introduction to partial di erential...

Introduction

Introduction to Partial Differential Equationspart of EM, Scalar and Vector Fields module (PHY2064)

Richard Sear

November 11, 2019

Richard Sear Introduction to Partial Differential Equations

Introduction

Recommended textbooks

Advanced Engineering Mathematics, Erwin Kreyszig

Advanced Engineering Mathematics, Kenneth Stroud andDexter Booth

Mathematical methods in the physical sciences, Mary Boas

Essential mathematical methods for the physical sciences,Riley and Hobson (bit more advanced that other three)

These books have extra problems for you to do.


Introduction

Course summary:

Week 1 Introduction to PDEs: differential equations withmore than one variable. Common PDEs, linear homogeneousPDES, BCs.

Week 2 The wave equation (in one dimension).

Week 3 The diffusion equation (in one dimension).

Week 4 Laplace’s equation (in two dimensions).

Week 5 PDEs in spherical and circular polar coordinates,e.g., Schrodinger equation for the hydrogen atom.

Week 6 θ and φ-dependent solutions of the Schrodingerequation for the hydrogen atom


Introduction

Course structure: Each week has one lecture to introduce material,plus two class tutorials where you learn the material by doingproblems.

One lecture plus three (one on Tues + two on Thurs) hoursof tutorials in Engineering for Health buildingTwo hours on Thurs are joint with Solid State course

Tutorials:

There are two question sheets a week, on the topic covered inthat week.

Class tutorials are where you receive feedback & help.

Tables along edges to work as a group, and to discuss physics(only).

You are welcome to step out of the session, to take a break.


Introduction

Context: Maxwell’s equations & Schrodinger’s equation

Much of physics involves PDEs and vector calculus:

∇.E =ρ

ε0∇.B = 0

∇× E = −∂B

∂t∇× B = µ0

(J + ε0

∂E

∂t

)

−1

2

h2ε0

πme2∇2ψ − 1

rψ = E

(4πε0/e

2)ψ

Why and how does H absorb light at a wavelength of 122 nm?Need to solve both Schrodinger’s and Maxwell’s equations toanswer this and many many more questions in physics.


Introduction

This week

What are PDEs?

Intro to the (simplest & commonest) PDEs in physics

Superposition (for linear PDEs)

What solutions often look like

BCs for time-independent PDE (Laplace’s)

BCs for time-dependent PDE (wave and diffusion)


Introduction

What are PDEs?

Partial differential equations (PDEs) differ from ordinarydifferential equations (ODEs) in that there is more than onevariable. Thus the derivatives are partial derivatives, hence thename partial differential equations.

In physics when we have more than one variable this is almostalways either because the functions depend on more than onespatial dimension, e.g., on say both x and y , or because theydepend on space and time, e.g., x and t.

Here we will only consider linear PDEs. Nonlinear PDEs aretypically solved using a computer.


Introduction

What are PDEs?

In this part of the course, we will introduce the most commonsimple PDEs in physics, and show how to solve them.

Note that for both ODEs and PDEs, solving means finding afunction.

As it happens these are all second-order, i.e., contain secondderivatives, and so we will only study second-order PDEs.


Introduction

Different types of solutions & BCs

Three types of solutions for PDEs

General solution: general enough to include any particularsolution, i.e., any BCs

A solution: something that when you substitute in the PDEgives you LHS = RHS.

A particular solution: A solution of the PDE that satisfies allthe BCs.

For PDEs, a general solution often requires Fourier series or similar.


Introduction

The Laplacian ∇2

Most PDEs in physics involve the Laplacian, ∇2, which is

∇2 =d2

dx2in 1D

∇2 =∂2

∂x2+

∂2

∂y2in 2D

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2in 3D

This is Cartesian coordinates, will do spherical polars later.


Introduction

Time-independent PDEs

We will study the very common time-independent PDE, namedafter Laplace. For a scalar function u(x , y , z), Laplace’s equationis:

Laplace’s equation ∇2u = 0

For example, both Maxwell’s equations and the diffusion equationcan simplify to Laplace’s equation. If the right hand side is notzero, but is a function, then that PDE is known as Poisson’sequation.Thus in three dimensions Laplace’s equation is

∇2u =∂2u

∂x2+∂2u

∂y2+∂2u

∂z2= 0


Introduction

Schrodinger’s time-independent PDE

Another important time-independent PDE that you have alreadycome across is Schrodinger’s time-independent equation for thequantum mechanical wavefunction ψ, of a particle in a potentialV (x , y , z),

− ~2

2m∇2ψ(x , y , z) + V (x , y , z)ψ(x , y , z) = Eψ(x , y , z)


Introduction

Time-dependent PDEs

Time dependent PDEs need to be solved for the function ofposition and time v(x , y , z , t).

The diffusion equation is

diffusion equation D∇2v =∂v

∂t

where D is a constant, the diffusion constant. D has dimensions ofa length squared over time. The wave equation is

wave equation ∇2v =1

c2

∂2v

∂t2

c is the speed of the wave and has dimensions of length over time.

Difference between the two PDEs is that the diffusion equation hasthe first derivative with respect to time, while the wave equationhas the second derivative with respect to time.


Introduction

Linear homogeneous PDEs

All PDEs in the course (except Poisson’s) are linear andhomogeneous.

Linear homogeneous PDEs are analogous to linear homogeneousODEs, which you have already studied.

Linear PDEs are PDEs in which every term is linear in F or∂F/∂x , ∂2F/∂x2, etc, and so there are no F 2, F 3, exp(F ),F (∂F/∂x), etc terms.

Linear homogeneous PDEs are PDEs in which every single termis either linear in F or its derivatives (or is zero). There are noterms that are just functions of x , y , z or t, and there are no F 2,etc, non-linear terms.


Introduction

Poll feedback test on linear homogeneous PDEs


Introduction


Linear homogeneous PDEs have the same useful property as linearhomogeneous ODEs. The sum of any two solutions of a linearhomogeneous PDE is also a solution to the same PDE, i.e., ifF1(x , y , z , t) and F2(x , y , z , t) are both solutions to a PDE, then

F (x , y , z , t) = a1F1(x , y , z , t) + a2F2(x , y , z , t)

is also a solution to the same PDE. This is true for any values ofthe constants a1 and a2.

This also means that if you multiply a solution F (x , y , z , t) by aconstant, call it a, then aF (x , y , z , t) is also a solution.


Introduction


This property of being able to add solutions to make anothersolution is extremely useful.

For example, it allows solutions to be constructed from Fourierseries, where each term in the series is a solution to the PDE.

In physics this piece of maths corresponds to what is often calledthe principle of superposition.


Introduction

Characteristic solutions of a particular PDE

A particular PDE, eg, wave PDE, has solutions that arecharacteristic of that PDE, eg, sine and cosine waves for the wavePDE:

W (x , t) = sin (kx − ωt)

and for Laplace’s PDE in two dimensions, an example solution is

F (x , y) = sin (kx) exp (−ky)

The question sheets have further examples of solutions to the PDEsin the course, that will help you get used to what they look like.


Introduction

Boundary conditions & Particular solutions

Just as with ODEs, to find a particular solution for a physicalsystem, we need both the PDE and the BCs.

Usually for linear homogeneous PDEs we determine functions thatsatisfy the PDE (e.g., often sines and cosines for the wave PDE)and then construct a particular solution by summing (which can dofor linear homogeneous PDEs) many of these functions, eg as aFourier series.


Introduction

Boundary conditions for time-independent PDEs

In two dimensions the region we want to solve the PDE in, is anarea and so the boundaries of this area are the curves that form itsedge.

y

x

area A

BC is value off(x,y) alongperimeter

In two dimensions BC is the value of f (x , y) around the perimeterof the area.


Introduction

Boundary conditions for time-dependent PDEs: Diffusionequation in one dimension

A possible solution of the diffusion equation, in one dimension,consists of a cosine standing wave, with an amplitude that decayswith time:

W (x , t) = A exp(−0.5k2t

)cos(kx)

for a diffusion constant D = 0.5 m2/s. Because the wave equationis first order with respect to the time derivative, BCs that areinitial BCs need to specify only the function at t = 0 s.

Note that as the wave PDE is second-order with respect to time,there the initial BCS are the function at t = 0 s, and the timederivative of this function.


Introduction

Boundary conditions for time-dependent PDEs: Diffusionequation in one dimension

As an example, let us impose the BC

W (x , t = 0 s) = 10 cos(2πx)

What is then the particular solution?[Poll everywhere]


Introduction

This week

What are PDEs?

Intro to the (simplest & commonest) PDEs in physics

Superposition (for linear PDEs)

What solutions often look like

BCs for time-independent PDE (Laplace’s)

BCs for time-dependent PDE (wave and diffusion)


introduction to partial di erential...

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