’s than for o.d.e.’s. The example in the next section illustratesthis for the case of well-posedness.

1.4. Ill-Posed versus Well-Posed Problems

The problem of ill-posedness is best appreciated by a simple example.

Example 1.4.1. Consider the following Laplace equation with Dirichlet b.c.’s in n=2 dimensions: ∆u = 0 on Ω = R+0 × R

u(0, y) = fε(y)ux(0, y) = 0

with data term fε(y) = ε sin (y/ε) for some ε > 0. This problem has a unique solution (the proof relies on theTheorem of Cauchy-Kowaleska, cf. Chapter 2), and is not difficult to establish, viz.

uε(x, y) = ε sin(yε

)cosh

(xε

).

You may verify that this is indeed a solution, that it is analytical on Ω, and that it has no upper bound. Moreover,the data term is bounded in the sense that ‖fε‖∞ = ε, which can be made arbitrarily small by suitable choiceof 0 < ε 1. Curiously, the solution uε develops a singularity in the limit ε ↓ 0, showing that this problem isill-posed. For a well-posed problem you would have expected the solution to converge to the null function on Ω(the unique solution for vanishing data term) in any reasonable norm.

The ill-posedness of the problem in the example is not an inherent problem of the linear p.d.o. ∆ as such, butof the formulation of the overall system including b.c.’s. We will encounter b.c.’s that do make the problemwell-posed.

1.5. Generalised Solutions

A classical solution to Eqs. (1.1–1.4) is one that satisfies the system in the strict sense of an N -fold continuouslydifferentiable function fulfilling the stated differential conditions. The a priori regularity condition required may,however, be too restrictive, as it excludes “obvious” generalisations. Consider e.g. the system

∂u

∂t= c

∂u

∂xfor (x, t) ∈ R× R, (1.5)

with constant parameter c ∈ R, subject to the initial condition

u(x, 0) = f(x) , (1.6)

for some specified function f : R→ K. It is easy to verify through a change of variables,ξ = x+ ctτ = t

(1.7)

6

1. INTRODUCTION 1.6. CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS

that v(ξ(x, t), τ(x, t)) = u(x, t) satisfies ∂v/∂τ = 0, so that v(ξ, τ) = v(ξ, 0), whence u(x, t) = v(ξ(x, t), 0) =v(x+ ct, 0). Imposing the initial condition, Eq. (1.6), yields

u(x, t) = f(x+ ct) , (1.8)

revealing a travelling wave with profile f(x) (snapshot at t=0) propagating with velocity |c|. For this to representa classical solution we need to require f ∈C1(R), so that u∈C1(R×R), yet this seems merely a technicality. Forinstance, imagine a sequence fnn∈N of C1(R)-functions, with a non-differentiable limit f=limn→∞ fn. Thusalthough f 6∈C1(R), f can be arbitrarily well approximated by a classically admissible wave profile fN ∈C1(R)for N large enough. It is undesirable to exclude such limiting profiles f 6∈ C1(R) and (a fortiori) corresponding,“generalised solutions” u given by Eq. (1.8). The theory of distributions resolves this dilemma, cf. Chapter 4.

1.6. Classification of Partial Differential Equations

P.d.e.’s can be classified according to different criteria, such as their order, their linear or nonlinear character,and—in case of linear p.d.e.’s—the constant or variable nature of the coefficients of the p.d.o.

For linear p.d.e.’s with constant coefficients a more detailed classification is obtained in terms of the so-calledsymbol. It is based on the observation that functions of the form e(x; ξ) = eix·ξ are eigenfunctions of a linearp.d.o. for any parameter ξ ∈ Rn:

Le(x; ξ) =∑|α|≤N

cα∇αe(x; ξ) =∑|α|≤N

cα(iξ)αe(x; ξ)def= p(ξ)e(x; ξ) . (1.9)

Definition 1.6.1. Let L =∑|α|≤N cα∇α be a linear p.d.o.

• The polynomial p(ξ) =∑|α|≤N

cα(iξ)α is called the symbol of L.

• The homogeneous polynomial of degree N , pN (ξ) =∑|α|=N

cα(iξ)α, is called the main symbol of L.

Example 1.6.1. In n = 2 dimensions a second order linear p.d.o. with constant coefficients has the form

L = a∂2

∂x2+ 2b

∂2

∂x∂y+ c

∂2

∂y2+ d

∂

∂x+ e

∂

∂y+ f ,

with a, b, c, d, e, f ∈ R. Its main symbol defines a real quadratic form:

p2(ξ, η) = a(iξ)2 + 2b(iξ)(iη) + c(iη)2 = −(aξ2 + 2bξη + cη2) = − (ξ, η)

(a bb c

)(ξη

). (1.10)

The eigenvalues of the coefficient matrix

A =

(a bb c

)are real by virtue of symmetry of the matrix. Three scenarios are possible:

• detA > 0: L is called elliptic (the eigenvalues of A have equal signs);

• detA = 0: L is called parabolic (exactly one eigenvalue of A vanishes);

• detA < 0: L is called hyperbolic (the eigenvalues of A have opposite signs).

7

1.7. OVERVIEW 1. INTRODUCTION

The classification in this two-dimensional example can be extended to higher dimensions. However, the signof the determinant detA = λ1 . . . λn is then no longer conclusive, and one needs to inspect the individualeigenvalues λ1, . . . , λn ∈ R of the n×n symmetric coefficient matrix of the main symbol.

Example 1.6.2. In n > 2 dimensions a second order linear p.d.o. with constant coefficients has the form

L =

n∑i,j=1

aij∂2

∂xi∂xj+

n∑i=1

ai∂

∂xi+ a0 .

Its main symbol defines a real quadratic form:

p2(ξ) =

n∑i,j=1

aij(iξi)(iξj) = −n∑

i,j=1

aijξiξj . (1.11)

By virtue of symmetry, the eigenvalues of the coefficient matrix

A =

a11 . . . a1n

...an1 . . . ann

are real, say λ1, . . . , λn ∈ R. One now distinguishes the following qualitative categories:

• all λi 6= 0 and of equal signature: L is called elliptic;

• all λi 6= 0, with exactly one eigenvalue of opposite signature relative to the rest: L is called hyperbolic;

• at least one λi is zero: L is called parabolic;

• all other scenarios (relevant if n ≥ 4): L is called ultrahyperbolic.

Linear p.d.o.’s with variable coefficients can be classified pointwise, i.e. after freezing the argument x = x0 ∈ Ωof the coefficient functions cα(x0), and subsequently treating these coefficients as constants. The classificationmay then of course vary with the choice of x0.

1.7. Overview

Part I

In the remainder of Part I we address some of the conceptual issues touched upon in this introductory chapter inmore detail:

• Any attempt to solve a p.d.e. system should be accompanied by considerations of existence and uniqueness,recall Section 1.3. An important result in this context is formulated in Chapter 2.

• Chapter 3 discusses the variational principle often used in scientific models to generate p.d.e. systems,recall Section 1.3.

• In Chapter 4 we consider the concept of generalised solutions of a p.d.e. system, a motivating example ofwhich was given in Section 1.5.

• Chapter 5 introduces Fourier theory, providing important tools for linear shift invariant systems.

• Chapter 6 reviews some important results on differentiability from complex analysis.

8

1. INTRODUCTION 1.7. OVERVIEW

Part II

Unfortunately, given existence and uniqueness, there is no single method of general applicability for solvingp.d.e. systems. On the positive side, there do exist a few methods that are, to some extent, generic.

In Part II we focus on a number of generic techniques, de-emphasizing applications:

• Chapter 7 exploits the Fourier theory of Chapter 5 to solve p.d.e. systems.

• Chapter 8 introduces the method of characteristics in the context of first order p.d.e.’s.

• Chapter 9 explains the method of separation of variables.

• Chapter 10 introduces the method of Green’s functions, or fundamental solutions, using Chapter 4.

Part III

Finally, in Part III, we put things together, applying the concepts and techniques of the foregoing parts to study aselection of prevalent p.d.e. systems:

• Chapter 12 considers hyperbolic systems, illustrated by the wave equation.

• Chapter 13 considers parabolic systems, illustrated by the heat equation.

• Chapter 14 considers elliptic systems, illustrated by the Laplace equation.

We concentrate on analytical tools. Numerical and perturbative schemes constitute another important class ofsolution methods, especially for systems that cannot be handled analytically (the typical case). Such methods are,however, beyond the scope of this course. Nevertheless, the most powerful solution strategies tend to be of a “hy-brid” nature, with perturbative expansions around an analytically known “zeroth order” system, and/or powerfulnumerical schemes constructed based on slick use of analytical properties of the system. In this sense one mayview these lecture notes as an important stepping stone towards numerical and perturbative p.d.e. methods.

9

1.7. OVERVIEW 1. INTRODUCTION

10

2EXISTENCE AND

UNIQUENESS

2.1. Theorem of Cauchy-Kowalewski

The Theorem of Cauchy-Kowalewski plays an important role in establishing existence and uniqueness of solu-tions for a broad class of p.d.e. systems. In this chapter we state the main result for later convenience. You arereferred to the literature for a rigorous proof (the pivot of which is Taylor’s Theorem).

Definition 2.1.1. Let L =∑|α|≤N cα∇α be a linear p.d.o. on the n-dimensional region Ω. An analytical (n−1)-

dimensional submanifold V ⊂Ω is called non-characteristic at x0 ∈ V relative to L if there exist local coordinatesy = (y1, . . . , yn) ∈ Rn in a neighbourhood of x0, with x− x0 = Ry for some rotation R, s.t.

• yn=0 describes the tangent plane to V at y = 0 (i.e. at x=x0), and

• L can be written as L(y) = c(y)∂N

∂yNn+ . . . with c(y) 6= 0 on a neighbourhood of y= 0, with the ellipsis

denoting terms not involving ∂N/∂yNn .

Thus V is non-characteristic at x0 ∈ V if the p.d.o. has a nontrivialN -th order term transversal to V at that point.

Theorem 2.1.1 (Cauchy-Kowalewski). Recall Eqs. (1.1–1.2):

Lu = f on Ω , (2.1)

11

2.2. EXAMPLES 2. EXISTENCE AND UNIQUENESS

with L =∑|α|≤N cα∇α. We assume that the b.c.’s take the form

u = g0 ,∂u

∂n= g1 , . . . ,

∂uN−1

∂nN−1= gN−1 on V , (2.2)

in which V ⊂ Ω is an (n−1)-dimensional analytical manifold, and in which∂

∂n= n · ∇ denotes derivation

perpendicular to V . We furthermore assume that

• all coefficients cα, |α| ≤ N , as well as f are analytical on Ω,

• gk is analytical on a neighbourhood of x0 ∈ V for each k = 0, . . . , N − 1,

• V is non-characteristic at x0 relative to the p.d.o. L.

Then there exists a unique, analytical solution of Eqs. (2.1–2.2) on a neighbourhood of x0.

It is important to keep in mind that Theorem 2.1.1 does not tell us anything about well-posedness of the system.Thus an “analytical solution” is not necessarily tantamount to a “nice solution”. Recall Example 1.4.1.

2.2. Examples

Example 2.2.1. Consider the following p.d.e. with i.c.’s (cf. Chapter 12, notably Section 12.1): 2u = h for (x, t) ∈ Ω ⊂ Rn × R+,u|t=0 = φ ,ut|t=0 = ψ ,

The plane V : t=0 on which the i.c.’s are specified has a normal in t-direction everywhere and the p.d.o. containsa corresponding second order term of the form ∂2u/∂t2. In other words, V is non-characteristic relative to 2. Incase of analytical h, φ and ψ, Theorem 2.1.1 therefore guarantees a unique analytical solution near t = 0.

Example 2.2.2. Consider the following p.d.e. with i.c. (cf. Chapter 13, notably Section 13.1):ut −∆u = 0 for (x, t) ∈ Ω ⊂ Rn × R+,

u|t=0 = f .

The plane V : t = 0 on which the i.c. is specified has a normal in t-direction everywhere, but the p.d.o. doesnot contain a corresponding second order term ∝ ∂2/∂t2 in that direction. In other words, V is everywherecharacteristic relative to ∂t −∆. Thus in this case Theorem 2.1.1 is not applicable.

Example 2.2.3. Consider the following p.d.e. with b.c.’s (cf. Chapter 14, notably Section 14.1):∆u = 0 for x ∈ Ω ⊂ Rn,u|V = φ ,

∂u

∂n

∣∣∣∣V

= ψ ,

in which V : n ·x = 0 represents a plane with normal n ∈ Rn on which the b.c.’s are specified and ∂/∂n denotesthe derivative normal to V . By virtue of rotation invariance of ∆ a rotation may be performed s.t. it containsan explicit second order normal derivative ∂2/∂n2. Therefore V is non-characteristic relative to ∆. In case ofanalytical φ and ψ, Theorem 2.1.1 thus guarantees a unique analytical solution near x=0 (or any other x ∈ V ).

Example 2.2.4. Recall Example 1.4.1. This special case is covered by the previous example and thus has aunique, analytical solution near x=0, with explicit form given in that example. Clearly, Theorem 2.1.1 does notguarantee a well-posedness solution!

12

3CALCULUS OF

VARIATIONS

3.1. Origin

In science p.d.e. systems often emerge as necessary and/or sufficient conditions from physical laws originallystated in integral forms, collecting “stuff” inside some closed region. Variational problems constitute a partic-ularly important class. These are often used to describe dynamical systems characterised by some conservedquantity, such as energy-momentum, or by some stationary quantity, such as the so-called action functional.

3.2. Basic Technique

Consider a fixed region Ω ⊂ Rn, together with a function F : Ω×K×Kn → R : (x, u,∇u) 7→ F (x, u,∇u), inwhich x is shorthand for the n independent variables x = (x1, . . . , xn) ∈ Ω as usual, and in which u : Ω→ K :x 7→ u(x) and∇u : Ω→ Kn : x 7→ ∇u(x) denote some dependent variable and its first order partial derivatives,respectively. (Although it does cover a wide class of relevant problems, the restriction to first order is merelyimposed to simplify the discussion and could be dropped.)

Next, consider the functional J : U (Ω)→ R : u 7→ J [u], given by

J [u] =

∫Ω

F (x, u,∇u) dx , (3.1)

13

3.2. BASIC TECHNIQUE 3. CALCULUS OF VARIATIONS

in which dx is shorthand for the elementary volume measure dx = dx1 . . . dxn, and U (Ω) ⊂ D ′(Ω) denotessome function subspace. In a nutshell, the variational principle entails the study of how this functional behavesunder sufficiently small but otherwise arbitrary variations δu of the function u, seeking those instances of u forwhich the functional is stationary, i.e. for which the “infinitesimal” variations δJ [u] = J [u + δu] − J [u] vanishidentically (to first order) for all δu.

To cope with the ∞-dimensional nature of the vector space U (Ω), consider 1-dimensional variations aroundu ∈ U (Ω) of the form δu = εv along some arbitrarily fixed element v ∈ U (Ω), with ε ∈ R “infinitesimally”small. The variational problem can then be recast into a standard calculus problem of finding stationary points ofJ [u + εv] interpreted as a function of ε ∈ R at ε = 0, indicating the change induced by moving along the fixed“direction” v ∈ U (Ω) at the fiducial “point” u ∈ U (Ω), viz.

d

dεJ [u+ εv]ε=0 = 0 for all v ∈ U (Ω). (3.2)

The condition that this should hold for all v ∈ U (Ω) is essential, as it renders the problem independent of theprecise direction, or “mode of variation”. As we will see below, Eq. (3.2) yields a p.d.e. for u, with solutionu = u∗ (say).

Of course, the integrand F in Eq. (3.1) must be subject to certain conditions in order to guarantee existence and,if desired, uniqueness of a stationary point u∗. Additional “convexity” constraints are needed for such a pointto correspond to a (local) minimum of J . For our purposes we require F ∈ C2(Ω × K × Kn) as a necessarycondition, de-emphasizing the details of the other abovementioned aspects.

Let us apply the recipe of Eq. (3.2) to Eq. (3.1). In that case we obtain

J [u+ εv] =

∫Ω

F (x, u+ εv,∇u+ ε∇v) dx . (3.3)

A straightforward expansion up to first order in ε produces

J [u+ εv] =

∫Ω

F (x, u,∇u)dx+ ε

∫Ω

[Fu(x, u,∇u) v +

n∑i=1

Fui(x, u,∇u)∂v

∂xi

]dx+ h.o.t. (3.4)

in which the subscripts u and ui denote derivatives w.r.t. the scalar argument u and the i-th component ∂iu ofthe vector argument ∇u (i = 1, . . . , n), respectively. The term “h.o.t.” stands for higher order terms in ε. Thex-dependence of the dependent variables u = u(x) and v = v(x) is suppressed for the sake of brevity.

In order to factor out the arbitrary function v in the first order term of the integrand, apply partial integration,exploiting the chain rule

Fui(x, u,∇u)∂v

∂xi=

∂

∂xi(Fui(x, u,∇u) v)− ∂Fui

∂xi(x, u,∇u) v . (3.5)

Together with the definition of J [u], Eq. (3.1), this yields

J [u+εv] = J [u]+ε

∫Ω

[Fu(x, u,∇u)−

n∑i=1

∂Fui∂xi

(x, u,∇u)

]v dx+ε

∫Ω

n∑i=1

∂

∂xi(Fui(x, u,∇u) v) dx︸︷︷︸∗

+h.o.t.

(3.6)The ∗-integral can be rewritten with a result from differential geometry known as Stokes’ Theorem, recall page vii.The specific instance actually needed here is known in vector calculus as Green’s Theorem or the DivergenceTheorem. The divergence of a vector field w : Ω→ Kn : x 7→ w(x) is defined as

divw =

n∑i=1

∂wi∂xi

. (3.7)

14

3. CALCULUS OF VARIATIONS 3.3. EXAMPLES

The consequence of Stokes’ Theorem is that the volumetric ∗-integral can be expressed as a surface integral:∫Ω

n∑i=1

∂

∂xi(Fui(x, u,∇u) v) dx =

∫∂Ω

n∑i=1

Fui(x, u,∇u) ni v dσ . (3.8)

All in all we have obtained

limε→0

J [u+ εv]− J [u]

ε=

∫Ω

[Fu(x, u,∇u)−

n∑i=1

∂Fui∂xi

(x, u,∇u)

]v dx+

∫∂Ω

n∑i=1

Fui(x, u,∇u) ni v dσ .

(3.9)The condition of stationarity entails that this should vanish for all v ∈ U (Ω). This implies

Fu(x, u,∇u)−n∑i=1

∂Fui∂xi

(x, u,∇u) = 0 on Ω, (3.10)

n∑i=1

Fui(x, u,∇u) ni = 0 on ∂Ω. (3.11)

To see this, first consider the subset of all modes v ∈ U0(Ω) that vanish on the boundary ∂Ω. Then the secondterm on the r.h.s. of Eq. (3.9) is absent, implying that the part in-between square brackets in the integrand of thefirst term must be identically zero, which proves Eq. (3.10). Subsequently use this fact in Eq. (3.9), now alsoincluding all nontrivial modes v ∈ U (Ω) that do not vanish on the boundary. This establishes Eq. (3.11).

Eq. (3.10) is known as the Euler-Lagrange equation associated with the functional Eq. (3.1), furnished with anatural b.c., viz. Eq. (3.11).

3.3. Examples

We illustrate the variational principle by some examples taken from classical mechanics. The first exampleillustrates the principle in the simplest context of an action functional for the path of a single point particle, whichtakes the form of a time integral of a so-called Lagrangian function (kinetic minus potential energy associated withthe particle’s trajectory x(t)). This case is extended to continuum mechanics in the second example, illustratinghow a wave phenomenon for a field quantity u(x, t) emerges from a dynamical system described in terms of anaction functional that takes the form of a spatiotemporal integral of a Lagrangian density (kinetic minus potentialenergy densities “stored” in a field configuration u(x, t)).

Example 3.3.1. For a single point particle the action functional over the time interval [0, T ] takes the form

S[x] =

∫ T

0

L(t, x, x) dt ,

in which x = x(t) ∈ Rn and x = x(t) ∈ Rn are parametrisations of the particle’s position and velocity at timet ∈ [0, T ].

The action principle entails that S[x] is stationary for a physical path x = x(t) ∈ Rn, i.e.

δS[x] =

∫ T

0

δL(t, x, x) dt =

∫ T

0

(n∑i=1

∂L

∂xiδxi +

n∑i=1

∂L

∂xiδxi

)dt = 0 .

Via partial integration this is seen to be equivalent to

δS[x] =

n∑i=1

∫ T

0

(∂L

∂xi− d

dt

∂L

∂xi

)δxidt+

n∑i=1

∫ T

0

d

dt

(∂L

∂xiδxi

)dt = 0 .

15

3.3. EXAMPLES 3. CALCULUS OF VARIATIONS

The last term is a boundary term:

∫ T

0

d

dt

(∂L

∂xiδxi

)dt =

[∂L

∂xiδxi

]T0

=∂L

∂xiδxi

∣∣∣∣t=T

− ∂L

∂xiδxi

∣∣∣∣t=0

.

Since each instantaneous δxi(t), t ∈ [0, T ], is a priori independent, stationarity implies

(•)

∂L

∂xi− d

dt

∂L

∂xi= 0 ,

∂L

∂xi

∣∣∣∣t=0

= 0 ,

∂L

∂xi

∣∣∣∣t=T

= 0 .

The o.d.e. obtained in this way is known as the “Euler-Lagrange equation” corresponding to the action S[x]. Theexpression pi = ∂L/∂xi is known in the trade as the (i-th component of the) “momentum”. Thus the “natural”b.c.’s are provided by vanishing momenta.

In classical mechanics, the Lagrangian for a one-particle system represents the difference L= T−U of kinetic(T ) and potential energy (U ) of the particle. In particular, for a particle of mass m>0 attached to an ideal springwith elastic modulus κ>0 we have

S[x] =1

2

∫ T

0

[mx2(t)− κx2(t)

]dt .

For N -particle systems one introduces a phase-space pair (xν(t), xν(t)) for each particle, ν = 1, . . . , N . Onethen adds up their joint kinetic and potential energy contributions to build the corresponding Lagrangian.

The next example illustrates the continuum limit for an N -particle system as N → ∞, replacing discrete pointtrajectories xν(t) by a dense spatiotemporal field u(x, t).

Example 3.3.2. Consider the following functional on (n+1)-dimensional flat spacetime Rn×R+, with ρ, τ > 0:

S[u] =1

2

∫R+

∫Rn

[ρut(x, t)

2 − τ‖∇u(x, t)‖2]dx dt .

The procedure of Section 3.2, notably Eqs. (3.10–3.11), produces the following Euler-Lagrange system (wedisregard the boundary at infinity):

ρutt − τ∆u = 0 on Rn × R+,

ut = 0 for (x, t) ∈ Rn × 0.

This is the wave equation, recall the d’Alembertian operator introduced in Section 1.2 on page 4, with c2 = τ/ρ.

If one takes the form of the Lagrangian density for granted, the above example explains the origin of the associatedwave phenomenon, which the “deus ex machina” p.d.e. system as such fails to do. But where did the specificLagrangian density come from? The next example, adapted from Gelfand and Fomin [3, Chapter 36], illustratesthis for the simplified case n=1, by making the explicit connection between a discrete N -particle system (in theappropriate limit N →∞) and its continuum counterpart formulated in terms of a physical field.

16


Figure 3.1: Vibrating string. Vertical displacement and stretching are highly exaggerated for the sake of clarity.

Example 3.3.3. We consider the transverse displacement of a homogeneous string, elastically fastened at the endpoints x = 0 and x = `, meaning that the end points feel a restoring force proportional to their displacementsu(0, t) and u(`, t). The setup is sketched in Fig. 3.1 for some fiducial moment t ∈ [0, T ] in time.

More specifically, the string is attached at the end pointst x= 0 and x= ` to ideal springs constrained to allowstretching only in vertical direction. Unstretched, these strings have zero length, while their elastic moduli aregiven by the constants κ1>0 and κ2>0, respectively. The string has horizontal length ` and its equilibrium stateis u(x, t) = 0 for all x ∈ [0, `] and t ∈ [0, T ], say. During vibration it will be assumed that the overall stretchingof the string remains small, so that the length of the string is approximately constant, i.e. roughly equal to `.

In order to construct the Lagrangian for our continuous string (excluding the springs at the end points) we maydivide it into N horizontal pieces of equal width ∆x (so that `=N∆x), and consider each element as a particlewith mass ∆mν = ρ∆x, in which ρ>0 is the string’s (constant) linear mass density. Freezing position x ∈ [0, `]along the horizontal axis, the kinetic energy of the particle is determined by its velocity ut(x, t), and equals, tolowest order in ∆x,

∆T ≈ 1

2ρut(x, t)

2∆x .

If the displacements u(x, t) and u(x+ ∆x, t) of the string at the ends of the interval (x, x+ ∆x) differ, then, bythe Pythagorean rule, the difference ∆u = u(x + ∆x, t) − u(x, t) induces a (small) stretching of the string byan amount ∆` =

√∆x2 + ∆u2 −∆x = ∆x(

√1 + (∆u/∆x)2 − 1) = 1

2ux(x, t)2∆x + h.o.t. Assuming zeropotential energy of the string in equilibrium, the stretched piece of string then acquires a potential energy equal tothe work needed for the stretching, viz. ∆U = τ∆`, in which τ is the string’s (approximately) constant tension(magnitude of forces applied by the ends of the string once stretched). To lowest order in ∆x we thus find:

∆U ≈ 1

2τux(x, t)2∆x .

Adding up all contributions ∆L = ∆T −∆U for the string pieces and taking the limit ∆x → 0 (and N → ∞while keeping ` = N∆x fixed), we obtain the action functional as a time-integral of the Lagrangian L, whichitself becomes a spatial integral of a Lagrangian density L :

(∗) S[u] =1

2

∫ T

0

∫ `

0

L (x, t, u, ux, ut) dx dt ,

with(?) L (x, t, u, ux, ut) =

1

2

[ρut(x, t)

2 − τux(x, t)2].

Upon varying the function u by δu we obtain a lowest order variation δL of L in (?) given by

δL = ρutδut − τuxδux = − [ρutt − τuxx] δu+∂

∂t[ρutδu]− ∂

∂x[τuxδu] ,

17


which in turn induces a variation δS of S in (∗) given by

δS =

∫ T

0

∫ `

0

δL dx dt = −∫ T

0

∫ `

0

[ρutt − τuxx] δu dx dt+

∫ T

0

∫ `

0

∂

∂t[ρutδu] dx dt︸︷︷︸

−∫ T

0

∫ `

0

∂

∂x[τuxδu] dx dt︸︷︷︸

.

The last two terms are boundary terms:

()∫ T

0

∫ `

0

∂

∂t[ρutδu] dx dt = ρ

∫ `

0

[ut(x, T )δu(x, T )− ut(x, 0)δu(x, 0)] dx ,

respectively

() −∫ T

0

∫ `

0

∂

∂x[τuxδu] dx dt = −τ

∫ T

0

[ux(`, t)δu(`, t)− ux(0, t)δu(0, t)] dt .

These terms vanish identically when restricting variations to δu(x, T ) = δu(x, 0) = δu(`, t) = δu(0, t) = 0. TheEuler-Lagrange equations, essentially already derived in the previous example (take n=1), thus become:

(••) ρutt − τuxx = 0 on (0, `)× (0, T ).

The system’s natural b.c.’s are found by independent variation of δu(x, T ) and δu(x, 0), considered as functionsof x ∈ (0, `), respectively of δu(`, t) and δu(0, t), considered as functions of t ∈ (0, T ):

(†)ut(x, 0) = ut(x, T ) = 0 on (0, `),ux(0, t) = ux(`, t) = 0 on (0, T ).

These do not necessarily reflect realistic constraints, since the physical details of the fastening of the end pointsof the string have been ignored (sofar).

In the final example we account for the contributions of the end points as well.

Example 3.3.4. The Euler-Lagrange equation (••)—or rather the analysis, still to be done, of its solutions—clarifies the potential string dynamics if no boundary or initial/final conditions, such as (†), are imposed. Inrealistic scenarios the b.c.’s must reflect the way in which the end points are fastened, so that we need to includethe contribution of the springs in Fig. 3.1 into an extended action functional, which is the sum of (∗) for the stringitself and two additional terms, one for each end point. Moreover, unlike the b.c.’s, the temporal conditions areartifacts of our a priori arbitrary choice of temporal interval. For this reason T →∞ in Example 3.3.2, admittingonly an initial condition, which may likewise be dispensed with by taking the time interval to be all of R.

Let us investigate the natural physical boundary conditions by including the springs into the system. In order topull the end points away from equilibrium (u=0) towards their actual fixation points, u(0, t) and u(`, t), the workexpended will be “remembered” in the form of potential energy. Recall that the springs are ideal, i.e. they havezero mass, and zero length if unloaded, with elastic moduli κ1 > 0 and κ2 > 0. Again assuming zero potentialenergy for unloaded springs, and defining u1(t) = u(0, t) and u2(t) = u(`, t), we thus have

U1 =1

2κ1u1(t)2 and U2 =

1

2κ2u2(t)2 ,

for their respective potential energies, with corresponding actions

S1[u1] = −1

2

∫ T

0

κ1u1(t)2 dt and S2[u2] = −1

2

∫ T

0

κ2u2(t)2 dt .

Added to (∗) this yields the action functional for the entire setup including b.c.’s:

Stotal[u] = S[u] + S1[u1] + S2[u2] .

18


Recall that (••) arises from the variational principle by considering stationarity of the action under all variationsδu of u with δu(x, T ) = δu(x, 0) = δu(`, t) = δu(0, t) = 0. Thus when varying Stotal[u] under variations δuincluding nontrivial variations δu(0, t) 6= 0 and δu(`, t) 6= 0, then (••) is a necessary but insufficient condition.Imposing this, we may restrict our attention to the variation of the boundary actions:

δStotal[u] = () + () + δS1[u1] + δS2[u2] .

Discarding () by imposing δu(x, T )=δu(x, 0)=0 (since we have not specified any a priori physical constraintsfor the initial and final configuration of the system), and observing that

δS1[u1] = −κ1

∫ T

0

u1(t)δu1(t) dt = −κ1

∫ T

0

u(0, t) δu(0, t) dt ,

respectively

δS2[u2] = −κ2

∫ T

0

u2(t)δu2(t) dt = −κ2

∫ T

0

u(`, t) δu(`, t) dt ,

we end up with

δStotal[u] = −∫ T

0

[τux(`, t) + κ2u(`, t)] δu(`, t) dt+

∫ T

0

[τux(0, t)− κ1u(0, t)] δu(0, t) dt .

The action principle dictates that this should vanish for all δu(`, t) and δu(0, t), so that we finally obtain thefollowing b.c.’s for t ∈ (0, T ):

−κ1u(0, t) + τux(0, t) = 0 ,κ2u(`, t) + τux(`, t) = 0 .

These “mixed” b.c.’s include the limiting cases of free end points (κ1, κ2 → 0, implying ux(0, t) =ux(`, t) = 0)as well as of rigidly fixed end points (κ1, κ2 →∞, implying u(0, t)=u(`, t)=0.)

19


20

4DISTRIBUTION

THEORY

4.1. Motivation

A classical p.d.o., recall Eqs. (1.1–1.2), requires differentiability of its operand. If we wish to admit non-classicalsolutions, such as Eq. (1.8) for the p.d.e. system Eqs. (1.5–1.6) with f 6∈ C1(R), then this raises a paradox.

The resolution is provided by the theory of distributions. This theory employs auxiliary, so-called test functions,which are, by construction, differentiable in classical sense. The problem statement of Eqs. (1.1–1.2) is thenreformulated into one in which the p.d.o. is “transposed” to these test functions.

To appreciate the idea, consider the simplest o.d.e. in n = 1:

u′ = f on R. (4.1)

For a classical solution to exist, we need to require u ∈ C1(R) and f ∈ C0(R). Assuming, for the moment, thatthis is the case, multiply left and right hand sides with a test function φ ∈ S (R), and integrate:∫ ∞

−∞u′(x)φ(x)dx =

∫ ∞−∞

f(x)φ(x)dx for all φ ∈ S (R). (4.2)

The class S (R) is explained in Section 4.2. For the present case all we need to know is that φ ∈ C1(R) andφ(x) → 0 “sufficiently fast” as x → ±∞. Since we require Eq. (4.2) to hold for all φ ∈ S (R), it is equivalentto the classical formulation Eq. (4.1).

21

4.1. MOTIVATION 4. DISTRIBUTION THEORY

Next, perform a partial integration step on the l.h.s. The result is

−∫ ∞−∞

u(x)φ′(x)dx =

∫ ∞−∞

f(x)φ(x)dx for all φ ∈ S (R). (4.3)

Note that there is no effective boundary term on the l.h.s., since by construction1 u(x)φ(x) → 0 wheneverx→ ±∞.

Given the original regularity constraints on f and u, Eq. (4.3) is again equivalent to Eq. (4.1), as well as toEq. (4.2). However, if we take Eq. (4.3) as our point of departure, i.e. if we would have formulated the o.d.e. interms of this equation a priori, then there is apparently no need for insisting on differentiability or even continuityof the functions u and f , as long as the integral expressions are well-defined.

We may mimic the original o.d.e. formulation, Eq. (4.1), by redefining the classical functions

f : R→ K : x 7→ f(x) , (4.4)u′ : R→ K : x 7→ u′(x) , (4.5)

in Eq. (4.1) through “function overloading”, as follows:

f : S (R)→ K : φ 7→ f(φ) =

∫ ∞−∞

f(x)φ(x)dx , (4.6)

u′ : S (R)→ K : φ 7→ u′(φ) = −∫ ∞−∞

u(x)φ′(x)dx . (4.7)

These are the distributional counterparts of the classical functions, Eqs. (4.4–4.5), originally intended in Eq. (4.1).No confusion between a function and its corresponding distribution is likely to arise as long as one is clear aboutthe applicable prototype, i.e. either R → K (for a classical function) or S (R) → K (for a distribution). Thedistributional counterpart of Eq. (4.1) may thus be written as

u′ = f on S (R). (4.8)

As a shorthand we say that f ∈ S ′(R) if f : S (R)→ K is a continuous linear functional of the type Eq. (4.6).A few considerations arise:

• Can all continuous linear functionals of type S (R) → K be represented by an integral formula such asEq. (4.6)? (Lemma 4.2.1 and Theorem 4.2.2, v.i.)

• For those cases that can be represented by Eq. (4.6), is the correspondence with the classical representationone-to-one? (Notation 4.2.1 and Definition 4.2.5, v.i.)

These issues (for which the strict answer is negative in both cases) will be considered in the next section.

The example of “function overloading” illustrated by Eq. (4.1) and Eqs. (4.4–4.7) generalizes to arbitrary p.d.e.’sin any dimension n, and to any domain of definition Ω ∈ Rn, provided the test functions are properly constructed.The formal requirements are given in Section 4.2.

Observation 4.1.1. The lesson is thus that we may, without formal notational distinction, interpret any p.d.e.either in the “classical” sense, in which operands and data are ordinary functions satisfying differentiabilityconditions compatible with the p.d.e., or in a “distributional” sense, in which operands and data represent linearfunctionals requiring only weak regularity conditions. The latter interpretation admits generalised solutions.

1This explains why we need “sufficiently fast” convergence: the amplification factor u(x) might otherwise spoil convergence!

22

4. DISTRIBUTION THEORY 4.2. DISTRIBUTIONS FORMALISED

4.2. Distributions Formalised

We start by considering the multivariate case with unbounded domain Rn.

Definition 4.2.1. The class S (Rn) of smooth test functions of rapid decay, φ : Rn → K, a.k.a. Schwartzfunctions, is defined as follows:

φ ∈ S (Rn) iff φ ∈ C∞(Rn) and supx∈Rn

|xα∇βφ(x)| <∞ for all multi-indices α and β.

Thus a Schwartz function is an infinitely differentiable function which is “essentially compact”, meaning thatit decays rapidly in any direction towards infinity. Rapid decay applies to any of its derivatives and means thatconvergence of function values towards zero is faster than any monomial factor (or more generally, any functionof polynomial growth) would be able to counteract.

Definition 4.2.2. An a.e. defined function f :Rn→K is called a function of polynomial growth, f ∈P(Rn), ifthere exist constants c>0 and m≥0 such that |f(x)| ≤ c

(1 + ‖x‖2

)ma.e.

Definition 4.2.3. The topological dual S ′(Rn) of S (Rn) is the class of continuous linear functionals T : S (Rn)→ K,a.k.a. the class of tempered distributions.

Theorem 4.2.1. If T ∈ S ′(Rn), then there exist a constant c>0 and multi-indices α, β such that

|T (φ)| ≤ c supx∈Rn

|xα∇βφ(x)| .

Two tempered distributions T1, T2 ∈ S ′(Rn) are equal iff T1(φ) = T2(φ) for all φ ∈ S (Rn). T ∈ S ′(Rn) iscalled positive if T (φ) > 0 for all positive test functions φ ∈ S (Rn). Of special interest are the so-called regulartempered distributions, as these can be associated (non-uniquely) with classical functions.

Definition 4.2.4. The regular tempered distribution Tf ∈ S ′(Rn) associated with a function of polynomialgrowth, f ∈P(Rn), is the tempered distribution

Tf : S (Rn) −→ K : φ 7→∫Rnf(x)φ(x) dx .

One says that the regular tempered distribution Tf is associated with the “function f under the integral”. A regulartempered distribution Tf is often whimsically identified with the function f despite ambiguity. This is just the“function overloading” principle introduced in Section 4.1. Although convenient, one needs to remain cautious.

Notation 4.2.1. If Tf = Tg , i.e. if Tf (φ) = Tg(φ) for all φ ∈ S (Rn), we say that f and g are equivalent: f ∼ g.

Observation 4.2.1. Equivalent functions may differ on a set of measure zero, i.e. w.r.t. physically void details.

Definition 4.2.5. It is customary to write f = g instead of f ∼ g for two equivalent functions of polynomialgrowth, f, g ∈P(Rn), associated with the regular tempered distributions Tf , Tg ∈ S ′(Rn).

Recall that classically, i.e. outside the context of distributions, functions are identified if and only if they agreein all arguments: f = g iff f(x) = g(x) for all x ∈ Rn. It should be clear from the context which of the twoequivalence concepts are applicable.

The space of tempered distributions is strictly larger than the space of regular tempered distributions.

Lemma 4.2.1. Let φ ∈ S (Rn), and a ∈ Rn any given point. Then

δa : S (Rn) −→ K : φ 7→ φ(a)

defines a tempered distribution, called the Dirac point distribution at point a ∈ Rn.

23

4.2. DISTRIBUTIONS FORMALISED 4. DISTRIBUTION THEORY

The Dirac point distribution δ0 is usually written as δ.

Definition 4.2.6. One tends to abuse the notation of Definition 4.2.4, writing any tempered distribution T in theform Tf , even if no such function f ∈ P(Rn) exists. In particular, the Dirac point distribution δa is associatedwith a hypothetical “function under the integral” with the same symbol, δa : Rn → K : x 7→ δa(x) = δ(x− a),s.t.

δa(φ)def=

∫Rnδa(x)φ(x) dx

def= φ(a) .

Notice, however, that, as a “function under the integral”, we have δ(x)=0 for all x 6=0, and that it is meaninglessto ask for the value δ(0).

Observation 4.2.2. Dimensional analysis reveals that, from a physicist’s point of view, one should interpret thefunction under the integral δa : Rn → K as a (singular) density function (normalised to unity upon integration).

The deliberate malpractice of notational ambiguity is carried over to all tempered distributions, cf. Theorem 4.2.2,v.i.

Notation 4.2.2. One often declines to make a notational distinction between functions and distributions. Thus itshould be clear from the context whether a symbol like f denotes a classical function f : Rn −→ K as it mightappear “under the integral” (i.e. f ∈P(Rn)) or a proper distribution f : S (Rn) −→ K (or f ∈ S ′(Rn)).

Except for a few instances where it might lead to confusion, we will adhere to this convenient abuse of notation,as it will allow us to interpret a p.d.e. in either classical or generalised sense, whichever is appropriate.

The core of distribution theory is that we can define arbitrary derivatives of tempered distributions in a well-posedway, independent of operationally meaningless “infinitesimal” details.

Definition 4.2.7. For any nonnegative multi-index α the derivative ∇αT ∈ S ′(Rn) of T ∈ S ′(Rn) is thetempered distribution defined by

∇αT : S (Rn) −→ K : φ 7→ (∇αT ) (φ)def= T (∇ †αφ) ,

in which the transposed p.d.o. is given by∇ †α = (−1)|α|∇α.

The reason for the minus signs is that in the subspace of regular tempered distributions Tf for which f is suffi-ciently smooth, the above definition boils down to the classical definition of differentiation. This can be verifiedby repeated partial integrations using the integral expression on the right hand side of Definition 4.2.4 and thefact that f is smooth and of polynomial growth. For let f ∈ C∞(Rn) ∩P(Rn), then on the one hand we have

∇αTf (φ)def= Tf (∇ †αφ) = (−1)|α|

∫Rnf(x)∇αφ(x) dx ,

by Definition 4.2.7. On the other hand, we may consider the regular tempered distribution T∇αf associated withthe function ∇αf , and apply repetitive partial integrations, i.e.

T∇αf (φ) =

∫Rn∇αf(x)φ(x) dx

p.i.= (−1)|α|

∫Rnf(x)∇αφ(x) dx .

All boundary terms that show up in this process vanish as a result of the rapid decay of the test function (recall theexample in Section 4.1). Comparing the last two results we observe that T∇αf = ∇αTf , a reasonable consistencyrequirement. This explains why Definition 4.2.7 is the natural way to define ∇αT for any tempered distribution.In particular, we may differentiate the Dirac point distribution.

Result 4.2.1. Recall Definition 4.2.7. Let φ ∈ S (Rn), a ∈ Rn any given point, and α a multi-index. Then

∇αδa : S (Rn) −→ K : φ 7→ ∇ †αφ(a) = (−1)|α|∇αφ(a) .

24

4. DISTRIBUTION THEORY 4.2. DISTRIBUTIONS FORMALISED

It can be shown that a distribution whose support is a single point is a finite linear combination of derivatives ofthe Dirac distribution at that point.

Definition 4.2.8. The support supp f of a function f : Ω → K is the closure of the set of all x ∈ Ω for whichf(x) 6= 0.

Theorem 4.2.2. If f ∈ S ′(Rn) is a general point distribution, with supp f = a for some point a ∈ Rn, thenthere exists an order m ∈ Z+

0 such that

f =∑|α|≤m

cα∇αδa .

Point distributions with m = 0 are also referred to as point measures.

Adaptations of the theory of Schwartz exist for arbitrary subdomains Ω ⊂ Rn. Such adaptations are useful forp.d.e. systems defined on an arbitrary region Ω ⊂ Rn.

Definition 4.2.9. The class D(Ω) of smooth compact test functions, φ : Ω→ K, is defined as follows:

φ ∈ D(Ω) iff φ ∈ C∞(Ω) and φ has compact support suppφ ⊂ Ω.

Note that the supremum condition in Definition 4.2.1 is obsolete, its role (viz. the suppression of boundary terms)is automatically fulfilled by the condition of compact support.

A question that might arise is whether nonzero test functions φ ∈ D(Ω) actually exist. If not, the stipulatedequivalence of Eqs. (4.1) and (4.3) would fail for a general domain Ω. Example 4.4.1 illustrates how one mayconstruct a smooth function of arbitrary compact support for the one-dimensional case.

Definition 4.2.10. The topological dual D ′(Ω) of D(Ω) is the class of continuous linear functionals T : D(Ω)→ K,a.k.a. the class of distributions on Ω.

Results obtained for S (Rn) and S ′(Rn) may be adapted to D(Ω) and D ′(Ω) for any subset Ω ∈ Rn, includingΩ = Rn.

Limits on D ′(Ω) (and a fortiori on S ′(Rn), v.i. Theorem 4.2.3) are introduced as follows.

Definition 4.2.11. Let u, un ∈ D ′(Ω), n ∈ N. We say that u = limn→∞

un if u(φ) = limn→∞

un(φ) for all φ ∈ D(Ω).

It is not difficult to show that the acts of differentiation and of taking a limit can be interchanged.

Lemma 4.2.2. Recall Definition 4.2.11. If u = limn→∞

un, then ∇αu = limn→∞

∇αun for all multi-index orders α.

We end with a theorem that sheds light on how the various function spaces are nested. It clarifies in a most conciseway why we may think of the elements of D ′(Ω) as truly generalised functions.

Theorem 4.2.3 (Inclusion Theorem). The following inclusions hold:

D(Rn) ( S (Rn) ( L1(Rn) ( S ′(Rn) ( D ′(Rn) .

Moreover, for compact Ω ∈ Rn we have

Lq(Ω) ⊂ Lp(Ω) whenever 1 ≤ p ≤ q .

25

4.3. REPARAMETRISATIONS OF THE DIRAC POINT DISTRIBUTION 4. DISTRIBUTION THEORY

Note that the deliberate “confusion” of Notation 4.2.2 actually renders the first inclusion chain meaningful, as itrelies on our identification of equivalence classes of functions with (regular) tempered distributions. The secondinclusion chain tells us that, restricted to compact domains, L1(Ω) is the largest and L∞(Ω) the smallest of allLp-type spaces. A compact space is a normed space in which all points lie within some fixed distance of eachother. A closed and bounded subset of Rn (equipped with its standard norm) provides an example of a compactspace. Beyond the realm of Euclidean geometry we may think of intrinsically compact universes U, such asSn−1 = x ∈ Rn | ‖x‖=1, the unit sphere (conceptually embedded in an n-dimensional Euclidean space).

To conclude, a p.d.e. and its solution may be interpreted either in classical or in distributional sense. The merit ofthe notational ambiguity introduced above is that the standard formulation of a linear p.d.e. needs no modification,as it admits both views simultaneously.

Result 4.2.2. The linearN th order p.d.e. Lu = f on Ω may be interpreted either as Lu(x) = f(x) for all x ∈ Ω,or as Lu(φ) = f(φ) for all φ ∈ D(Ω). In the former case it is assumed that u ∈ CN (Ω), f ∈ C(Ω). The lattercase requires only very weak assumptions, viz. u, f ∈ D ′(Ω). In that case it is understood that, by definition,Lu(φ) = u(L†φ). More specifically, if L =

∑|α|≤N cα∇α, then L† =

∑|α|≤N cα∇†α =

∑|α|≤N (−1)|α|cα∇α.

4.3. Reparametrisations of the Dirac Point Distribution

A generalisation of Definition 4.2.6 is particularly useful in applications that call for curvilinear coordinates, suchas polar, cylindrical, or spherical coordinates. Suppose x= x(ξ) represents a (diffeomorphic) reparametrisationof Cartesian coordinates x∈Rn in terms of curvilinear coordinates ξ∈Rn, and a=x(α)∈Rn, say.

Lemma 4.3.1. In Cartesian coordinates, the n-dimensional Dirac delta function, δa(x) = δ(x−a), is separable:

δ(x− a) =

n∏i=1

δ(xi − ai) ,

in which each factor on the r.h.s. represents a one-dimensional Dirac delta function.

Theorem 4.3.1. We have the formal identity

δ(x− a) =δ(ξ − α)

|J(ξ)|in which J = det

∂x

∂ξdenotes the Jacobian determinant.

To see this, note that, on the one hand,

φ(a) =

∫δ(x− a)φ(x) dx =

∫δ(x(ξ)− x(α))φ(x(ξ)) |J(ξ)| dξ ,

in which the r.h.s. is obtained by a change of variables, x = x(ξ). On the other hand we also have

φ(a) = φ(x(α)) =

∫δ(ξ − α)φ(x(ξ)) dξ .

Comparison with the previous representation establishes Theorem 4.3.1. With our conventions for polar, cylin-drical and spherical coordinates (cf. Preliminaries) we obtain the following.

Result 4.3.1. In polar, cylindrical, respectively spherical coordinates in R2/R3 we have

δ(x− x0)δ(y − y0) =δ(r −R)δ(φ− Φ)

r,

δ(x− x0)δ(y − y0)δ(z − z0) =δ(r −R)δ(φ− Φ)δ(ζ − Z)

r,

δ(x− x0)δ(y − y0)δ(z − z0) =δ(r −R)δ(φ− Φ)δ(θ −Θ)

r2 sin θ.

26

4. DISTRIBUTION THEORY 4.4. EXAMPLES

The constant parameters R, Φ, Z and Θ pertain to the appropriate polar, cylindrical, respectively sphericalcoordinates r, φ, z and θ of the fixed points (x0, y0) ∈ R2, respectively (x0, y0, z0) ∈ R3.

The proof is left as an exercise. Cf. Example 4.4.6 for the case of an affine reparametrisation.

4.4. Examples

Example 4.4.1. For c > 0 consider ψ : R→ R : x 7→ ψ(x) given by

ψ(x) =

exp

(1

x2 − c2

)if x ∈ (−c, c)

0 if x 6∈ (−c, c)

It is clear that limx→±c ψ(x) = 0, and in fact, limx→±c ψ(k)(x) = 0 for any k-th order derivative of ψ. Thus if

we set ψ(k)(±c) = 0, then ψ ∈ C∞(R). The function ψ is said to be a smooth function of compact support, a.k.a.as a “bump function”. The support of ψ is the closure of the set of x ∈ R for which ψ(x) 6= 0, i.e. the closedinterval [−c, c] in this example. Notation: ψ ∈ D([−c, c]). We may generate new smooth functions of compactsupport via convolution with an integrable function χ of compact support, say suppχ = [a, b], with a < b:

φ(x) =

∫ ∞−∞

χ(y)ψ(x− y) dy .

The function φ inherits its smoothness from ψ, and its compact support (viz. [a − c, b + c]) from both ψ and χ.Slick choice of parameters will thus allow you to construct a smooth function of any desirable compact support.

Example 4.4.2. The following function fails to be differentiable at x=0 in classical sense:

f : R −→ R : x 7→ f(x) = |x|

However, interpreting it as a regular tempered distribution via the identification f ∼ Tf ∈ S ′(R), i.e.

Tf (φ) =

∫Rf(x)φ(x) dx = −

∫ 0

−∞xφ(x) dx+

∫ ∞0

xφ(x) dx ,

for all φ ∈ S (R), renders it infinitely differentiable in distributional sense. Indeed we have

T ′f (φ) = −∫Rf(x)φ′(x) dx =

∫ 0

−∞xφ′(x) dx−

∫ ∞0

xφ′(x) dx

= [xφ(x)]x=0x→−∞ − [xφ(x)]

x→∞x=0 −

∫ 0

−∞φ(x) dx+

∫ ∞0

φ(x) dx =

∫R

sgn (x)φ(x) dx = Tsgn (φ) ,

in which sgn : R→ R : x 7→ sgn (x) is the sign function:

sgn : R −→ R : x 7→ sgn (x) =

−1 als x < 0,0 als x = 0,1 als x > 0.

Thus in a precise sense we have T ′f = Tsgn . Identifying the r.h.s. with Tf ′ —which strictly holds only forclassically differentiable f—amounts to the “pseudo-classical” statement f ′(x) = sgn (x), i.e.

d

dx|x| = sgn (x) ,

27

4.4. EXAMPLES 4. DISTRIBUTION THEORY

but note that the (classically ill-defined) value of f ′(0) = sgn (0) is irrelevant and has been arbitrarily assignedhere (to ensure antisymmetry). Moreover,

T ′′f (φ) =

∫Rf(x)φ′′(x) dx = −

∫ 0

−∞xφ′′(x) dx+

∫ ∞0

xφ′′(x) dx

= − [xφ′(x)]x=0x→−∞ + [xφ′(x)]

x→∞x=0︸︷︷︸

0

+

∫ 0

−∞φ′(x) dx−

∫ ∞0

φ′(x) dx

= [φ(x)]x=0x→−∞ − [φ(x)]

x→∞x=0 = 2φ(0) = 2δ(φ) .

Thus T ′′f = 2δ. Again adopting the interpretation f ∼ Tf , we may state “pseudoclassically” that f ′′ = 2δ. Thefactor 2 reflects the jump in slope from−1 to +1 of the graph of f at the origin. Higher order derivatives are nowstraightforward: T (k+1)

f ∼ f (k+1) = 2δ(k−1), k ∈ N.

Example 4.4.3. Let U ∈ S ′(R) be a tempered distribution satisfying the following distributional o.d.e.:

(†) U ′′ = δ .

Clearly this o.d.e. does not admit a classical solution U ∈ C2(R), for then U ′′ ∈ C0(R), contradicting the r.h.s.Therefore, stipulate a solution of the form U = Tu ∈ S ′(R) for some function u : R→ R. Example 4.4.2 showsthat

u(x) =1

2|x|

is a generalised solution. Indeed, direct substitution of

U(φ) = Tu(φ) =

∫ ∞−∞

u(x)φ(x)dx

in (†) yields

U ′′(φ) = U(φ′′) =

∫ ∞−∞

u(x)φ′′(x)dx =1

2

∫ ∞−∞|x|φ′′(x)dx = −1

2

∫ 0

−∞xφ′′(x)dx+

1

2

∫ ∞0

xφ′′(x)dx

= −1

2[xφ′(x)]

0−∞ +

1

2

∫ 0

−∞φ′(x)dx+

1

2[xφ′(x)]

∞0 −

1

2

∫ ∞0

φ′(x)dx

=1

2[φ(x)]

0−∞ +

1

2[φ(x)]

∞0 = φ(0) = δ(φ) .

Since this holds for all φ ∈ S (R) we indeed find U ′′ = δ.

Note that this solution is not unique, since we may add a general solution of the corresponding homogeneouso.d.e. Thus the general solution of (†) is given by (the regular tempered distribution corresponding to)

U(x) =1

2|x|+ ax+ b .

Example 4.4.4. The staircase function f : R→ R : x 7→ f(x), given by f(x) = bxc, i.e. the entier of x ∈ R, orthe largest integer k ∈ Z such that k ≤ x, has infinitely many discontinuities, cf. Figure 4.1.

Since f has a discontinuity at each integer, and constant values in-between integers, its classical derivative is

f ′(x) =

0 if x ∈ R\Zill-defined if x ∈ Z .

ThusT ′f 6= Tf ′ ,

28

4. DISTRIBUTION THEORY 4.4. EXAMPLES

Figure 4.1: Graph of y = f(x) = bxc. The red bullets at the discontinuities are not part of the graph.

the r.h.s. being trivial if f ′ : R\Z → R is interpreted as the classical derivative of f , with arbitrarily assignedvalues at the discontinuities (e.g. according to the most parsimonious infilling f(k)=0 for all k ∈ Z), v.s.:

Tf ′(φ) =

∫Rf ′(x)φ(x) dx =

∫R\Z

f ′(x)φ(x) dx = 0 .

The last two equalities hold by virtue of the zero measure of the set Z. For the l.h.s., on the other hand, we find

T ′f (φ) = −T (φ′) = −∫ ∞−∞

f(x)φ′(x) dx ,

which clearly does not vanish for all φ ∈ S (R). Interpreted as a generalised function f ∼ Tf , however, we find

T ′f (φ) = −Tf (φ′) = −∫Rbxcφ′(x)dx = −

∑k∈Z

∫ k+1

k

kφ′(x)dx = −∑k∈Z

k(φ(k+1)−φ(k)) =∑k∈Z

φ(k) =∑k∈Z

δk(φ) ,

for any φ ∈ S (R), whenceT ′f =

∑k∈Z

δk .

Thus, recalling Definition 4.2.6, we may loosely write

d

dxbxc =

∑k∈Z

δk(x) =∑k∈Z

δ(x− k) = III(x) ,

where on the r.h.s. we have introduced the so-called “Dirac comb” III, a.k.a. “impuls train” or “sampling function”in physics and engineering disciplines.

Example 4.4.5. Consider the o.d.e. u′ + u = 0 for x ∈ R. The function f : R→ R : x 7→ f(x), given by

f(x) =

0 x < 0e−x x ≥ 0 ,

clearly provides a solution “almost everywhere”, viz. for x ∈ R\0. Obviously f 6∈C1(R) is not a classical solu-tion due to the discontinuity at the origin. Let us therefore verify whether its associated distribution Tf ∈ D ′(R)satisfies the o.d.e. in distributional sense. (Note that f is not of polynomial growth, which is why we must lookbeyond S ′(R).) This is in fact nearly the case, in the sense that the reservation “almost everywhere” for the clas-sical validity domain is obsolete in the distributional view, and a certain “correction term” that has no classicalcounterpart enters the picture. Taking

Tf (φ) =

∫ ∞−∞

f(x)φ(x) dx =

∫ ∞0

e−x φ(x) dx ,

a direct verification shows that

T ′f (φ)∗= −Tf (φ′) = −

∫ ∞−∞

f(x)φ′(x) dx = −∫ ∞

0

e−x φ′(x) dx?=[−e−x φ(x)

]∞0−∫ ∞

0

e−x φ(x) dx = φ(0)−Tf (φ) .

29

4.4. EXAMPLES 4. DISTRIBUTION THEORY

The equality marked by ∗ holds by definition of distributional differentiation, the one marked by ? follows bypartial integration. (Note that for the last step to hold we must have suitably defined test functions; the choiceφ ∈ D(Rn) made here will do.) Using the definition of the Dirac point distribution, δ(φ) = φ(0), we may rewritethe result as

T ′f (φ) = δ(φ)− Tf (φ) ,

which shows that Tf in fact satisfies the inhomogeneous o.d.e. u′ + u = δ in D ′(R)-distributional sense. Noticethat no restrictions on the domain of definition need to be imposed, and that the result is consistent with theclassical result given the reservation on the classical validity domain, since δ(x) = 0 for x ∈ R\0.

Example 4.4.6. Let A be a constant invertible n×n-matrix and b ∈ Rn a constant vector parameter. Then

() δA,b(x)def= δ(Ax+ b) =

1

|detA|δ(x+A−1b) =

1

|detA|δ−A−1b(x) .

The crucial thing to note is the appearance of the scaling factor. There is no “classical” view to see this (onesometimes sees physically void statements s.a. “δ(0) = ∞”). A true explanation requires a strict distributionalapproach. To this end, let φ ∈ S (Rn), then

δA,b(φ)def=

∫Rnδ(Ax+ b)φ(x) dx

∗=

1

|detA|

∫Rnδ(y)φ(A−1(y − b)) dy

?=

1

|detA|φ(−A−1b)

?=

1

|detA|

∫Rnδ(x+A−1b)φ(x) dx

∗=

1

|detA|

∫Rnδ−A−1b(x)φ(x) dx .

In ∗ a change of variables has been carried out: y = Ax + b, whence x = A−1(y − b) and dy = |detA| dxwith invariant integration boundaries. In ? we have used the definition of the standard Dirac delta function, δ(y),and in ∗ the one for a shifted Dirac delta function, δa(x) with a = −A−1b, recall Definition 4.2.6. From theseobservations, using the fact that φ ∈ S (Rn) is arbitrary, it follows that

δA,b =1

|detA|δ−A−1b ,

for which () provides the “function under the integral”.

30

5FOURIER

TRANSFORMATION

5.1. Introduction

The spaces S (Rn), S ′(Rn), and L2(Rn) are closed under Fourier transformation1. These cases are covered inSections 5.2–5.4, respectively. Section 5.5 summarizes some useful Fourier theorems.

Various definitions of the Fourier transform are encountered in the literature. To appreciate how these are related,we introduce the (a, b)-parametric convention, which covers all cases of practical interest. In this convention,(a, b) ∈ R\0 × R+ are fixed parameters.

Our default will be (a, b) = (1, 1), which is fairly common in pure mathematics and systems engineering.

5.2. The Fourier Transform on S (Rn)

Definition 5.2.1. Fourier transformation is a continuous, linear, invertible mapping F(a,b) : S (Rn) −→ S (Rn) :φ 7→ F(a,b)(φ), defined as follows:

F(a,b)(φ)(ω)def= b

∫Rne−iaω·x φ(x) dx .

1In particular, every tempered distribution f ∈ S ′(Rn) has a Fourier transform. This is not the case for every f ∈ D ′(Rn).

31

5.2. THE FOURIER TRANSFORM ON S (RN ) 5. FOURIER TRANSFORMATION

Notation 5.2.1. For our default convention, (a, b) = (1, 1), we will write F (φ) or φ instead of F(1,1)(φ).

Conjecture 5.2.1. Cf. Definition 5.2.1. The inverse Fourier transformation is given by

F−1(a,b)(ψ)(x) =

|a|n

(2π)nb

∫Rneiaω·x ψ(ω) dω .

Observation 5.2.1. Forward and inverse Fourier transforms are formally related by an (a, b)-reparametrization:

T : R\0 × R+ −→ R\0 × R+ : (a, b) 7→ T (a, b)def= (a′, b′) given by

a′ = −a

b′ =|a|n

(2π)nb.

In other words: F−1(a,b) = F(a′,b′). Note that the transformation T = T −1 equals its own inverse, allowing us

to “toggle” between forward and backward Fourier transforms.

The class of test functions is closed under differentiation. The question arises of how the Fourier transform of aderivative is related to that of the test function itself.

Lemma 5.2.1. Let φ ∈ S (Rn), and let F(a,b)(φ) ∈ S (Rn) be its Fourier transform according to Defini-tion 5.2.1. Then for integer orders k1, . . . , kn ≥ 0, and ω = (ω1, . . . , ωn) ∈ Rn, we have

F(a,b)

(∂k1+...+knφ

∂xk1 . . . ∂xkn

)(ω) = (iaω1)k1 . . . (iaωn)kn F(a,b)(φ)(ω) .

The next theorem follows from Lemma 5.2.1 by linear extension.

Theorem 5.2.1. Let φ ∈ S (Rn), and let F(a,b)(φ) ∈ S (Rn) be its Fourier transform according to Defini-tion 5.2.1. Moreover, let P (c,∇) =

∑α,|α|≤k cα∇α be a k-th order linear p.d.o. with constant coefficients

cα ∈ K, in which the sum extends over all multi-indices α of order |α| = 0, . . . , k. Then we have

F(a,b) (P (c,∇)φ) (ω) = P (c, iaω) F(a,b)(φ)(ω) .

Proof. The basic observation underlying Theorem 5.2.1 is the fact that ∇αeiaω·x = (iaω)α eiaω·x, from whichit follows by linearity that P (c,∇) eiaω·x = P (c, iaω) eiaω·x. In the terminology of linear algebra this statesthat the complex “planar wave” function x 7→ eiaω·x—considered as a function of x with parameter ω—is aneigenfunction of the linear p.d.o. P (c,∇) with eigenvalue P (c, iaω). Using this, it follows that

P (c,∇)φ(x) =|a|n

(2π)nb

∫RnP (c,∇) eiaω·x F(a,b)(φ)(ω) dω =

|a|n

(2π)nb

∫Rneiaω·x P (c, iaω) F(a,b)(φ)(ω) dω .

The first equality follows by Conjecture 5.2.1 upon interchanging differentiation and integration. Inspection ofthe result, again with the help of Conjecture 5.2.1, shows that the function ω 7→ P (c, iaω) F(a,b)(φ)(ω) is theFourier transform of x 7→ P (c,∇)φ(x). 2

Example 5.2.1. Let us consider a few simple cases for n = 1. According to Lemma 5.2.1 we have, for ω ∈ R,

F(a,b)

(dkφ

dxk

)(ω) = (iaω)k F(a,b) (φ) (ω) .

In higher dimensions results are similar, only now we have a frequency vector ω ∈ Rn rather than a scalarfrequency, ω = (ω1, . . . , ωn). Also, we may now consider any linear combination of partial derivatives withrespect to the n spatial coordinates x = (x1, . . . , xn).

32

5. FOURIER TRANSFORMATION 5.3. THE FOURIER TRANSFORM ON S ′(RN )

Example 5.2.2. Recall Lemma 5.2.1, and suppose n = 2. Writing ω = (ωx, ωy) ∈ R2 in this case, Lemma 5.2.1yields:

F(a,b)

(∂p+qφ

∂xp∂yq

)(ωx, ωy) = (iaωx)p (iaωy)q F(a,b) (φ) (ωx, ωy) .

Example 5.2.3. Linear combinations of partial derivatives map to corresponding linear combinations of theirFourier transforms. As an example, consider the Laplacian ∆φ of φ. Fourier transformation yields

F(a,b)

(∂2φ

∂x2+∂2φ

∂y2

)(ωx, ωy) = −a2

(ω2x + ω2

y

)F(a,b) (φ) (ωx, ωy) .

Example 5.2.4. It is beyond our scope to deal rigorously with so-called fractional powers of differential operators,but their significance is most intuitive in Fourier space. The previous example reveals that one may view F (−∆)as a multiplication operator in Fourier space, viz. F (−∆) =

(ω 7→ f(ω)) 7→ (ω 7→ ‖ω‖2f(ω)

). By the same

token, it appears reasonable to define the multiplication operator F (√−∆) = (ω 7→ f(ω)) 7→ (ω 7→ ‖ω‖f(ω)).

In this way a seemingly irreducible second order p.d.o. like the d’Alembertian may be factorised into two conju-gate first order pseudo-p.d.o.’s:

2 =1

c2∂2

∂t2−∆ =

(1

c

∂

∂t± i√−∆

)(1

c

∂

∂t∓ i√−∆

).

As an aside: Paul Dirac (1902–1984) was the first to realize that in four-dimensional spacetime the d’Alembertian(more generally, the Helmholtz operator) can in fact be written as the square of a first order (non-fractional) p.d.o.∑3µ=0 c

µ∂µ, provided the constant coefficients cµ (in the convention of physics: cµ = i~γµ) in this p.d.o. areinterpreted as 4×4 matrices, and (thus) the operand as a 4-component so-called spinor instead of a scalar function.This brilliant “technical triviality” led to the discovery of antimatter (and earned Dirac a Nobel Prize in 1933).

5.3. The Fourier Transform on S ′(Rn)

Formally, distribution theory allows one to extend Fourier transformation to generalised functions, as follows.

Definition 5.3.1. Fourier transformation is a continuous, linear, invertible mapping F(a,b) : S ′(Rn) −→S ′(Rn) : T 7→ F(a,b)(T ), defined as follows:

F(a,b)(T )(φ)def= T (F(a,b)(φ)) for all φ ∈ S (Rn),

recall Definition 5.2.1.

Informally, we have seen that tempered distributions may be loosely identified with “functions under the integral”,provided we adopt suitable notational conventions to deal with the Dirac point distribution and related non-regulartempered distributions, recall Definition 4.2.6 and Theorem 4.2.2. Accepting this level of rigor, which is what weshall do henceforth, the Fourier transform and its inverse take the same form as before.

As an exercise you may convince yourself of the fact that for a regular tempered distribution Fourier transforma-tion in the formal sense of Definition 5.3.1 agrees with our informal definition below for the defining functionunder the integral, assuming that the Fourier integral expression for this function is well-defined. Do bear inmind, however, that functions involved should, strictly speaking, be interpreted as “functions under the integral”.

Definition 5.3.2. Fourier transformation is a continuous, linear, invertible mapping F(a,b) : S ′(Rn) −→S ′(Rn) : f 7→ F(a,b)(f), defined as follows:

F(a,b)(f)(ω)def= b

∫Rne−iaω·x f(x) dx .

33

5.3. THE FOURIER TRANSFORM ON S ′(RN ) 5. FOURIER TRANSFORMATION


F−1(a,b)(g)(x) =

|a|n

(2π)nb

∫Rneiaω·x g(ω) dω .

Example 5.3.1. Let us : R −→ R : ω 7→ us(ω) = e−s|ω|, in which s > 0 is a positive constant. Then

us(x) =1

2π

∫ ∞−∞

eiωx us(ω) dω =1

2π

∫ ∞−∞

e−s|ω|+iωx dω =1

2π

(∫ 0

−∞e(s+ix)ω dω +

∫ ∞0

e(−s+ix)ω dω

).

We have ∫ 0

−∞e(s+ix)ω dω =

[1

s+ ixe(s+ix)ω

]ω=0

ω→−∞=

1

s+ ix.

Similarly, ∫ ∞0

e(−s+ix)ω dω =

[1

−s+ ixe(−s+ix)ω

]ω→+∞

ω=0

=1

s− ix,

so that we finally obtain

us(x) =1

2π

(1

s+ ix+

1

s− ix

)=

1

π

s

x2 + s2.

That we actually obtain a real-valued solution is no coincidence. You should be able to figure out yourself howwe might have anticipated this. (Hint: us(ω) = us(−ω) ∈ R.)

Lemma 5.3.1. The Fourier transform of the Dirac point distribution at the origin, x 7→ δ(x), is a constantfunction:

F(a,b) (δ) (ω) = b .

Note that the symbol b here actually stands for b : Rn −→ K : ω 7→ b.

Proof. Apply the definition of the Dirac point distribution2, Lemma 4.2.1, Page 23.

Lemma 5.3.2. The inverse Fourier transform of a constant function is proportional to the Dirac point distributionat the origin, δ(x):

F−1(a,b) (b) (x) = δ(x) .

We refrain from a proof of Lemma 5.3.2. Note that Conjectures 5.2.1 and 5.3.1 follow from this lemma.

The lemma is equivalent to the following result, which is useful to memorize, as it does not depend on a particularFourier convention: ∫

Rne±iω·x dω = (2π)n δ(x) . (5.1)

You may verify the following formal scaling property, valid for any regular n×n-matrix A and vector B ∈ Rn:

δ(Ax+B) =1

|detA|δ(x+A−1B) . (5.2)

2Note that eiωx is not a Schwartz function. Nevertheless, it is clear that the Dirac point distribution can be generalised to any functionspace in which point evaluation makes sense.

34

5. FOURIER TRANSFORMATION 5.4. THE FOURIER TRANSFORM ON L2(RN )

5.4. The Fourier Transform on L2(Rn)

Among all Lp(Rn)-spaces (p≥1 or p=∞) the space L2(Rn) is rather special in the sense that it is isomorphic toits own dual. In fact L2(Rn) is a so-called Hilbert space, i.e. a complete inner product space. This means that it isequipped with an inner product ( , ) : L2(Rn)×L2(Rn)→ K, which in turn induces a norm ‖ ‖ : L2(Rn)→ R,and that relative to this norm every Cauchy sequence fn converges to an element of L2(Rn). Recall that fnis a Cauchy sequence if for all ε> 0 there exists an N ∈ N such that ‖fm − fn‖ < ε whenever m,n>N . Thiselegant property of L2(Rn) is also reflected in its closure under Fourier transformation. Technically we may usethe same definitions as for the spaces S (Rn) and S ′(Rn) (Definition 5.2.1 and Conjecture 5.2.1, respectivelyDefinition 5.3.2 and Conjecture 5.3.1), but with the same caveat as for the latter, viz. that one should interpretelements f ∈ L2(Rn) as equivalence classes, with f ∼ g iff f(x) = g(x) a.e.

Definition 5.4.1. Fourier transformation is a continuous, linear, invertible mapping F(a,b) : L2(Rn) −→L2(Rn) : f 7→ F(a,b)(f), defined as follows:

F(a,b)(f)(ω)def= b

∫Rne−iaω·x f(x) dx .


F−1(a,b)(g)(x) =

|a|n

(2π)nb

∫Rneiaω·x g(ω) dω .

5.5. Fourier Theorems

Recall Definitions 5.2.1, 5.3.2, 5.4.1 and Conjectures 5.2.1, 5.3.1, 5.4.1. In what follows u1,2 : Rn −→ Kand u1,2 : Rn −→ K are suitably defined functions, meaning that all stipulated products and convolutions arewell-defined, with well-defined (inverse) Fourier transforms. (A ˜ indicates that the corresponding function isconceived of as living in Fourier space.)

Theorem 5.5.1 (Fourier Transform of a Function Product).

F(a,b)(u1u2) =|a|n

(2π)n bF(a,b)(u1) ∗F(a,b)(u2) .

In particular, following our convention (a, b) = (1, 1), we have

F (u1u2) =1

(2π)nF (u1) ∗F (u2) .

This theorem explains the popularity of the Fourier convention in which (a, b) = (±2π, 1), or (a, b) = (±1, 1/(2π)n).

Proof. For notational convenience we set u ≡ u1u2 and F(a,b)(u) ≡ u in this proof. We have

u(ω)def= b

∫Rnu1(x)u2(x) e−iaω·x dx .

Substituting the respective inverse Fourier transforms for u1,2, i.e.

u1,2(x) =|a|n

(2π)n1

b

∫Rneiaω·x u1,2(ω) dω ,

35

5.5. FOURIER THEOREMS 5. FOURIER TRANSFORMATION

yields

u(ω) = b

∫Rn

[|a|n

(2π)n1

b

∫Rneiaω1·x u1(ω1) dω1

] [|a|n

(2π)n1

b

∫Rneiaω2·x u2(ω2) dω2

]e−ia·ωx dx .

We may rewrite this expression as

u(ω) =|a|n

(2π)n1

b

∫Rn

|a|n

(2π)n1

b

∫Rnu1(ω1) u2(ω2)

[b

∫Rneia(ω1+ω2−ω)·x dx

]dω1 dω2 ,

provided we are allowed to interchange the order of integrations. (We take this for granted.) We recognize theinnermost integral as a variant of Eq. (5.1). Thus we conclude that

u(ω) =|a|n

(2π)n b

∫Rnu1(ω1) u2(ω − ω1) dω1 ,

in other words,

u(ω) =|a|n

(2π)n b(u1 ∗ u2) (ω) .

The rest follows by inspection of this result. 2

An immediate result of Theorem 5.5.1 is the following.

Theorem 5.5.2 (Fourier Transform of a Convolution Product).

F(a,b) (u1 ∗ u2) =1

bF(a,b)(u1)F(a,b)(u2) .


F (u1 ∗ u2) = F (u1)F (u2) .

Proof. Set v1,2 = F(a,b)(u1,2). Theorem 5.5.1 applied to the pair v1,2 states that

F(a′,b′)(v1v2) =|a′|n

(2π)n b′(F(a′,b′)(v1) ∗F(a′,b′)(v2)

).

for all (a′, b′) ∈ R\0 × R+. By Observation 5.2.1, Page 32, together with the definitions of v1,2, we mayrewrite this as

F−1(a,b)

(F(a,b)(u1)F(a,b)(u2)

)= b

(F−1

(a,b)(F(a,b)(u1)) ∗F−1(a,b)(F(a,b)(u2))

)= b (u1 ∗ u2) .

Application of F(a,b) to left and right hand sides completes the proof. 2

Theorems 5.5.1–5.5.2 are powerful results that are often used in the analysis of linear shift invariant systems. Adiagrammatic representation of Theorem 5.5.1 is

(u1, u2) −−−−→ u1 u2

F(a,b)

y xF−1(a,b)

(F(a,b)(u1),F(a,b)(u2)) −−−−→ |a|n

(2π)nbF(a,b)(u1) ∗F(a,b)(u2)

(5.3)

Theorem 5.5.2 can be depicted as

(u1, u2) −−−−→ u1 ∗ u2

F(a,b)

y xF−1(a,b)

(F(a,b)(u1),F(a,b)(u2)) −−−−→ 1

bF(a,b)(u1) F(a,b)(u2)

(5.4)

36

5. FOURIER TRANSFORMATION 5.5. FOURIER THEOREMS

In each diagram the upper horizontal route expresses an algebraic operation in the spatial domain (product, resp.convolution). The lower horizontal route expresses the equivalent operation in Fourier space. For each operation(moving from one corner in the diagram to another) there are always two distinct but equivalent alternatives. Notethat you may reverse the vertical arrows if you replace F(a,b) by F−1

(a,b), vice versa.

The diagrams suggest how one could efficiently implement numerical schemes for convolution (either in thespatial domain or in Fourier space), viz. by avoiding the convolution corner in the applicable diagram. Thereason for such a detour would be that whereas convolution is a multi-local operation, multiplication acts point-wise, which potentially saves us a significant amount of computation time. Of course, the concomitant cost ofthe forward and inverse Fourier transformations encountered along the detour has to be taken into considerationas well. Still, in certain cases this may be the most efficient way to compute a convolution product.

Other, equivalent formulations of Theorem 5.5.1 and 5.5.2 are stated in the following results.

Result 5.5.1 (Fourier Transform of a Function Product in Fourier Space).

F−1(a,b)(u1)F−1

(a,b)(u2) =|a|n

(2π)n bF−1

(a,b) (u1 ∗ u2) .


F−1(u1)F−1(u2) =1

(2π)nF−1 (u1 ∗ u2) .

Proof. This is basically Theorem 5.5.1, with u1,2 = F−1(a,b)(u1,2), and with F−1

(a,b) applied to both sides. 2

Result 5.5.2 (Fourier Transform of a Convolution Product in Fourier Space).

F−1(a,b)(u1) ∗F−1

(a,b)(u2) =1

bF−1

(a,b) (u1u2) .


F−1(u1) ∗F−1(u2) = F−1 (u1u2) .

Proof. This is Theorem 5.5.2, subject to the same adaptation as in the previous proof. 2

Diagrams similar to Diagrams (5.3–5.4) can be constructed for the latter two results.

37

5.5. FOURIER THEOREMS 5. FOURIER TRANSFORMATION

38

6COMPLEX

ANALYSIS

6.1. Introduction

Although complex numbers z = x+ iy ∈ C may be viewed as points (x, y) ∈ R2 in the plane, it is important torealize that C-differentiability is a stronger requirement than R2-differentiability. In this chapter we recall a fewrelevant definitions and results from complex analysis.

6.2. C-Differentiability and Holomorphic Functions

Definition 6.2.1. Let Ω ⊂ C be an open subset, and f : Ω→ C a complex-valued function. Then f is said to bedifferentiable at z0 ∈ Ω if

f ′(z0)def= lim

z→z0

f(z)− f(z0)

z − z0∈ C

exists, i.e. if

|f(z)− f(z0)− f ′(z0)(z − z0)| = o(z − z0) (z → z0) .

Theorem 6.2.1 (Cauchy-Riemann Equations). Let f : Ω→ C : z 7→ f(z) = u(z)+ iv(z), with z = x+ iy ∈ C,

39

6.2. C-DIFFERENTIABILITY AND HOLOMORPHIC FUNCTIONS 6. COMPLEX ANALYSIS

u(z) = Re f(z) ∈ R, v(z) = Im f(z) ∈ R, be differentiable at z0 = x0 + iy0 ∈ Ω ⊂ C. Then at z = z0 we have∂u

∂x=

∂v

∂y∂v

∂x= −∂u

∂y

or, equivalently, i∂f

∂x=∂f

∂y.

Definition 6.2.2. Let Ω ⊂ C be an open subset, and f : Ω → C a complex-valued function. The function f iscalled holomorphic at z0 ∈ Ω ⊂ C if it is C-differentiable at z0, and holomorphic on Ω if it is holomorphic atevery point of Ω.

Example 6.2.1. Examples of holomorphic functions (with z = x+ iy):

• z 7→ zk−1, k ∈ N is holomorphic on C,

• z 7→ exp(z) is holomorphic on C,

• z 7→ z−k, k ∈ N, is holomorphic on C\0.

Example 6.2.2. Counterexamples:

• z 7→ z∗ = x− iy is nowhere holomorphic,

• z 7→ |z| =√zz∗ =

√x2 + y2 is nowhere holomorphic,

• z 7→ Re z = (z + z∗)/2 = x is nowhere holomorphic,

• z 7→ Im z = (z − z∗)/(2i) = y is nowhere holomorphic.

Note that the counterexamples can all be written as functions of z and z∗, treated formally as independent vari-ables. As a rule of thumb, such functions are not holomorphic if they depend nontrivially on z∗.

That C-differentiability is a strong condition also becomes apparent from the following theorems.

Theorem 6.2.2. A holomorphic function on an open set Ω ⊂ C is infinitely differentiable on Ω.

Theorem 6.2.3. A holomorphic function f : Ω → C on an open set Ω ⊂ C has a Taylor series expansion atevery point z0 ∈ Ω, which converges on any open ball β ⊂ Ω centered at z0. In particular, if Ω = C, then fadmits a globally convergent Taylor series expansion at every point.

Holomorphic functions of the latter type are called entire functions. An interesting result by Cauchy (but namedafter Liouville) states that if an entire function is bounded, then it is constant. Thus if an entire function is non-trivial in the sense of not being constant, then in some direction (but not necessarily in all directions) it must tendto infinity as |z| → ∞.

Another remarkable property of holomorphic functions is the following.

Theorem 6.2.4 (Cauchy’s Integral Theorem). Let Ω ⊂ C be a simply connected open subset, and f : Ω → C aholomorphic function. Then ∮

γ

f(z) dz = 0

for every closed loop γ ⊂ Ω.

Two types of singularities may occur at isolated points, “poles” and “essential singularities”.

40

6. COMPLEX ANALYSIS 6.2. C-DIFFERENTIABILITY AND HOLOMORPHIC FUNCTIONS

Definition 6.2.3. If f : Ω\a → C is holomorphic and limz→a|f(z)| = ∞, then a ∈ C is called a pole of f . If

this limit cannot be sensibly defined, then a ∈ C is called an essential singularity.

Example 6.2.3. As an example, consider f : C\0 → C : z 7→ exp(1/z) near its singular point 0 ∈ C. This isan essential singularity, because if you approach it in a particular way, e.g. zn = 1/(w + 2πin)→ 0 as n→∞for some arbitrary w ∈ C, then |f(zn)| = | exp(w + 2πin)| = | exp(w)| does not diverge, whereas for someother sequence zn → 0, e.g. zn = 1/n, it does. In fact, in each δ-neighbourhood of the singularity you will beable to find any function value f(z) = exp(w) = λ ∈ C (even at infinitely many points z, |z| < δ 1).

Thus holomorphic functions exhibit a certain “decent” behaviour outside of an isolated pole, as the followingtheorem testifies.

Theorem 6.2.5 (Cauchy’s Differentiation Formula). Let f : Ω → C be a holomorphic function on a simplyconnected open subset Ω ⊂ C, which is continuous on the closure Ω of Ω. Then for every a ∈ Ω we have

f (n)(a) =n!

2πi

∮γ

f(z)

(z − a)n+1dz ,

in which the contour integral is taken counter-clockwise.

This theorem plays an important role in the following representation of a holomorphic function with singularities.

Theorem 6.2.6 (Laurent Series Expansion). Let Ω ⊂ C be the annulus given by Ω = z ∈ C | ρ1 < |z−a| < ρ2,with ρ1, ρ2 ∈ R+ ∪ 0,∞, and f : Ω→ C a holomorphic function. Then inside Ω, f admits a series expansionof the form

f(z) =∑n∈Z

an(z − a)n .

Moreover, the coefficients are given by

an =1

2πi

∮γ

f(z)

(z − a)n+1dz .

recall Theorem 6.2.5.

Note that an infinite annulus, Ω = C\a, is admitted. This theorem allows us to classify the singularity a, andassign an “order” to it if it happens to be a pole:

• If ak = 0 for all k < −m < 0, and a−m 6= 0, then a is a pole of order m ∈ N.

• If for all m ∈ N there exists a nontrivial ak 6= 0 with k < −m, then a is an essential singularity.

A pole of order m = 1 is called a simple pole.

Definition 6.2.4. Recall Theorem 6.2.6. The coefficient a−1 ∈ C is called the residue of f at a ∈ C, notation:a−1 = Res(f ; a).

Theorem 6.2.7 (Residue Theorem). Let Ω ⊂ C be a simply connected open subset, and f : Ω\z1, . . . , zn → C,in which all zi ∈ Ω, i = 1, . . . , n, are distinct. Then∮

γ

f(z) dz = 2πi

n∑i=1

Res(f ; zi) Indγ(zi) ,

in which Indγ(ζ) is the winding number of the closed path γ around ζ ∈ C\γ:

Indγ(ζ) =1

2πi

∮γ

dz

z − ζ.

41

6.2. C-DIFFERENTIABILITY AND HOLOMORPHIC FUNCTIONS 6. COMPLEX ANALYSIS

The winding number gives the number of times the curve γ turns around the point ζ, with a counter-clockwiseturn counting as positive, and a clockwise turn as negative. This is best illustrated with a picture, cf. Fig. 6.1.

Figure 6.1: Illustration of the winding number. Source: Wolfram MathWorld.

42

IIPDE SYSTEMS:TECHNIQUES &

EXAMPLES

43

7THE FOURIER

METHOD


In this chapter we sketch the method of Fourier transformation to solve certain linear p.d.e. systems with constantcoefficients. A tacit assumption made throughout is the existence of a well-defined Fourier representation ofthe solution u of a p.d.e. system, recall Eqs. (1.1–1.2) and the discussion in Sections 5.2–5.4. Furthermore, thedomain of definition is either n-dimensional space, Ω = Rn, or (n+1)-dimensional spacetime, Ω = I ×Rn, thelatter in the case of a distinguished evolution coordinate t ∈ I ⊂ R alongside the spatial coordinates x ∈ Rn.Finally, the coefficients cα will be assumed to be constants, and the b.c.’s trivial in the sense that the solutionvanishes towards infinity in any direction, or is globally bounded.

Taking Ω = Rn and applying Theorem 5.2.1 to Eqs. (1.1–1.2) yields

∑|α|≤N

cα(iω)α u(ω) = f(ω) on Rn, (7.1)

in which f : Rn → K is the Fourier transform of f : Rn → K. This is a simple algebraic equation, so that weobtain the explicit solution to our p.d.e. in Fourier space:

u(ω) =f(ω)∑

|α|≤N cα(iω)αon Rn. (7.2)

45

7.2. EXAMPLES 7. THE FOURIER METHOD

Fourier inversion then yields the solution in spatial representation,

u(x) =1

(2π)n

∫Rneiω·x

f(ω)∑|α|≤N cα(iω)α

dω . (7.3)

Explicit evaluation of this integral tends to be the hardest part (“there ain’t no such thing as a free lunch”).

If I = R one may treat t essentially as a coordinate, so that Eqs. (7.1–7.3) apply with n replaced by n+1. Often,however, the evolution coordinate t ∈ I is treated as a parameter, and Fourier transformation is applied only inthe spatial domain. The result is an equation which is algebraic w.r.t. the spatial frequency parameters, but stillcontaining t-derivatives, in other words, an o.d.e.

7.2. Examples

Example 7.2.1. Solve ∂u

∂t−∆u = 0 for (x, t) ∈ Rn × R+

u(x, 0) = f(x)

in which f ∈ S ′(Rn).

Solution: Substitute

u(x, t) =1

(2π)n

∫Rneiω·x u(ω, t) dω . (7.4)

Note that we treat t as a parameter. As a result we obtaindu(ω, t)

dt+ ‖ω‖2 u(ω, t) = 0 ,

u(ω, 0) = f(ω) .

This o.d.e. system is readily solved:u(ω, t) = f(ω) φt(ω) ,

withφt(ω)

def= e−t‖ω‖

2

.

The final step is Fourier inversion. Things simplify a bit in view of the Fourier theorems of Section 5.5. Forsuppose, for the moment, that f(ω) = 1 for all ω ∈ Rn. In that case, Eq. (7.4) yields

u(x, t) =1

(2π)n

∫Rneiω·x−t‖ω‖

2

dωdef= φt(x) ,

in which(?) φ(x, t)

def= φt(x) =

1√

4πtn e− ‖x‖

2

4t .

The details are left as an exercise. Hint: Rewrite

iω · x− t ‖ω‖2 = −∥∥∥∥√t ω − 1

2ix√t

∥∥∥∥2

− ‖x‖2

4t.

and use the following theorem, a result from complex analysis:

Lemma 7.2.1. For all ξ ∈ Rn we have∫Rne−‖ω+iξ‖2 dω =

√πn

.

46

7. THE FOURIER METHOD 7.2. EXAMPLES

Finally, use Theorem 5.5.2 or Result 5.5.2, to obtain the solution for arbitrary data function f ∈ S ′(Rn):

u(x, t) = (f ∗ φt) (x) .

In mathematics the function φt is called the fundamental solution (i.c. of the heat equation). In linear systemstheory φt is usually referred to as the point spread function, and its Fourier counterpart φt as the transfer function.

Example 7.2.2. Cf. Example 13.2.2 in Chapter 13 for a comment.

Example 7.2.3. Consider the following inhomogeneous system with homogeneous i.c.∂u

∂t−∆u = h for (x, t) ∈ Rn × R+

u(x, 0) = 0

in which h ∈ S ′(Rn × R) s.t. ht(x)def= h(x, t) = 0 for t < 0. Spatial Fourier transformation according to

Eq. (7.4) yields du(ω, t)

dt+ ‖ω‖2 u(ω, t) = ht(ω)

u(ω, 0) = 0

in whichht(ω) =

∫Rne−iω·x h(x, t) dx .

The homogeneous o.d.e. has been solved in Example 7.2.1. Let us write this solution as

u0(ω, t) = A(ω) φt(ω) ,

with φt(ω) as defined in Example 7.2.1. Stipulate a particular inhomogeneous solution of the form

up(ω, t) = A(ω, t) φt(ω) .

Substitution into the inhomogeneous o.d.e. yields

dA(ω, t)

dtφt(ω) = ht(ω) ,

from which we obtain a solution for A(ω, t) in the form (note that 1/φt(ω) = φ−t(ω))

A(ω, t) =

∫ t

0

hτ (ω) φ−τ (ω) dτ .

With this choice, up(ω, t) fulfils the homogeneous i.c., thus

u(ω, t) =

∫ t

0

hτ (ω) φt−τ (ω) dτ .

Fourier inversion yields (using definition (?) in Example 7.2.1)

u(x, t) =

∫ t

0

(hτ∗φt−τ )(x) dτ =

∫ t

0

∫Rnh(ξ, τ)φ(x−ξ, t−τ) dξdτ =

∫Rn+1

h(ξ, τ) θ(t−τ)φ(x−ξ, t−τ) dξdτ ,

in which ∗ denotes the spatial convolution operator, recall Theorem 5.5.2, or equivalently, Result 5.5.2, and inwhich θ is the indicator function on R+, i.e. θ(t) = 0 for t < 0 and θ(t) = 1 for t > 0.

47

7.2. EXAMPLES 7. THE FOURIER METHOD

48

8THE METHOD OF

CHARACTERISTICS


The method of characteristics is generally applicable to first order p.d.e.’s. These take the following form:n∑i=1

ai∂u

∂xi+ cu = f , on Ω ⊂ Rn. (8.1)

in which ai, c, f : Ω → K are sufficiently smooth functions. Typically the b.c. is prescribed on an (n−1)-dimensional submanifold V ⊂ Ω (a special case is V = ∂Ω):

u = ψ on V , (8.2)

with ψ : V → K sufficiently smooth.

To appreciate the method of characteristics, notice that the function a : Rn → Kn defines a vector field, and thatn∑i=1

ai∂

∂xi= a · ∇ (8.3)

represents directional derivation along the vector field a. For this reason it is natural to study the solution ofEqs. (8.1–8.2) along the integral curves of this vector field.

Definition 8.1.1. The characteristic ` through x0 ∈ Ω is the integral curve x = x(t;x0) uniquely determined bythe o.d.e.

` : x = a(x) for t ∈ I(x0).

49

8.1. BASIC TECHNIQUE 8. THE METHOD OF CHARACTERISTICS

with initial condition x(0, x0) = x0. The existence interval of the solution is indicated here by I(x0).

Below we are particularly interested in characteristics through points x0 ∈ V . If ` : x = x(t) is any characteristic,and u`(t) = u(x(t)), c`(t) = c(x(t)), f`(t) = f(x(t)), then

u` + c`u` = f` . (8.4)

If V is transversal to the characteristics, and Ω0 ⊂ Ω is a region of interest such that each point lies on exactlyone characteristic that intersects V at x(0)=x0, say, then the initial condition for Eq. (8.4) becomes

u`(0) = ψ(x0) . (8.5)

Eqs. (8.4–8.5) admit a unique solution.

In the case of constant a ∈ Kn, the characteristics form a family of parallel lines parametrised by x0 ∈ Rn:

` : x(t, x0) = x0 + at . (8.6)

The transversality assumption, nV · a 6= 0 at x0 ∈ V , implies that any neighbouring characteristic sufficientlyclose to ` intersects V in a unique point close to x0 ∈ V . Let us denote the neighbourhood of V within thep.d.e.’s domain of definition Ω on which the transversality condition holds by Ω0⊂Ω, cf. Fig. 8.1. The heuristicprinciple of the method of characteristics is that any point x ∈ Ω0 is uniquely determined by a parameter pair(x0, t), in which x0 singles out a unique characteristic and t a unique point on that characteristic, viz. x =x(t, x0). Therefore this equation can be inverted so as to express the parameters into the original coordinatesof the problem, (t, x0) = (t(x), x0(x)) say. In this way the solution along each of the covering characteristicsimplies the solution of the p.d.e. problem, u(x) = u`(t(x), x0(x)), at least on Ω0 ⊂ Ω.

Figure 8.1: The region Ω0 ⊂ Ω is a union of non-intersecting characteristics ` : x(t, x0) = x0 +at, each uniquelydefined by a transverse intersection x0 ∈ V . In other words, ∀x ∈ Ω0 ∃x0 ∈ V, t ∈ I(x0) s.t. x = x(t, x0).

For the sake of simplicity let us assume that also the coefficient c is a constant, then the solution of Eqs. (8.4–8.5)may be obtained by the method of “variation of constant”, as follows:

• Note that u`,0(t) = Ae−ct solves the homogeneous o.d.e. u` + cu` = 0, in which A ∈ K is a constant.

• Replace A by A(t) and insert u`(t) = A(t) e−ct into the inhomogeneous o.d.e. u` + cu` = f` to find A(t).

• Disambiguate the solution by accounting for the initial condition.

50

8. THE METHOD OF CHARACTERISTICS 8.2. EXAMPLES

The result is

u`(t, x0) = ψ(x0) e−ct +

∫ t

0

f(τ) e−c(t−τ) dτ . (8.7)

Thus u` is a function of the n independent degrees of freedom (x0, t) ∈ Ω0, which can be re-expressed in termsof the n original coordinates x ∈ Ω0 to obtain the explicit solution u(x) on Ω0.

8.2. Examples

Example 8.2.1. Consider the following system:

(?)

ux − uy = 0 on Ω : y > 0,u(x, 0) = ψ(x) on ∂Ω : y = 0.

Consider the parametrised curve ` : (x, y) = (ξ(t), η(t)). Denote the solution along this curve by u` = u `.Take the curve ` to be a characteristic:

` :

ξ = 1 ,η = −1 .

In explicit form: ` : (x, y) = (ξ0 + t, η0 − t). If we leave (ξ0, η0) unconstrained, then the intersection points` ∩ ∂Ω are obtained at t∂Ω = η0, with corresponding points (ξ0 + η0, 0) ∈ ∂Ω. Alternatively we may requiret∂Ω = 0 by assuming η0 = 0, with corresponding intersection points (ξ0, 0) ∈ ∂Ω. We will adopt the latterchoice henceforth:

(2) ` : (x, y) = (ξ0 + t,−t) .

It is clear from its orientation that every characteristic ` intersects the boundary ∂Ω transversally.

The p.d.e. system (?) for u reduces to a simple o.d.e. system for u`, viz.

(∗)

u` = 0 ,u`(0) = ψ(ξ0) .

The solution is the constant functionu`(t) = ψ(ξ0) .

In order to obtain the desired solution of (?) we must relate the parameters (t, ξ0) to (x, y) using (2). Substituting(x, y) for (ξ0 + t,−t) in u`(t) = u(ξ0 + t,−t) = ψ(ξ0), using ξ0 = x+ y, yields

u(x, y) = ψ(x+ y) .

Example 8.2.2. Let us consider the special case of Eq. (8.1) with variable c = c(x), keeping all other coefficientsconstant. Then the same line of reasoning as in the previous section applies with respect to the characteristics.Also the procedure that led to Eq. (8.7) still holds as such, but produces a more general result, viz.

u`(t, x0) = ψ(x0) e−∫ t0c`(τ)dτ +

∫ t

0

f`(τ) e−∫ tτc`(s)ds dτ . (8.8)

For constant c this is clearly consistent with Eq. (8.7).

The next example illustrates the case of a non-constant vector field a = a(x).

Example 8.2.3. Consider the following system:

(??)

xux + yuy + u = 0 on Ω = R2\0, 0,u(cos θ, sin θ) = ψ(θ) on V : (x, y) = (cos θ, sin θ) with θ ∈ [0, 2π).

51

8.2. EXAMPLES 8. THE METHOD OF CHARACTERISTICS

Consider the parametrised curve ` : (x, y) = (ξ(t), η(t)). Denote the solution along this curve by u` = u `.Take the curve ` to be a characteristic:

` :

ξ = ξ ,η = η .

In explicit form: ` : (x, y) = (ξ0et, η0e

t), i.e. a family of half lines emanating from the origin (t → −∞) inradial direction through the point (ξ0, η0). It is clear from its orientation that every characteristic ` intersects theunit sphere transversally. The value t = 0 corresponds to intersection points (ξ0, η0) ∈ V provided we assumeξ20 + η2

0 = 1, which we will do henceforth:

(22) ` : (x, y) = (ξ0et, η0e

t) subject to ξ20 + η2

0 = 1.

The p.d.e. system for u reduces to the following o.d.e. system for u`:

(∗∗)u` + u` = 0 ,u`(0) = ψ(arg(ξ0 + iη0)) .

The solution isu`(t) = ψ(arg(ξ0 + iη0)) e−t .

To obtain the desired solution of (??) we must eliminate the parameters (t, ξ0, η0)—two of which are independent—using (22) in favour of (x, y). Note that

• arg(ξ0 + iη0) = arg((ξ0 + iη0) et) = arg(ξ(t) + iη(t)) = arg(x+ iy), and

• et =√

(ξ20 + η2

0) e2t =√ξ(t)2 + η(t)2 =

√x2 + y2.

Therefore

u(x, y) =ψ(arg(x+ iy))√

x2 + y2.

Example 8.2.4. There is a more direct way to reduce the p.d.e. system of the previous example to an o.d.e.system, obviating the intermediate step via characteristics. It is based on a slick choice of coordinates adapted tothe symmetry of the problem. A close inspection reveals that (??) is symmetric under dilations (i.e. the form ofthe system does not change upon a transformation (x′, y′) = (λx, λy) with λ > 0), suggesting a polar coordinatetransformation (note that a dilation amounts to a 1-dimensional radial scaling):

x(r, θ) = r cos θ ,y(r, θ) = r sin θ ,

with (r, θ) ∈ R+ × [0, 2π). Setting v(r, θ) = u(x(r, θ), y(r, θ)), and observing that

∂

∂x= cos θ

∂

∂r− sin θ

r

∂

∂θ,

∂

∂y= sin θ

∂

∂r+

cos θ

r

∂

∂θ,

yields the following equivalent system, in which a ′ indicates d/dr:rv′ + v = 0 on Ω = R2\0, 0,v(1, θ) = ψ(θ) with θ ∈ [0, 2π).

This is an o.d.e. system in the variable r, in which θ enters as a parameter. The solution is

v(r, θ) =ψ(θ)

r.

Compare this to the result from the previous example.

52

9SEPARATION OF

VARIABLES


In this chapter we sketch another generic method to solve p.d.e. systems, recall Eqs. (1.1-1.2), based on separationof variables. Certain tacit assumptions will be made concerning the general conditions under which this methodis applicable, but again we will confine ourselves to specific examples to illustrate the main idea. An explicitassumption will be that the domain of definition Ωk ⊂ Rn−k for a subset of independent variables x1, . . . , xn−k(0 < k < n) is open and bounded, with boundary ∂Ωk.

The method stipulates the existence of solutions of the form

u(x1, . . . , xn) = u1(x1, . . . , xn−k)u2(xn−k+1, . . . , xn) , (9.1)

with 0 < k < n. In many cases of interest k = 1, e.g. in dynamic systems this ansatz is typically used to separatethe system’s behaviour in space (with coordinates x1, . . . , xn−1, say) and time (xn = t).

Applying this technique often results in a so-called Sturm-Liouville problem.

Definition 9.1.1. A (regular1) Sturm-Liouville problem is an eigensystem expressed in terms of a second ordero.d.e. with homogeneous b.c.’s of the following form: L u = λu on (a, b) ⊂ R,

αu(a) + βu′(a) = 0 with α, β ∈ R, α2 + β2 6= 0γu(b) + δu′(b) = 0 with γ, δ ∈ R, γ2 + δ2 6= 0

1In a singular Sturm-Liouville problem we have a = −∞ and/or b =∞, or the coefficient of u′′ diverges at a and/or b.

53

9.2. EXAMPLES 9. SEPARATION OF VARIABLES

in which the linear operator L has the form

(∗) L u = − 1

w((p u′)′ + q u) ,

for certain specified functions p, q, and positive “weight” function w > 0.

The Sturm-Liouville problem consists of finding those values of λ ∈ R for which nontrivial solutions exist.Under suitable conditions on the functions p, q and w (in particular p, p′, q, w ∈ C([a, b]), Sturm-Liouville theoryprovides us with the following result.

Theorem 9.1.1. Recall Definition 9.1.1. The eigenvalues λn, n ∈ N, are real and can be ordered s.t.

λ1 < λ2 < . . . < λn < . . . with limn→∞

λn =∞.

The eigenfunction φn corresponding to λn, a.k.a. the n-th fundamental solution of the Sturm-Liouville problem, isunique up to normalization, and has exactly n−1 zero crossings in the interval (a, b). Moreover, the (appropriatelynormalized) eigenfunctions form an orthonormal basis for the Hilbert space L2([a, b], w), in the sense that

(φm , φn)wdef=

∫ b

a

φ∗m(x)φn(x)w(x) dx = δmn .

Remark. An important property of the Sturm-Liouville operator in Definition 9.1.1 is (L u , v)w = (u ,L v)w.Among others, this implies λ ∈ R as well as orthogonality of the eigenfunctions φnn∈N. The proof is left asan exercise.

9.2. Examples

Example 9.2.1. Consider the following system, describing a vibrating string without a priori given initialization:

(∗)

utt − uxx = 0 for (x, t) ∈ (0, 1)× R+,u(0, t) = u(1, t) = 0 .

Suppose(†) u(x, t) = φ(x)χ(t) .

For the p.d.e. this yields φχ = φ′′χ, with dots and primes indicating temporal, respectively spatial differentiation.The b.c. reduces to φ(0) = φ(1) = 0, assuming a nontrivial solution. Ignoring denominator problems that mightoccur, this implies that χ/χ = φ′′/φ = −c2 constant, since l.h.s. and r.h.s. are functions of different independentvariables t and x. Note that the arbitrary constant −c2 (the notation betrays a modest amount of foresight) maybe non-positive (in which case take c≥ 0) or non-negative (in which case take c= iκ with κ≥ 0 and i2 =−1).Setting

χ+ c2χ = 0 ,

we obtain(?) χ(t) = Sc sin(ct) + Cc cos(ct) .

Here, Sc and Cc are constants that depend on the parameter c, which, at this stage, is left undetermined. Note thatsin(iκt) = i sinh(κt) and cos(iκt) = cosh(κt).

As a consequence we have the following Sturm-Liouville problem for φ (recall Definition 9.1.1 with p = 1,q = 0, w = 1, a = 0, b = 1, λ = c2):

φ′′ + c2φ = 0 for x ∈ (0, 1),φ(0) = φ(1) = 0 .

54

9. SEPARATION OF VARIABLES 9.2. EXAMPLES

The solution of the o.d.e. is formally similar to that of χ,

φ(x) = Ac sin(cx) +Bc cos(cx) .

The thing to note is that the spatial b.c.’s prevent c from taking just any value, at least for a nontrivial solution.To admit a nontrivial solution satisfying the b.c.’s we must apparently set Bc = 0 and take a real-valued c ∈ R(consistent with Theorem 9.1.1), viz.

c = kπ with k ∈ N.

Finally, without loss of generality, we may normalize the solution, since the integration constants sk = Skπ andck = Ckπ in the factor χ(t) (?) can still be freely chosen, e.g.

() φk(x) =√

2 sin(kπx) k ∈ N .

The normalization is such thatφk · φ` = δk`

for the standard inner product on L2((0, 1)). All in all

(•) u(x, t) =

∞∑k=1

(sk sin(kπt) + ck cos(kπt)) φk(x) .

Whether this formal series represents a classical or a generalised solution depends on the convergence propertiesof the particular sequences skk∈N and ckk∈N. Note that, generically, it is not separable, unlike the ansatz (†).

In the next example we exploit the freedom of choice of the coefficients to further restrict the system.

Example 9.2.2. Consider the p.d.e. system (∗) of Example 9.2.1 under the additional initial condition

(i.c.)

u(x, 0) = f(x) ,ut(x, 0) = g(x) .

We expect this to induce constraints on the coefficients sk and ck. Inspection of (•) for t = 0 and comparisonwith the i.c.’s reveals that

f(x) =

∞∑k=1

ckφk(x) ,

g(x) = π

∞∑k=1

kskφk(x) ,

in terms of the basis functions given by () in the previous example. To retrieve the coefficients, take innerproducts of f and g with φ`, using orthonormality, φk · φ` = δk`:

c` =

∫ 1

0

f(x)φ`(x) dx = f · φ` ,

s` =1

π`

∫ 1

0

g(x)φ`(x) dx =1

π`g · φ` .

All in all, recalling () and (•) from the previous problem,

(∗∗) u(x, t) =

∞∑k=1

(g ·φkπk

sin(kπt) + f ·φk cos(kπt)

)φk(x) .

In general, the result (∗∗) above represents a generalised solution, thus one should take point evaluation with agrain of salt. To appreciate this, let us write u(x, t) = limN→∞ UN (x, t) with UN (x, t) =

∑Nk=1 uk(x, t), with

self-explanatory definition of uk(x, t). This limit generally holds in the weak sense of distribution theory, recallDefinition 4.2.11:

u = limN→∞

UN iff u(φ) = limN→∞

UN (φ) for all φ ∈ D((0, 1)× R+). (9.2)

If f and g are sufficiently regular, stronger convergence results can be obtained that render u a classical solution.

55

9.2. EXAMPLES 9. SEPARATION OF VARIABLES

Example 9.2.3. Cf. Examples 12.2.5 –12.2.6 in Chapter 12 for more examples in the context of hyperbolicsystems.

Example 9.2.4. Cf. Example 13.2.1 in Chapter 13 for an application in the context of parabolic systems.

Example 9.2.5. Consider the Laplace equation with inhomogeneous b.c.’s on the unit square:

(∗)

uxx + uyy = 0 for (x, y) ∈ (0, 1)× (0, 1),u(x, 0) = u(x, 1) = u(0, y) = 0 ,

u(1, y) = 1 .

Suppose(†) u(x, y) = X(x)Y (y) .

The result isX ′′

X= −Y

′′

Y= c2 .

In view of the homogeneous b.c.’s on the horizontal edges, consider the system for Y first. As in Example 9.2.1we obtain a Sturm-Liouville problem:

Y ′′ + c2Y = 0 for y ∈ (0, 1),Y (0) = Y (1) = 0 .

which implies Y (y) = A sin(cy) +B cos(cy) for some constant c ∈ C. Imposing the homogeneous b.c.’s yieldsB = 0 and (given that A 6= 0) c = kπ, k ∈ N. Recall that the normalised family Yk(y) =

√2 sin(kπy)k∈N is

orthonormal relative to the standard inner product on L2([0, 1]).

Setting c = kπ, the resulting system forX isX ′′−k2π2X = 0. The single homogeneous b.c.X(0) = 0 impliesX(x) ∝ sinh(kπx). All in all we may stipulate a solution of the original problem the form

u(x, y) =

∞∑k=1

ck sinh(kπx) sin(kπy) .

The coefficients should follow by enforcing the final, inhomogeneous b.c., viz. u(1, y) = 1, or

∞∑k=1

ck sinh(kπ) sin(kπy) = 1 .

To isolate the coefficients, take the standard inner product with φ`(y) =√

2 sin(`πy), using φk · φ` = δk`. Forthe l.h.s. we then obtain c` sinh(`π)/

√2, for the r.h.s. we find

1 · φ` =

∫ 1

0

φ`(y) dy =

[−√

2

`πcos(`πy)

]1

0

=

0 if ` is even,2√

2

`πif ` is odd.

All in all

u(x, y) =

∞∑k=1,k odd

4

kπ sinh(kπ)sinh(kπx) sin(kπy) =

∞∑k=1

4

(2k−1)π sinh((2k−1)π)sinh((2k−1)πx) sin((2k−1)πy) .

56

10FUNDAMENTAL

SOLUTIONS &GREEN’S

FUNCTIONS


Solving the inhomogeneous linear p.d.e. Lu = f on Ω subject to Bu = g on ∂Ω defined by Eqs. (1.1–1.4) for-mally amounts to finding a left-inverse Linv of L incorporating the b.c.’s. This whimsical observation can be mademore rigorous using the concept of a Green’s function, a.k.a. fundamental solution or, in systems engineering,impulse response.

Throughout this chapter we will assume that the coefficients cα(x) = cα in Eqs. (1.1–1.2) are constant, althoughthis is not a necessary condition per se for the proposed method.

57

10.2. CONSTRUCTION 10. FUNDAMENTAL SOLUTIONS & GREEN’S FUNCTIONS

The concept of a fundamental solution relies on the fact that the domain of definition of the Dirac point measure,recall Lemma 4.2.1 and Definition 4.2.6, can be extended so as to include all functions for which point evaluationmakes sense. With Eqs. (1.1–1.2) in mind, suppose we are in possession of a (generalised!) symmetric functionψ ∈ D ′(Ω), s.t.

Lψ = δ on Ω. (10.1)

Let us define the shifted function ψξ in analogy with the shifted Dirac function δξ for ξ ∈ Ω, Definition 4.2.6:

ψξ(x) = ψ(x− ξ) . (10.2)

Symmetry entails ψ(x) = ψ(−x) for all x ∈ Ω, whence ψξ(x) = ψx(ξ). Note that ψξ is a solution of

Lψξ = δξ on Ω. (10.3)

For this reason L is said to be linear shift invariant. This implies that

u(x) =

∫Ω

ψξ(x) f(ξ) dξdef= (ψ ∗Ω f)(x) . (10.4)

satisfies Lu = f .

Proof: Lu(x) = L∫

Ωψξ(x) f(ξ) dξ =

∫ΩLψξ(x) f(ξ) dξ =

∫Ωδξ(x) f(ξ) dξ =

∫Ωδx(ξ) f(ξ) dξ = f(x).

Any generalised function ψ ∈ D ′(Ω) that obeys Eq. (10.1) is called a fundamental solution of the p.d.e. ViaEq. (10.2) a suitably chosen fundamental solution gives rise to a so-called Green’s function, a symmetric double-argument function Ψ : Ω × Ω → K : (x, ξ) 7→ Ψ(x, ξ) = ψξ(x), adapted to certain b.c.’s. Obviously ψ isdiscontinuous at the origin, which implies that Ψ is discontinuous at ‘diagonal’ points (x, x) ∈ Ω × Ω. Fun-damental solutions, and a fortiori Green’s functions, are not unique. One may add classical solutions of thehomogeneous p.d.e., which may be exploited to enforce specific boundary conditions.

10.2. Construction

10.2.1. Via Classical Homogeneous Solutions

To construct a Green’s function a useful observation is that outside the singularity it must have the same form asa classical homogeneous solution of the p.d.e. As an illustration of this principle we consider the construction ofa Green’s function for the Laplace/Poisson equation.

Theorem 10.2.1. A Green’s function Ψ : Rn × Rn → R : (x, ξ) 7→ Ψ(x, ξ) for the Laplacian ∆ is given by

Ψ(x, ξ) =1

2|x− ξ| if n = 1,

Ψ(x, ξ) =1

2πln ‖x− ξ‖ if n = 2,

Ψ(x, ξ) =−1

(n− 2) voln−1 Sn−1 ‖x− ξ‖n−2if n = 3,

in which voln−1 Sn−1 denotes the surface area of the unit sphere Sn−1 ⊂ Rn in n dimensions. That is, ifψξ(x) = Ψ(x, ξ), then ∆ψξ = δξ.

Note that the case n = 1 has already been handled in Example 4.4.3.

Proof. The proof for general n ∈ N requires a lemma.

58

10. FUNDAMENTAL SOLUTIONS & GREEN’S FUNCTIONS 10.2. CONSTRUCTION

Lemma 10.2.1. If u : Rn → K : x 7→ u(x) is rotationally symmetric w.r.t. the origing, then ∆u =1

rn−1∂r(rn−1∂ru

),

in terms of the radial coordinate r = ‖x‖ =√x2

1 + . . .+ x2n.

Proof of Theorem 10.2.1. With the lemma it is straightforward to prove that, for fixed ξ ∈ Rn, ψξ as conjecturedin Theorem 10.2.1 satisfies ∆ψξ = 0 for ‖x−ξ‖ > 0 (set r = ‖x−ξ‖). The point x = ξ needs special care. To thisend, consider the region Ωε = x ∈ Rn | ε<‖x− ξ‖<R ⊂ Ω with R>ε>0 and Ω = x ∈ Rn | ‖x− ξ‖<R.Then for any test function φ ∈ D(Ω) we have, by Stokes’ Theorem,∫

Ωε

(ψξ ∆φ︸︷︷︸I

− φ∆ψξ︸︷︷︸II

) dx =

∫∂Ωε

(ψξ∇φ︸︷︷︸III

− φ∇ψξ︸︷︷︸IV

) · n dσ .

We have just argued that II = 0 identically on Ωε. For III and IV only the innermost boundary with ‖x− ξ‖ = εyields potential contributions by definition of D(Ω). Let us consider them one by one.

For term III we observe that |ψξ∇φ · n| ≤ C(ε) max ‖∇φ‖, with constant C(ε) = |ψξ(x)|‖x−ξ‖=ε = O(ε2−n)if n 6= 2, and O(ln ε) if n = 2, and

∫‖x−ξ‖=ε dσ = εn−1 voln−1 Sn−1, so that

|III| = |∫‖x−ξ‖=ε

ψξ∇φ · n dσ| ≤ C(ε) max ‖∇φ‖∫‖x−ξ‖=ε

dσ =

O(ε) if n 6= 2,

O(ε ln ε) if n = 2.

That is, limε→0 III = 0.

As for term IV, we explicitly compute∇ψξ · n =∂ψξ∂n = −∂ψξ∂r on the ε-sphere ‖x− ξ‖ = ε in terms of the radial

coordinate r = ‖x− ξ‖:∂ψξ∂r

∣∣∣∣r=ε

=1

2if n = 1,

∂ψξ∂r

∣∣∣∣r=ε

=1

2πεif n = 2,

∂ψξ∂r

∣∣∣∣r=ε

=1

voln−1 Sn−1εn−1if n ≥ 3.

(The latter formula thus in fact covers all cases including n = 1, 2.) Noting that |φ(x) − φ(ξ)| ≤ εmax ‖∇φ‖,and writing φ(x) = φ(ξ) + (φ(x)− φ(ξ)), we find

IV = −∫‖x−ξ‖=ε

φ∇ψξ ·n dσ = −(φ(ξ)+O(ε))

∫‖x−ξ‖=ε

∇ψξ ·n dσ = (φ(ξ)+O(ε))

∫‖x−ξ‖=ε

∂ψξ∂r

∣∣∣∣r=ε

dσ .

We conclude that limε→0 IV = φ(ξ) = δξ(φ), and therefore, combining all of the above observations,∫Ω

ψξ ∆φdx = limε→0

∫Ωε

ψξ ∆φdx = δξ(φ) ,

i.e. ∆ψξ = δξ in distributional sense. This concludes the proof of Theorem 10.2.1.

10.2.2. Via Separation of Variables

If, via the method of separation of variables (Chapter 9), Eq. (10.1), furnished with homogeneous b.c.’s, canbe reduced to a Sturm-Liouville problem (recall Definition 9.1.1 and Theorem 9.1.1), then there is a systematicway to construct a Green’s function that satisfies the b.c.’s. This construction also works in higher dimensions,

59


but let us consider the Sturm-Liouville problem (∗) of Definition 9.1.1 for the sake of simplicity: Let φnn∈Nbe a complete orthonormal family for the Sturm-Liouville problem in Definition 9.1.1, then we may stipulate aGreen’s function of the form

ψξ(x) = Ψ(x, ξ) =∑n∈N

an(ξ)φn(x) .

Insertion into Eq. (10.3) yields the following result.

Result 10.2.1. Recall Definition 9.1.1 for the definition of L . A Green’s function for the systemLψξ = δξ on (a, b),

αψξ(a) + βψ′ξ(a) = 0 with α, β ∈ R, α2 + β2 6= 0

γψξ(b) + δψ′ξ(b) = 0 with γ, δ ∈ R, γ2 + δ2 6= 0

is given by

ψξ(x) = Ψ(x, ξ) =∑n∈N

w(ξ)φ∗n(ξ)φn(x)

λn,

in which φnn∈N is an orthonormal basis ofL2([a, b], w) induced by the Sturm-Liouville problem in Definition 9.1.1,recall Theorem 9.1.1.

Proof. Expand ψξ(x) =∑n∈N an(ξ)φn(x) and substitute this into Lψξ = δξ to determine the coefficients

an(ξ). Using L φn = λnφn we find ∑n∈N

an(ξ)λn φn(x)def= δξ(x) .

Multiplying both sides with φ∗m(x)w(x) and integrating over x ∈ (a, b) yields for the l.h.s.

∑n∈N

an(ξ)λn

∫ b

a

φ∗m(x)φn(x)w(x) dx =∑n∈N

an(ξ)λn (φm , φn)w = λm am(ξ) ,

by virtue of orthonormality (φn , φm)w = δnm, and for the r.h.s.∫ b

a

δ(x− ξ)φ∗m(x)w(x) dx = w(ξ)φ∗m(ξ) ,

Thus

am(ξ) = w(ξ)φ∗m(ξ)

λm.

The extension to genuine p.d.e.’s in n dimensions leads to the following result.

Result 10.2.2. Let Lψξ = δξ on Ω with homogeneous b.c.’s on ∂Ω. Let φαα∈Nn be a complete family oforthonormal eigenfunctions on L2(Ω, w) subject to given b.c.’s, i.e. Lφα = λαφα, with

(φα , φβ)wdef=

∫Ω

φ∗α(x)φβ(x)w(x) dx = δαβ .

Then

ψξ(x) = Ψ(x, ξ) =∑α∈Nn

w(ξ)φ∗α(ξ)φα(x)

λα.

60

10. FUNDAMENTAL SOLUTIONS & GREEN’S FUNCTIONS 10.2. CONSTRUCTION

10.2.3. Via Fourier Transformation

In Example 7.2.3 we have effectively constructed a fundamental solution and a fortiori Green’s function for theheat operator (with homogeneous i.c.) on Rn × R via the (spatial1) Fourier route:

∂u

∂t−∆u = δ for (x, t) ∈ Rn × Ru(x, 0) = 0

with a spatiotemporal Dirac point distribution δ ∈ S ′(Rn × R) on the r.h.s. Taking h(x, t) = δ(x)δ(t) in thatexample we obtain a fundamental solution

ψ(x, t) = θ(t)φ(x, t) =θ(t)√

4πtn exp

(−‖x‖

2

4t

),

in which θ ∈ S ′(R) is the Heaviside function (or indicator function on R+), defined by θ(t) = 0 for t < 0 andθ(t) = 1 for t > 0. The Green’s function for the above system including homogeneous b.c. is a shifted copy:

Ψ(x, t, ξ, τ) = ψξ,τ (x, t) = ψ(x− ξ, t− τ) = θ(t− τ)φ(x− ξ, t− τ) =θ(t− τ)√4π(t− τ)

n exp

(−‖x− ξ‖

2

4(t− τ)

).

Recall that the solution of the inhomogeneous system of Example 7.2.3 is obtained by spatiotemporal convolutionof the data term h : Rn×R : (x, t) 7→ h(x, t) (with h(x, t) = 0 identically if t < 0) and the fundamental solution:

u(x, t) = (h ∗ ψ)(x, t) =

∫ ∞−∞

∫Rnh(ξ, τ) Ψ(x, t, ξ, τ) dξdτ ,

in which ∗ now denotes spatiotemporal convolution.

1Alternatively one may apply spatiotemporal Fourier transformation on Rn × R.

61


62

IIIPDE SYSTEMS:

COMMON CASES

63

11FIRST ORDER

SYSTEMS

11.1. The Transport Equation

In this chapter we consider the inhomogeneous transport equation with i.c., recall Example 1.2.9 on page 5:ut − a · ∇u+ a0u = f for (x, t) ∈ Ω ⊂ Rn × R+,

u|t=0 = g ,(11.1)

with

a · ∇ =

n∑i=1

ai∂i . (11.2)

11.2. The Dirac Equation

A famous first order p.d.e. in quantum physics is the so-called Dirac equation, which was introduced as one ofthe attempts to replace the non-relativistic Schrödinger equation by a relativistic counterpart. The Klein-Gordonequation may be seen as another attempt, recall Example 1.2.7:

(2 +m2

)u = 0 (in natural units). Aiming for

a first order time-evolution equation, Paul Dirac stipulated (using different but equivalent notation) a first order

65

11.2. THE DIRAC EQUATION 11. FIRST ORDER SYSTEMS

p.d.o. in (3+1)-dimensional spacetime of the form

i

3∑µ=0

γµ ∂µ −m. (11.3)

Recall the relativistic Klein-Gordon operator:

2 +m2 = ∂tt −∆ +m2 .

Dirac imposed the constraint that the Dirac equation for a physical field must imply the Klein-Gordon equation.To see how this comes about, consider a field ψ satisfying Eq. (11.3). We then have

0 =

(−i

3∑µ=0

γµ ∂µ −m

)(i

3∑ν=0

γν ∂ν −m

)ψ =

(1

2

3∑µ=0

3∑ν=0

γµ, γν ∂µ∂ν +m2

)ψ ,

in which γµ, γν = γµ γν + γν γµ. Consistency with the Klein-Gordon equation thus produces the followingformal condition

(•) γµ, γν = 2 ηµν ,

in which η00 = 1, η11 = η22 = η33 = −1, and all other ηij = 0 (Minkowski metric). The conditions (•) cannotbe fulfilled if the coefficients are ordinary (commuting) numbers γµ ∈ K. The “anti-commutator” introducedhere, however, concisely captures Dirac’s brilliant idea, viz. that one can satisfy (•) by interpreting each γµ as aparticular 4×4-matrix, and, thus, ψ as a 4-component so-called “spinor” instead of a scalar field. You may verifythat the 4× 4-matrices

γ0 =

(I ∅∅ −I

)γ1 =

(∅ σ1

−σ1 ∅

)γ2 =

(∅ σ2

−σ2 ∅

)γ3 =

(∅ σ3

−σ3 ∅

),

with 2× 2-submatrices

∅ =

(0 00 0

)I =

(1 00 1

)σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)do fulfil the stated conditions (•). The three matrices σi, i = 1, 2, 3, are known as the Pauly spin matrices.These were already known from Wolfgang Pauli’s phenomenological theory of spin (Nobel Prize in physics,1945, “for the discovery of the Exclusion Principle, also called the Pauli Principle”). However, this theory wasconstructed on the basis of 2-component wave functions. Dirac’s 4-component spinor formalism introduced a newparadigm in physics, for which he received the Nobel Prize in physics in 1933 (jointly with Erwin Schrödinger)“for the discovery of new productive forms of atomic theory”. Among others it led to the prediction of antimatterto explain the theoretical possibility of electrons with negative energy states in his formalism, which wouldeffectively behave in electromagnetic interactions as if positively charged. These are nowadays known as anti-electrons or positrons and have been experimentally confirmed.

All in all, the Dirac equation takes the form (i/∂ −m

)ψ = 0 ,

in which /∂ =∑3µ=0 γ

µ∂µ is a commonly used abbreviation in physics. In this notation we have set ~ = c = 1.Reintroducing these fundamental constants, the Dirac equation takes the form

(i~/∂ −mc

)ψ = 0. Note that the

Klein-Gordon equation holds component-wise for the spinor, whereas the Dirac equation couples the componentsthrough the γµ-matrices.

The Dirac equation can be derived from a variational principle, involving besides the Dirac spinor ψ a second,so-called conjugate Dirac spinor ψ = ψ†γ0. The Dirac action functional is given by

S[ψ,ψ] =

∫ψ(x)(i

3∑µ=0

γµ∂µ −m)ψ(x) dx .

66

11. FIRST ORDER SYSTEMS 11.2. THE DIRAC EQUATION

Variation with respect to ψ (leaving ψ fixed) gives the Dirac equation in the form of Eq. (11.3). Variation withrespect to ψ (leaving ψ fixed) yields the conjugate Dirac equation for ψ = ψ†γ0:

i

3∑µ=0

∂µψγµ +mψ = 0 .

Note that ψ is still to the left of γµ, as in the action functional.

67

11.2. THE DIRAC EQUATION 11. FIRST ORDER SYSTEMS

68

12SECOND ORDER

HYPERBOLIC

SYSTEMS

12.1. The Wave Equation

In this chapter we consider the inhomogeneous wave equation with i.c.’s, recall Section 1.2: 2u = h for (x, t) ∈ Ω ⊂ Rn × R+,u|t=0 = φ ,ut|t=0 = ψ ,

(12.1)

with

2 =1

c2∂2

∂t2−∆ . (12.2)

Recall from Section 1.6 that this is an example of a hyperbolic system.

We focus explicitly on spatial dimensions n = 1, 2, 3. In all examples below, tacit assumptions are in effectconcerning such matters as regularity of the data, existence of well-defined Fourier transforms, interchangeabilityof differentiation and integration, et cetera, so as to justify all manipulations. Recall that generalised solutionsmay still exist if some of these assumptions fail to hold.

69

12.2. EXAMPLES 12. SECOND ORDER HYPERBOLIC SYSTEMS

Examples 12.2.1–12.2.6 illustrate the case n = 1. Examples 12.2.7–12.2.11 deal with the case n = 3. Exam-ple 12.2.12 considers the case n=2. The reason for this particular order will become apparent in due course.

12.2. Examples

We start with the homogeneous case in n=1 dimension.

Example 12.2.1. Consider the (1+1)-dimensional homogeneous wave equation with i.c.’s,

()

utt − c2uxx = 0 for (x, t) ∈ R× R+.u|t=0 = φ ,ut|t=0 = ψ .

As an ansatz we try the method proposed in Chapter 7. Since spatiotemporal Fourier transformation wouldscramble the time-localised i.c.’s, we consider spatial Fourier transformation, ignoring the time coordinate t:

u(x, t) =1

2π

∫ ∞−∞

eiωx u(ω, t) dω ,

Substituting into () yields the following o.d.e. system (with ω ∈ R interpreted as a parameter):

(∗)

d2u

dt2+ c2ω2 u = 0 for (ω, t) ∈ R× R+.

u|t=0 = φ ,du

dt

∣∣∣∣t=0

= ψ .

The general solution of the o.d.e. in (∗) takes the form

u(ω, t) = A+(ω) eiωct +A−(ω) e−iωct .

Imposing the i.c.’s in (∗) fixes the integration constants:

A±(ω) =1

2

(φ(ω)± 1

iωcψ(ω)

).

Fourier inversion yields

(•) u(x, t) =1

2π

∫ ∞−∞

φ(ω)

(eiω(x+ct) + eiω(x−ct)

2

)dω +

1

2π

∫ ∞−∞

ψ(ω)

iωc

(eiω(x+ct) − eiω(x−ct)

2

)dω .

In order to evaluate the second integral we need to recall a basic result from calculus. Setting

Ψ(x)def=

1

2π

∫ ∞−∞

ψ(ω)eiωx

iωdω ,

we observe that

Ψ′(x) =1

2π

∫ ∞−∞

ψ(ω)eiωx dω = ψ(x) ,

provided differentiation and integration may be interchanged, implying

Ψ(x) =

∫ x

x0

ψ(ξ) dξ ,

70

12. SECOND ORDER HYPERBOLIC SYSTEMS 12.2. EXAMPLES

for some arbitrary constant x0 ∈ R. It follows that

u(x, t) =1

2φ(x+ ct) +

1

2φ(x− ct) +

1

2c

∫ x+ct

x0

ψ(ξ) dξ − 1

2c

∫ x−ct

x0

ψ(ξ) dξ .

Note that the x0-dependence actually cancels out, since∫ b

x0

ψ(ξ) dξ −∫ a

x0

ψ(ξ) dξ =

∫ b

a

ψ(ξ) dξ .

All in all, the solution of () is given in semi-closed form by

() u(x, t) =1

2φ(x+ ct) +

1

2φ(x− ct) +

1

2c

∫ x+ct

x−ctψ(ξ) dξ .

Cf. Fig. 12.1 for an illustration.

Figure 12.1: Travelling wave solutions of () in one spatial dimension. Top row: With i.c.’s φ(x) = δ(x),ψ(x) = 0, the initial spike splits into two half-weight spikes, one travelling to the left and the other to the right.Each disturbance moves with constant velocity |c| without altering shape, and remains locally confined to a singlepoint. Bottom row: With i.c.’s φ(x) = 0, ψ(x) = δ(x), the initial (unmeasurable) disturbance develops two wavefronts, one travelling to the left and the other to the right. In-between the fronts, moving with constant velocity|c|, the solution has a constant amplitude.

Example 12.2.2. Another option for solving the system () in Example 12.2.1 via the Fourier route is to ignore,momentarily, the i.c.’s, extending the domain of definition to all of spacetime. That is, we first look for a globallydefined solution u : R× R→ K. In this case, the genuine two-dimensional spacetime Fourier transform,

(?) u(x, t) =1

(2π)2

∫ ∞−∞

∫ ∞−∞

eiωx+iνt u(ω, ν) dνdω ,

71


can be applied to transform the p.d.e. in () into the following equivalent algebraic equation (cf. the o.d.e. in (∗)):(−ν2 + c2ω2

)u(ω, ν) = 0 for (ω, ν) ∈ R2.

This implies u(ω, ν) = 0 for all (ω, ν) ∈ Ω =R2 | ν ± cω 6= 0

, and, a fortiori, that we must expect nontrivial

solutions to exist only in strict distributional sense, i.c. u ∈ S ′(R2). In physics, a constraint like C : ν2−c2ω2 =0 is known as a dispersion relation, whereas the support set Ω ⊂ R2 is sometimes referred to as a “shell”.

In view of the Theorem 4.2.2 we take as an ansatz

u(ω, ν) =∑±A±(ω, ν) δ(ν ∓ cω) .

Substitution into (?), and evaluation of the time-integral, yields

(??) u(x, t) =∑±

1

2π

∫ ∞−∞

α±(ω) eiω(x±ct) dω ,

in which we have redefined the coefficients:

α±(ω)def=

1

2πA±(ω,±cω) .

With this general solution we may finally turn to the i.c.’s. By considering

φ(x) =1

2π

∫ ∞−∞

φ(ω) eiωx dω and ψ(x) =1

2π

∫ ∞−∞

ψ(ω) eiωx dω ,

the i.c.’s take the form

φ(ω) = α+(ω) + α−(ω) respectively ψ(ω) = iωc (α+(ω)− α−(ω)) ,

so thatα±(ω) =

1

2φ(ω)± 1

2iωcψ(ω) .

Insertion into (??) produces exactly the same result as (•) in Example 12.2.1, from which the solution ()follows.

Example 12.2.3. Again, recall Example 12.2.1. Instead of the Fourier route we now employ another solutionstrategy, based on the following observation:

1

c2∂2

∂t2− ∂2

∂x2=

(1

c

∂

∂t− ∂

∂x

)(1

c

∂

∂t+

∂

∂x

),

i.e. the d’Alembertian in n= 1 spatial dimension is a composition of two commuting first order (“forward” and“backward”) advection p.d.o.’s (cf. Example 1.2.9 in Chapter 1). The form of the associated characteristics (recallChapter 8, notably Definition 8.1.1) argues for a strategy similar to the one outlined in Section 1.5, as follows.

Consider a coordinate transformation adapted to the characteristics of the two advection p.d.o.’s (“light conecoordinates”):

(••)ξ = x+ ctη = x− ct

and define v(ξ(x, t), η(x, t)) = u(x, t). Note that

∂

∂x=

∂

∂ξ+

∂

∂ηand

∂

∂t= c

∂

∂ξ− c ∂

∂η.

Now v(ξ, η) satisfies the p.d.e.∂2v

∂η∂ξ= 0 .

72


The solution is given by(?) v(ξ, η) = f(ξ) + g(η) ,

in which f, g : R → K are a priori arbitrary functions. The i.c.’s originally formulated for u(x, t) on the x-axis,t=0, translate to conditions for v(ξ, η) along the line η = ξ, viz.

v(ξ, ξ) = φ(ξ) respectively vξ(ξ, ξ)− vη(ξ, ξ) =1

cψ(ξ) .

Combined with the solution (?) this yields

f(ξ) + g(ξ) = φ(ξ) respectively f ′(ξ)− g′(ξ) =1

cψ(ξ) ,

whence

f(ξ) =1

2φ(ξ) +

1

2c

∫ ξ

ξ0

ψ(α) dα respectively g(ξ) =1

2φ(ξ)− 1

2c

∫ ξ

ξ0

ψ(α) dα .

Thus

v(ξ, η) =1

2φ(ξ) +

1

2φ(η) +

1

2c

∫ ξ

η

ψ(α) dα .

Transformed back from light cone to Cartesian coordinates using (••), u(x, t) = v(ξ(x, t), η(x, t)), we onceagain find the same result () as in Example 12.2.1 .

Example 12.2.4. Consider the analogue of the system in Example 12.2.1 on a finite spatial interval:

(∗)

utt − uxx = 0 for (x, t) ∈ (0, 1)× R+,

u(0, t) = u(1, t) = 0 ,u(x, 0) = f(x) ,ut(x, 0) = g(x) .

This system has been scrutinised in Examples 9.2.1 and 9.2.2 on pages 54ff. In particular, its solution is given by(∗∗) at the end of Example 9.2.2.

Example 12.2.5. Consider the inhomogeneous extension of the system in Example 12.2.4:

(∗∗)

utt − uxx = h for (x, t) ∈ (0, 1)× R+,

u(0, t) = u(1, t) = 0 ,u(x, 0) = f(x) ,ut(x, 0) = g(x) .

with nontrivial inhomogeneous term h : (0, 1)×R+ → K. A naive ansatz of the form (†) in Example 9.2.1 won’twork, since the function h cannot be assumed to be separable. Instead, we take inspiration from the homogeneousproblem to solve the inhomogeneous problem, as follows. Denoting the solution to the homogeneous problem byu0 for the sake of clarity, we have (recall (∗∗) at the end of Example 9.2.2)

u0(x, t) =

∞∑k=1

χk(t)φk(x) .

For this homogeneous solution we have χk(t) =g · φkπk

sin(kπt) + f · φk cos(kπt) and φk(x) =√

2 sin(kπx).

According to the Sturm-Liouville Theorem, recall Theorem 9.1.1, the spatial eigenfunctions φkk∈N spanL2((0, 1)). For this reason we stipulate a solution of the form

(‡) u(x, t) =

∞∑k=1

ψk(t)φk(x) ,

i.e. we hypothesize that the effect of the inhomogeneous term can be accounted for by a suitable adaptation ofthe time-dependent coefficients, a trick of the trade reminiscent of the “variation of constant” method used in the

73


context of inhomogeneous linear first order o.d.e.’s as outlined on page 50. Inserting (‡) into (∗∗) and using theeigenfunction properties, φ′′k(x) = −k2π2φk(x), yields

∞∑k=1

(ψk + k2π2ψk

)φk = h .

Multiplication by φ` and integration over x ∈ (0, 1), using orthonormality of φkk∈N relative to the standardinner product on L2((0, 1)), shows that

() ψ` + `2π2ψ` = h` ,

in which

h(x, t) =

∞∑`=1

h`(t)φ`(x) ,

with

h`(t) =

∫ 1

0

h(x, t)φ`(x) dx .

The o.d.e. () governs the time-dependent coefficients ψk(t) in (‡) up to a pair of integration constants. Writing

f(x) =

∞∑k=1

fk φk(x) and g(x) =

∞∑k=1

gk φk(x) ,

with

fk =

∫ 1

0

f(x)φk(x) dx respectively gk =

∫ 1

0

g(x)φk(x) dx ,

we find, by taking t=0 in (‡) and imposing the i.c.’s in (∗∗):

() ψ`(0) = f` respectively ψ`(0) = g` .

Taken together, (‡), () and () admit a unique solution for ψk(t) and thus for u(x, t) in (∗∗) in terms of theinhomogeneous term h(x, t) and the initial data f(x) and g(x).

Example 12.2.6. Consider the system (∗∗) in Example 12.2.5 for the special case in which the inhomogeneousterm is given by a time-periodic function, say

h(x, t) = α(x) cos(βt) .

The i.c.’s are immaterial to the following discussion, as we will not be interested in a specific choice of integrationconstants, so we consider the following system:

utt − uxx = h for (x, t) ∈ (0, 1)× R+,u(0, t) = u(1, t) = 0 ,

Recall (‡) and () in Example 12.2.5, in this particular case:

(o) ψ` + `2π2ψ` = h` ,

with

h`(t) = α` cos(βt) and α` =

∫ 1

0

α(x)φ`(x) dx .

The solution of o.d.e. (o) is the sum of its homogeneous solution ψ`0(t) and a particular solution ψ`p(t). We have

ψ`0(t) = A` cos(`πt) +B` sin(`πt) .

74


To find a particular solution, stipulate one of the form ψ`p(t) = a` cos(βt) + b` sin(βt). Substitution into (o)shows that one can indeed find one with nontrivial coefficients, provided β 6= `π, viz.

ψ`p(t) =α`

`2π2 − β2cos(βt) .

If β = `π, the eigenfrequency of the system (o), then a nontrivial particular solution can be found by stipulatinga form ψ`p(t) = (a` + b`t) sin(βt), viz.

ψ`p(t) =α` t

2`πsin(`πt) .

Thus u(x, t) =∑∞`=1 ψ`(t)φ`(x), with φ`(x) =

√2 sin(`πx) and

ψ`(t) = A` cos(`πt) +B` sin(`πt) + α` ×

cos(βt)

`2π2 − β2off resonance, i.e. for β 6= `π,

t sin(`πt)

2`πon resonance, i.e. for β = `π.

If desired, i.c.’s may be imposed to fix A` and B`, recall () in Example 12.2.5. A remarkable phenomenonbecomes apparent, known as resonance. If the “driving force” h(x, t) contains a mode (αL 6= 0 for some L∈N)with a temporal frequency β ≈ Lπ, then the corresponding mode uL(x, t) = ψL(t)φL(x) of the solution hasan amplitude ∝ 1/|L2π2 − β2| 1, exhibiting a singularity as β → Lπ. In the singular case, β = Lπ, weobserve an “envelope” that linearly increases without bound as t → ∞. The system is said to resonate at theeigenfrequency β=Lπ, triggered by the inhomogeneous term h.

Example 12.2.7. Consider the (3+1)-dimensional homogeneous wave equation with i.c.’s,

(†)

utt − c2 (uxx + uyy + uzz) = 0 for (x, y, z, t) ∈ R3 × R+.u|t=0 = f ,ut|t=0 = g .

We are interested in spherically symmetric solutions around some fiducial point P with Cartesian coordinates(ξ, η, ζ), i.e. solutions that take the form u(x, y, z, t) = v(r, t) with r =

√(x− ξ)2 + (y − η)2 + (z − ζ)2,

assuming, for the moment, that also the i.c.’s are spherically symmetric, with, by abuse of notation, f(x, y, z) =f(r), g(x, y, z) = g(r), say. This suggests the introduction of spherical coordinates around P:

(×)

x = ξ + r sin θ cosφ ,y = η + r sin θ sinφ for (r, φ, θ) ∈ R+ × [0, 2π)× [0, π],z = ζ + r cos θ .

With this transformation and under the given symmetry assumption, (†) is transformed into

(÷)

(rv)tt − c2(rv)rr = 0 for (r, t) ∈ R+ × R+.v|t=0 = f ,vt|t=0 = g .

By an argument similar to that used in Example 12.2.3 we thus find

v(r, t) =1

2rF (r + ct) +

1

2rF (r − ct) +

1

2cr

∫ r+ct

r−ctG(α) dα ,

with F (r) = rf(r) and G(r) = rg(r), i.e. a superposition of incoming and outgoing waves. The singularity atr=0 may be removed by slick choice of f and g.

Example 12.2.8. From a spherically symmetric solution of the three-dimensional wave equation we may con-struct new (non-symmetric) solutions, as follows. Let us consider an outgoing wave emanating from the fiducialorigin of the following form (in distributional sense, recall Chapter 4):

v(x, y, z, t) =δ(r − ct)

4πr,

75


with r≡√x2 + y2 + z2, the distance from the point of evaluation (x, y, z) relative to the origin. For sufficiently

smooth g : R3 → K,

h(x, y, z, t) = (g ∗R3 v)(x, y, z, t) =

∫R3

g(ξ, η, ζ) v(x− ξ, y − η, z − ζ, t) dξdηdζ ,

also satisfies the wave equation. We may interpret this solution as the constructive interference pattern caused bypoint disturbances propagating from their source points (ξ, η, ζ) ∈ R3 at t = 0 to (x, y, z) ∈ R3 at time t ≥ 0,where each point source contributes with an amplitude g(ξ, η, ζ) (“Huygens’ Principle”), cf. Fig. 12.2.

Figure 12.2: The solution at (x, y, z, t) is only affected by source beacons that have been instantaneously activeat t= 0 at points (ξ, η, ζ) a distance R= ct away from (x, y, z). Thus an observer at (x, y, z) “sees” the historyof the (aggregated) source activities delayed by t time units. Vice versa, a point source active at time t = 0 in(x, y, z) will only affect those points (ξ, η, ζ, t) in spacetime that lie on the “light cone” with tip (x, y, z) andradius R = ct. The propagation phenomenon of local disturbances is thus qualitatively similar to that in theone-dimensional case, recall Fig. 12.1 (top row).

Using spherical integration variables (R,Θ,Φ) relative to the fixed destination point (x, y, z),

(××)

ξ = x+R sin Θ cos Φ ,η = y +R sin Θ sin Φ for (R,Φ,Θ) ∈ R+ × [0, 2π)× [0, π],ζ = z +R cos Θ .

we obtain

h(x, y, z, t) =ct

4π

∫ 2π

0

∫ π

0

g(x+ ct sin Θ cos Φ, y + ct sin Θ sin Φ, z + ct cos Θ) sin Θ dΘ dΦ .

In other words,h(x, y, z, t) = ct µ(x,y,z,ct)[g] ,

in which µ(x,y,z,R) returns the average of its operand over a sphere of radius R centered at (x, y, z). Note that

limt→0

µ(x,y,z,ct)[g] = g(x, y, z) .

The rescaled function u(x, y, z, t)=h(x, y, z, t)/c= tµ(x,y,z,ct)[g] apparently solves the following system:

(††)

utt − c2 (uxx + uyy + uzz) = 0 for (x, y, z, t) ∈ R3 × R+.u|t=0 = 0 ,ut|t=0 = g .

76


Example 12.2.9. Consider now the homogeneous system with general i.c.’s:

(†)

utt − c2 (uxx + uyy + uzz) = 0 for (x, y, z, t) ∈ R3 × R+.u|t=0 = f ,ut|t=0 = g .

Using the fact that any t-derivative of u will be a solution of the homogeneous linear p.d.e. if u is itself a solution,one readily verifies that the solution to (†) including i.c.’s is given by

u(x, y, z, t) = tµ(x,y,z,ct)[g] +∂

∂t

(tµ(x,y,z,ct)[f ]

).

The proof relies on elementary calculus (such as the product rule and chain rule for differentiation) in addition tothe following observations, in which χ : R3 → R denotes a continuous function (either f or g):

• µ(x,y,z,0)[χ] = χ(x, y, z);

• ∂tµ(x,y,z,ct)[χ] |t=0 = c∇χ(x, y, z) · 1

4π

∫ 2π

0

∫ π

0

r(Θ,Φ) sin Θ dΘdΦ∗= 0.

Here r(Θ,Φ) = (sin Θ cos Φ, sin Θ sin Φ, cos Θ) is the unit vector in the direction specified by (Θ,Φ). Thecrucial step (∗) follows by evaluating the three components of the vector-valued integral, but should also beimmediately obvious once you recognize that it represents the average of all unit vectors over all directions.

Example 12.2.10. Consider the inhomogeneous system with homogeneous i.c.’s:

(••)

utt − c2 (uxx + uyy + uzz) = h for (x, y, z, t) ∈ R3 × R+.u|t=0 = 0 ,ut|t=0 = 0 ,

for some sufficiently smooth data term h : R3×R+ → K. Its solution can be obtained with the help of Duhamel’sprinciple, which provides an explicit solution of (••) in terms of the already obtained solution of the system (†)with slick choice of i.c.’s formulated for a shifted initial moment. Namely, consider vtt − c2 (vxx + vyy + vzz) = 0 for (x, y, z, t) ∈ R3 × R+.

v|t=τ = 0 ,vt|t=τ = h|t=τ .

This is basically system (††) in Example 12.2.8, subject to a time shift. As the solution depends on the auxiliaryparameter τ , let us write it as a pentavariate function v : R3 × R+ × R+ → K : (x, y, z, t, τ) 7→ v(x, y, z, t; τ).Then it follows straightforwardly by substitution that

u(x, y, z, t) =

∫ t

0

v(x, y, z, t; τ) dτ

solves (••). To see this, note that the i.c.’s are trivially fulfilled:

u(x, y, z, 0) = 0 ,

ut(x, y, z, 0) = v(x, y, z, t; t) = 0 ,

as a result of the first i.c. for v. Furthermore,

ut(x, y, z, t) = v(x, y, z, t; t) +

∫ t

0

vt(x, y, z, t; τ) dτ =

∫ t

0

vt(x, y, z, t; τ) dτ ,

v.s., and

utt(x, y, z, t) = vt(x, y, z, t; t) +

∫ t

0

vtt(x, y, z, t; τ) dτ = h(x, y, z, t) + c2∆

∫ t

0

v(x, y, z, t; τ) dτ

= h(x, y, z, t) + c2∆u(x, y, z, t) ,

as a result of the second i.c., respectively the homogeneous wave equation for v. Also note that we did not needany details about the specific form of the solution, thus this argument generalises to any dimension n.

77


Example 12.2.11. Now consider the general, inhomogeneous problem:

(•)

utt − c2 (uxx + uyy + uzz) = h for (x, y, z, t) ∈ R3 × R+.u|t=0 = f ,ut|t=0 = g ,

for some sufficiently smooth data term h : R3 × R+ → K. The solution is a sum of the solution of the homo-geneous system with i.c.’s as obtained in Example 12.2.9 and a particular solution of the inhomogeneous systemwith homogeneous i.c.’s considered in Example 12.2.10. In turn, recall from Example 12.2.10 that the latter canbe reduced to a homogeneous system of the form studied in Examples 12.2.8–12.2.9 by exploiting Duhamel’sprinciple. Thus we have

(÷) u = u+ U ,

with utt − c2 (uxx + uyy + uzz) = 0 for (x, y, z, t) ∈ R3 × R+.u|t=0 = f ,ut|t=0 = g ,

and Utt − c2 (Uxx + Uyy + Uzz) = h for (x, y, z, t) ∈ R3 × R+.U |t=0 = 0 ,Ut|t=0 = 0 .

Thus the solution (÷) is the sum of

u(x, y, z, t) = tµ(x,y,z,ct)[g] +∂

∂t

(tµ(x,y,z,ct)[f ]

)and, with v(x, y, z, t; τ) as constructed in Example 12.2.10,

U(x, y, z, t) =

∫ t

0

(t−τ) v(x, y, z, t; τ) dτ =

∫ t

0

(t− τ)µ(x,y,z,c(t−τ))[h|t=τ ] dτ

=

∫ t

0

(t−τ)

4π

∫ 2π

0

∫ π

0

h(x+ c(t−τ) sin Θ cos Φ, y + c(t−τ) sin Θ sin Φ, z + c(t−τ) cos Θ, τ) sin Θ dΘ dΦ dτ .

In the following example z-invariance is imposed as a ploy to reduce the three-dimensional system to a two-dimensional one.

Example 12.2.12. In this example we consider the homogeneous two-dimensional system utt − c2 (uxx + uyy) = 0 for (x, y, t) ∈ R2 × R+.u|t=0 = f ,ut|t=0 = g ,

The solution can be derived from that of the three-dimensional system in Example 12.2.9, viz. by assuming thei.c.’s f, g : R3 → K in that example to be independent of z, so that, by abuse of notation, we may equivalentlyconsider these as functions of type f, g : R2 → K as they enter in the i.c.’s of the problem at hand. With suchz-independent i.c.’s we may simplify the pseudo-three-dimensional solution found in Example 12.2.9:

u(x, y, t) = tµ(x,y,0,ct)[g] +∂

∂t

(tµ(x,y,0,ct)[f ]

),

i.e.

u(x, y, t) =t

4π

∫ 2π

0

∫ π

0

g(x+ ct sin Θ cos Φ, y + ct sin Θ sin Φ) sin Θ dΘ dΦ +

+∂

∂t

[t

4π

∫ 2π

0

∫ π

0

f(x+ ct sin Θ cos Φ, y + ct sin Θ sin Φ) sin Θ dΘ dΦ

].

78


A change of integration variables, r = ct sin Θ (the radial coordinate obtained after projecting points (ξ, η, ζ) onthe active 3-sphere with radius ct around (x, y, 0) onto the “shadow” 2-disk in the xy-plane), turns this expressioninto the following form (to see this, split the Θ-integral into two parts, one for the upper and one for the lowerhemisphere, i.e. Θ ∈ [0, π/2] respectively Θ ∈ [π/2, π]):

u(x, y, t) =1

2πc

∫ 2π

0

∫ ct

0

g(x+ r cos Φ, y + r sin Φ)r√

c2t2 − r2dr dΦ +

+∂

∂t

[1

2πc

∫ 2π

0

∫ ct

0

f(x+ r cos Φ, y + r sin Φ)r√

c2t2 − r2dr dΦ

].

That there are two equal contributions from each hemisphere should be obvious by mirror symmetry relative tothe xy-plane, which explains the factor 1/(2π) = 2 × 1/(4π). This geometric projection also explains whyresidual “echoes” of the sources f(ξ, η) and g(ξ, η) originating from all points (ξ, η) in the interior of the 2-disk,0 ≤ r=

√(x− ξ)2 + (y − η)2 ≤ ct, affect the signal u(x, y, t) in point (x, y) at time t, in marked contrast with

the cases n = 1 and n = 3, in which only the boundaries of the respective n-spheres play a role. The appearanceof an attenuation factor 1/

√c2t2 − r2 in the integrand shows that the dominant contribution still comes from the

“wave front” r = ct. Sources at distances r > ct do not affect the signal at time t, a common property for alldimensions n.

79


80

13SECOND ORDER

PARABOLIC

SYSTEMS

13.1. The Heat Equation

In this chapter we consider the homogeneous heat equation with i.c.’s, recall Section 1.2:

ut −∆u = 0 for (x, t) ∈ Ω ⊂ Rn × R+,

u|t=0 = f ,(13.1)

with, in Cartesian coordinates,

∆ =

n∑i=1

∂2

∂x2i

. (13.2)

Recall from Section 1.6 that this is an example of a parabolic system.

81

13.2. EXAMPLES 13. SECOND ORDER PARABOLIC SYSTEMS

13.2. Examples

Example 13.2.1. Consider the heat equation with i.c. on a finite interval for a real-valued function:

(∗)

ut − uxx = 0 for (x, t) ∈ (0, 1)× R+,u(x, 0) = f(x) ,

u(0, t) = u(1, t) = 0 .

This example resembles that of Example 9.2.1 in Chapter 9, so let us likewise stipulate a nontrivial separablesolution:

(†) u(x, t) = φ(x)χ(t) .

For the p.d.e. this yields φχ = φ′′χ, with dots and primes indicating temporal, respectively spatial differentiation.This implies

φ′′ + c2φ = 0 ,

for some constant c ∈ C, with b.c.’s φ(0) = φ(1) = 0, and

χ+ c2χ = 0 .

The latter has a general solution of the form

(?) χ(t) = C e−c2t .

The former has a general solution of the form

φ(x) = A sin(cx) +B cos(cx) ,

withB=0,A ∈ R arbitrary, and with a restriction on c, viz. c = kπ, k ∈ N. Adopting the standard normalizationon the unit interval we thus find a sequence of solutions, one for every k ∈ N:

φk(x) =√

2 sin(kπx) (k ∈ N) .

Taking the restriction on c into account, (?) takes the form

χk(t) = Ck e−k2π2t ,

with Ck ∈ R. A general solution of (∗) is a linear superposition

u(x, t) =∑k∈N

Ckφk(x) e−k2π2t .

The coefficients are determined by the initial data u(x, 0) = f(x). By setting t= 0 in the above expression andtaking the standard inner product with φm we obtain

f(x) =∑k∈N

Ckφk(x) with Ck =

∫ 1

0

f(x)φk(x) dx .

Putting things together, we may thus write

u(x, t) =∑k∈N

∫ 1

0

f(y)φk(y) dy φk(x) e−k2π2t .

Example 13.2.2. Consider the heat equation with i.c. without spatial boundaries:∂u

∂t−∆u = 0 (x, t) ∈ Rn × R+ ,

u(x, 0) = f(x) ,

82

13. SECOND ORDER PARABOLIC SYSTEMS 13.2. EXAMPLES

in which f ∈ S ′(Rn). This problem has been solved in Example 7.2.1, recall page 46:

u(x, t) =1

(4πt)n

∫Rnf(y) e−

‖x−y‖24t dy .

Note that this solution may also be written as

u(x, t) =1

(2π)n

∫Rn

∫Rnf(y) eiω·(x−y) e−t‖ω‖

2

dy dω .

In this form one recognizes the analogy with the “sum/integral” expression at the end of Example 13.2.1.

83

13.2. EXAMPLES 13. SECOND ORDER PARABOLIC SYSTEMS

84

14SECOND ORDER

ELLIPTIC SYSTEMS

14.1. The Laplace Equation

14.2. The Helmholtz Equation

14.3. The Poisson Equation

85

14.3. THE POISSON EQUATION 14. SECOND ORDER ELLIPTIC SYSTEMS

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BIBLIOGRAPHY

[1] L. C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American MathematicalSociety, Providence, Rhode Island, 1998.

[2] S. J. Farlow. Partial Differential Equations for Scientists and Engineers. Dover Publications, Inc., New York, 1993.

[3] I. M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, Inc., second edition, 2000.

[4] A. D. Snider. Partial Differential Equations: Sources and Solutions. Prentice-Hall, Upper Saddle River, New Jersey,1999.

[5] M. Spivak. Calculus on Manifolds. W. A. Benjamin, New York, 1965.

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