MAP4341 Elementary PDEs ISummary of Lectures

c©Dr. Xiaoming Wang

Florida State University

Fall 2014

Outline

Chapter 1: Heat equation

Chapter 2: Separation of variables

Chapter 3: Fourier series

Chapter 4: Wave equation: vibrating strings and membranes

Chapter 5: Sturm-Liouville Eigenvalue Problems

Chapter 7: Higher Dimensional PDEs

Chapter 8: Nonhomogeneous Problems

Aug. 26th Lecture (sect. 1.1 and 1.2)

I What are PDEs?I Why do we care about PDEs?I What are the basic issues (existence, uniqueness and continuous dependence)?I Derivations of the 1D heat equation.

I thermal energy density e(x, t): heat energy per unit volumeI temperature u(x, t)I specific heat / heat capacity c: heat energy needed to raise the temperature of a unit

mass substance by one unitI e = cρuI heat flux φ(x, t): rate of thermal energy flowing to the right per unit mass per unit timeI heat source Q(x, t): heat energy generated per unit volume per unit time.I conservation of heat energy: changing rate of heat energy= heat flux in - heat flux out +

heat energy generated inside per unit timeI Fourier’s law for heat conduction

φ(x, t) = −K0∂u∂x

(x, t)

thermal conductivity K0I 1D heat equation

∂u∂t

= κ∂2u∂x2

+ q

thermal diffusivity κ =K0cρ

I (read the part on diffusion of chemical pollutant, Fick’s law of diffusion.)

Aug. 28th lecture (sect. 1.3, 1.4 and 1.5.)

I 1d heat equation: ∂u∂t = κ ∂

2u∂x2 + q

I initial condition for the 1d heat equationI boundary condition for the 1d heat equation

I prescribed temperatureu(0, t) = uB(t) (Dirichlet),I prescribed flux −K0(0) ∂u

∂x (0, t) = φ(t) (Neumann),I Newton’s law of cooling −K0(0) ∂u

∂x (0, t) = −H[u(0, t)− ub(t)] (Robin) (heat transfercoefficient H)

I Equilibrium temperature in 1D with prescribed temperature or insulatedboundaries, convergence to equilibrium solution

I Derivation of the heat equation in 2 and 3 D via conservation of heat energy

∂u∂t

= κ∆u + q

(notation difference: ∆ = ∇ · ∇ = ∇2)I steady state

I Poisson’s equation∆u = f

I Laplace equation (f = 0) (potential equation)

∆u = 0

Sept. 2 (sect. 2.1, 2.2 and 2.3, separation of variables)separation of variables for 1d homogeneous heat equation

∂u∂t

= κ∂2u∂x2

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

I simple special solution via separation of variables: u(x , t) = G(t)φ(x)I associated time-dependent equationI associated boundary value problem, and eigenvalue and eigenfunctions

problem −φ′′(x) = λφ(x), φ(0) = φ(L) = 0I special solution un(x , t) = sin nπx

L e−κ(nπ/L)2t

I strategy for general solution (sect. 2.2)I linear operator, linear (homogeneous) equationI principle of superposition for linear homogeneous equation

I construction of (formal) general solution via principle of superposition

u(x , t) =∞∑

n=1

Bn sinnπx

Le−κ(nπ/L)2t ,Bn =

2L

∫ L

0f (x) sin

nπxL

dx

(and convergence to steady state)I Fourier sine series f (x) ∼

∑∞n=1 Bn sin nπx

L , completeness (form a basis),orthogonality of the sines

I L2 inner product: (f , g)def=∫ L

0 f (x)g(x), L2 length: ‖f‖L2 =√

(f , f ). (not in thetextbook)

I Example: initial temperature = 100◦ constant: Bn = 0 if n is even, and Bn = 400nπ if

n is odd. (as a HW)I Issues: completeness (convergence)? basis? differentiability? (postponed)

Sept. 4th lecture (sect. 2.4)1d homogeneous heat equation ∂u

∂t = κ ∂2u∂x2 , u(x , 0) = f (x) with different b.c.

I with insulated ends ∂u∂x

∣∣x=0,L = 0

I separation of variables and the associated eigenvalue-eigenfunction problemI Fourier cosine series f (x) ∼ A0 +

∑∞n=1 An cos nπx

LI formal solution (and convergence to steady state)

u(x, t) = A0 +∞∑

n=1

An cosnπx

Le−κ(nπ/L)2 t

,

A0 =1L

∫ L

0u(x, 0) dx, An =

2L

∫ L

0u(x, 0) cos

nπxL

dx

I (completeness) and orthogonality of the Fourier cosine seriesI f (x) = x,An = 2L

(nπ)2 (cos nπ − 1), n = 1, 2, · · · ,A0 = L2

I in a thin circular ring and periodic boundary condition(u(−L, t) = u(L, t), ∂u

∂x (−L, t) = ∂u∂x (L, t))

I separation of variables and the associated eigenvalue-eigenfunction problemI Fourier series f (x) ∼ a0 +

∑∞n=1 an cos nπx

L +∑∞

n=1 bn sin nπxL

I formal solution (and convergence to steady state)

u(x, t) = a0 +∑∞

n=1 an cos nπxL e−κ(nπ/L)2 t +

∑∞n=1 bn sin nπx

L e−κ(nπ/L)2 t ,

a0 = 12L

∫ L−L u(x, 0) dx, an = 1

L

∫ L−L u(x, 0) cos nπx

L dx

bn = 2L

∫ L0 u(x, 0) sin nπx

L dx

I (completeness) and orthogonality of the Fourier seriesI f (x) = 0, x < L

2 ; f (x) = 1, x > L2 ,

a0 = 14 , an = 1

nπ (sin nπ − sin nπ2 ), bn = 1

nπ ( nπ2 − cos nπ)

Sept. 9th lecture, (sect. 2.4 + eigenfunction expansions)Summary of various eigenvalue-eigenfunction problems −ϕ′′ = λϕ + b.c.

BC Dirichlet (prescribed ...) Neumann (insulated ends) periodic (1d ring)E’val (nπ/L)2, n = 1, 2, 3, · · · (nπ/L)2, n = 0, 1, 2, 3, · · · (nπ/L)2, n = 0, 1, 2, 3, · · ·E’fnct sin nπx

L cos nπxL sin nπx

L , cos nπxL

series f (x) =∑∞

n=1 Bn sin nπxL f (x) =

∑∞n=0 An cos nπx

L f (x) = a0+∑∞

n=1 an cos nπxL

+∑∞

n=1 bn sin nπxL

coeff Bn = 2L

∫ L0 f (x) sin nπx

L A0 = 1L

∫ L0 f (x) a0 = 1

2L

∫ L−L f (x)

An = 2L

∫ L0 f (x) cos nπx

L an = 1L

∫ L−L f (x) cos nπx

Lbn = 1

L

∫ L0 f (x) sin nπx

L

I Questions: convergence? completeness? differentiability?I What about mixed boundary conditions, say Dirichlet at one end, and Neumann at

the other end? (optional)I Alternative view via eigenfunction expansion (assume orthogonality)

finite dim’l infinite dim’leq d~u

dt + A~u = 0, ~u∣∣t=0 = ~u0 ∈ Rn ∂u

∂t + κAu = 0, u∣∣t=0 = f

e’val& e’vec (e’fnct) λj , ~vj , (A~vj = λj~vj ), j = 1, · · · , n λj , ϕj , j = 1, 2, · · ·special soln ~vj e−λj t ϕj (x)e−κλj t

general soln∑n

j=1 cj~vj e−λj t∑∞

j=1 cjϕj (x)e−κλj t

u(x , t) =∞∑j=1

Cj (t)ϕj (x), u(x , 0) = f (x)⇒ Cj (t) = cj e−κλj t , cj =(f , ϕj )

(ϕj , ϕj )

Sept. 11th lecture ( sect. 2.5.1)Laplace equation ∆u = 0 inside a rectangle

I Boundary conditions:

BC1(left) : u(0, y) = g1(y);

BC2(right) : u(L, y) = g2(y);

BC3(bottom) : u(x , 0) = f1(x);

BC4(top) : u(x ,H) = f2(x)

I Divide and conquer u = u1 + u2 + u3 + u4 with uj takes care of BCj (different fromthe textbook!)

I Example: BC1, i.e., u1 via separation of variables u1(x , y) = h(x)φ(y)

u1(x , y) =∞∑

n=1

An sinnπyH

sinhnπ(x − L)

H

An =2

H sinh nπ(−L)/H

∫ H

0g1(y) sin

nπyH

dy

Special case of g1(y) = 1, An = 8nHπ sinh nπ(−L)/H for odd n. An = 0 otherwise.

I Alternative view via eigenfunction expansion (covered on Sept. 9th)

u1(x , y) =∞∑

n=1

Cn(x) sinnπyH⇒ C′′n (x)−(

nπH

)2Cn(x) = 0 b.c.⇒ Cn(x) = An sinhnπ(x − L)

H

Sept. 11th lecture continued ( sect. 2.5.2)Laplace’s equation ∆u = 0 on a circular disk with b.c. u

∣∣r=a = f (θ)

I Laplace operator in polar coordinates

∆u =1r∂

∂r

(r∂u∂r

)+

1r2

∂2u∂θ2

I separation of variable u(r , θ) = φ(θ)G(r)I associated eigenvalue-eigenfunction problem: −φ′′ = λφ plus periodic b.c.I radial function equation r d

dr (rG′(r))− n2G(r) = 0 (Euler type)I solution formula

u(r , θ) =∞∑

n=0

Anrn cos nθ +∞∑

n=1

Bnrn sin nθ

A0 =1

2π

∫ π

−πf (θ) dθ

(= u(~0)

)Anan =

1π

∫ π

−πf (θ) cos nθ dθ

Bnan =1π

∫ π

−πf (θ) sin nθ dθ

I Example: f (θ) = sin θ. An = 0, Bn = 0 if n 6= 1, B1 = 1a , u(x , y) = y

a .I Alternative view via eigenfunction expansion

u(r , θ) = G0(r) +∞∑

n=1

(Gns(r) sin nθ + Gnc(r) cos nθ)

Sept. 16th lecture ( section 2.5.4)

I Qualitative property of harmonic functions (solution to Laplace equation)I statement and understanding of mean value theorem u(~0) = 1

2π

∫ π−π u(a, θ) dθ

I maximum principle for harmonic functionsI uniqueness of solution to Poisson equation and continuous dependence on the

boundary data (in L∞ norm)

I solvability (compatibility) condition for Poisson equation in the Neumann BC case.I Energy method for well-posedness of 1d heat equation (postponed)

I L2 length and Fourier coefficients: ‖∑

Bn sin nπxL ‖L2 =

√L2∑

B2n (postponed)

My mathematical pedigree according to Mathematics Genealogy Project

Euler

d’Alembert

Lagrange

Laplace

Poisson

Fourier

...

Chasles

Dirichlet

Liouville

Darboux

H.A.Newton

Borel

Picard

...

Gibbs

Valiron

Lebesgue

...

Schwartz

...

Lions

Grothendieck

...

Temam

...

Yours truly

...

My mathematical lineage

Joseph-Louis Lagrange (student of Leonhard Euler) joint with Pierre-Simon Laplace(student of Jean Le Rond d’Alembert)⇒Siméon Denis Poisson (Ph.D. École Polytechnique 1800) + Jean Baptiste JosephFourier + ...⇒Michel Chasles ( Ph.D. École Polytechnique 1814) + Johann Peter Gustav LejeuneDirichlet (joint with Fourier) + Joseph Liouville⇒Jean Gaston Darboux (Ph.D. École Normale Supérieure Paris 1866) + another 1⇒Félix Edouard Justin Émile Borel (Ph.D. École Normale Supérieure Paris, 1893) + ...⇒Georges Valiron (Ph.D. Université de Paris, 1914) + ...⇒Laurent Schwartz (Ph.D. Université Louis Pasteur - Strasbourg I, 1943, Fields medalist1950) + ...⇒Jacques-Louis Lions (Ph.D. Université Henri Poincaré Nancy 1, 1954) + AlexanderGrothendieck + ...⇒Roger M. Témam (Ph.D. Université de Paris, 1967) + ...⇒Yours truly (Ph.D. Indiana University - Bloomington, 1996) + ...

Sept. 18th lecture, ( Sect. 3.1, 3.2, Fourier series and convergencestatement)

I Fourier series (defines a function if it converges)

a0 +∞∑

n=1

an cosnπx

L+∞∑

n=1

bn sinnπx

L= f (x)

I Fourier coefficients (for a function f )

a0 =1

2L

∫ L

−Lf (x) dx , an =

1L

∫ L

−Lf (x) cos

nπxL

dx , bn =1L

∫ L

0f (x) sin

nπxL

dx

f (x) ∼ a0 +∞∑

n=1

an cosnπx

L+∞∑

n=1

bn sinnπx

L(=?)

I piecewise smooth function, jump discontinuity, periodic extensionI Point-wise convergence Theorem: assume piece-wise smoothness; converges to

the average at a jump discontinuity 12 [f (x+) + f (x−)]

I Sketching Fourier series: 1. periodic ext.; 2. mark an "x" at the average of the twovalues at any jump discontinuity.

I Example: f (x) = 1 if x > L/2; f (x) = 0 otherwise.a0 = 1

4 , an = 1nπ (sin nπ − sin nπ

2 ), bn = 1nπ (cos nπ

2 − cos nπ)

I L2 theory (not covered in the textbook)I condition for convergence:

∑∞n=1(a2

n + b2n) <∞

I condition on the function for convergence: ‖f‖L2 <∞

I relationship between a function and its Fourier series: f L2= its Fourier series,

‖f‖2L2

def=∫ L−L f 2(x) = L(2a2

0 +∑∞

n=1(a2n + b2

n))

Sept. 18th lecture, ( sect. 3.3.1, sine series)

I Fourier sine series f (x) ∼∑∞

n=1 Bn sin nπxL ,Bn = 2

L

∫ L0 f (x) sin nπx

LI Odd function, odd extension, odd part of a function and the associated Fourier

sine seriesI Point-wise convergence Theorem: Assume piece-wise smoothness with jump

discontinuity, the Fourier sine series converges to 12 [f (x+) + f (x−)] for the

extended (odd, then periodic) function.I Sketching Fourier sine series: 1. odd extension; 2. periodic ext.; 3. mark an "x" at

the average of the two values at any jump discontinuity.I Example with Gibbs phenomenon: f (x) = 100,Bn = 400

nπ for odd n, 0 otherwise.Gibbs phenomenon (9% overshoot for f (x) = 100 with matlab code, or low modes ,

51 modes , figures on p.103)I More examples f (x) = x ,Bn = 2L

nπ (−1)n+1; f (x) = cos πxL ,Bn = 4n

π(n2−1)for even

n.I L2 theory (not covered in the textbook)

I condition for convergence:∑∞

n=1 B2n <∞

I condition on the function for convergence: ‖f‖L2 <∞

I relationship between a function and its Fourier series: f L2= its Fourier sine series,

‖f‖2L2 = L

2∑∞

n=1 B2n

Gibbs phenomenon

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

x/L

1

3 57

Gibbs phenomenon. page 103

Figure : Fourier sine series for f (x) = 100 with1, 3, 5, and 7 terms

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

x/L

51

Gibbs phenomenon, 51 terms, p. 103

Figure : Fourier sine series for f (x) = 100 with51 terms

Sept. 23rd lecture, ( sect. 3.3.2, 3.3.4, cosine series)

I Fourier cosine seriesf (x) ∼ A0 +

∑∞n=1 An cos nπx

L ,A0 = 1L

∫ L0 f (x),An = 2

L

∫ L0 f (x) cos nπx

LI Point-wise convergence Theorem: piece-wise smooth function with jump

discontinuity only, F-cosine series converges to 12 [f (x+) + f (x−)] of the extended

function (even then periodic)I Sketching Fourier cosine series: 1. even extension; 2. periodic ext.; 3. mark an "x"

at the average of the two values at any jump discontinuity.I Examples: f (x) = x ; f (x) = − L

2 sin πxL if x < 0, f (x) = x for 0 < x < L

2 ,f (x) = L− x if x > L

2 .

I L2 theory (not covered in the textbook)I condition for convergence:

∑∞n=0 A2

n <∞I condition on the function for convergence: ‖f‖L2 <∞

I relationship between a function and its Fourier series: f L2= its Fourier cosine series,

‖f‖2L2 = L

2 (2A20 +∑∞

n=1 A2n)

I Even part of a function and the associated Fourier cosine seriesI Fourier series = Fourier sine series for the odd part + Fourier cosine series for the

even part.

Sept. 23 continued (3.3.5, 3.6, cont F-series and term-by-term diff)I Continuous Fourier series

I general F-series: piecewise smooth + continuous + periodicityI F-cosine: piecewise smooth + continuousI F-sine: piecewise smooth + continuous + f (0) = f (L) = 0

I term-by-term differentiation of Fourier seriesI Counter-example to term-by-term differentiation x = 2L

∑∞n=1

(−1)n+1

nπ sin nπxL , but

1 6= 2∑∞

n=1(−1)n+1 cos nπxL

I point-wise senseI general F-series: continuity of the F-series + piecewise smoothness of f ′.

Alternative form on f : continuity of f + periodicity + piecewise smoothnessI F-cosine: piecewise smoothness of f ′ + continuity of the F-cosine series.

Alternative form on f : piecewise smoothness of f ′+ continuity of f .I F-sine: piecewise smoothness of f ′ + continuity of F-sine series.

Alternative: piecewise smoothness of f ′ + continuity of f + f (0) = f (L) = 0I F-sine: piecewise smoothness of f ′ + continuity of f (but no f (0) = f (L) = 0) (Postponed)

f ′(x) ∼1

L[f (L)− f (0)] +

∞∑n=1

[nπ

LBn +

2

L((−1)n f (L)− f (0))] cos

nπx

L

I Example: f (x) = x . F-cosine: x = L2 −

4Lπ2∑

n odd1

n2 cos nπxL .

I Example f (x) = x . F-sine series (w/o f (0) = f (L) = 0): x ∼∑∞

n=1L

nπ (−1)n+1 sin nπxL .

dfdx ∼ 1

I L2 sense differentiability (NOT in the textbook)∞∑

n=1

n2(a2n + b2

n) <∞

I Application to 1d homogeneous heat equation: solution is infinitely many timesdifferentiable w.r.t. x for any positive time.

Sept. 25 and 30 lecture, (3.4, diff w.r.t parameter, L2 theory)

I L2 sense differentiability (NOT in the textbook)

∞∑n=1

n2(a2n + b2

n) <∞

I differentiation with respect to parameters, L2 theory only (not in the textbook)

∞∑n=1

(|a′n(t)|2 + |b′n(t)|2) <∞

I Application of the L2 theory to 1d homogeneous heat equation with zerotemperatures at the ends (not in the textbook)

I solution is infinitely many times differentiable w.r.t. x and t for any positive time withf ∈ L2

u(x, t) =∞∑

n=1

Bn sinnπx

Le−κ(nπ/L)2 t

,Bn =2L

∫ L

0f (x) sin

nπxL

dx

I solution does solve the heat equationI solution is unique (via energy method)I solution continuously depends on the initial data (via energy method)

I Other boundary conditions and comparisons

Sept. 30 and Oct. 2nd lecture, ( 3.5, 3.6, integration, complex form)I term-by-term integration of Fourier series: always okay and always converges

(s3.5)∫ x

−Lf (t)dt = a0(x + L) +

∞∑n=1

[an

nπ/Lsin

nπxL

+bn

nπ/L(cos nπ − cos

nπxL

)]

I Example f (x) = 1 ∼ 4π

(sin πxL + 1

3 sin 3πxL + 1

5 sin 5πxL + · · · )

x ∼4Lπ2

(1 +132

+152

+ · · · )−4Lπ2

(cosπxL

+132

cos3πx

L+

152

cos5πx

L+ · · · )

4Lπ2

(1 +132

+152

+ · · · ) =1L

∫ L

0xdx =

L2

I complex form of Fourier series and complex orthgonality (s3.6)

f (x) ∼∞∑

n=−∞cne−inπx/L, cn =

12L

∫ L

−Lf (x)einπx/L, (c0 = a0, cn =

an + ibn

2, n > 0)

I complex orthogonality:∫ L−L e−inπx/L e−imπx/L dx = 0 if n 6= m

I convergence in the L2 sense:∑|cn|2 <∞

I once-differentiability in the L2 sense:∑

n2|cn|2 <∞I Example: f (x) = |x|.

c0 = L2 , c2k−1 = − 2L

(k−1)2π2 , c2k = 0, k = 1, 2, · · · , c−n = cn(bn = 0)

Oct. 7 lecture, (1D wave equation: derivation and solution 4.1, 4.2, 4.3)I derivation of the equation (1D wave) of a vertically vibrating string via Newton’s

lawI vertical displacement: u(x, t)I mass density: ρ0(x)I tensile force: T (x, t) = tension for perfectly flexible stringsI angle between the horizontal and the string: θ(x, t), ( ∂u

∂x = dydx = tan θ ≈ sin θ)

I Newton’s law in the vertical directionρ0(x)δx ∂

2u∂t2

= T (x + δx, t) sin θ(x + δx, t)− T (x, t) sin θ(x, t) + ρ0(x)δxQ(x, t)I wave equation:

∂2u∂t2

= c2 ∂2u∂x2

, c =

√T0

ρ0(x)(wave speed)

I boundary conditionsI fixed ends: u(0, t) = u(L, t) = 0 (more generally: prescribed ends) (Dirichlet)I elastic T0

∂u∂x (0, t) = k(u(0, t)− l). (y(t) = u(0, t): attached to a spring with spring

constant k , m d2ydt2

= −k(y(t)− l) + T0∂u∂x (0, t) + fext . For 1� m, fext = 0, leads to

elastic)I free: T0

∂u∂x (0, t) = 0 (Neumann).

I solution of the 1d wave equation via separation of variables

u(x , t) =∞∑

n=1

(An sinnπx

Lcos

nπctL

+ Bn sinnπx

Lsin

nπctL

)

An =2L

∫ L

0f (x) sin

nπxL, u(x , 0) = f (x),Bn

nπcL

=2L

∫ L

0g(x) sin

nπxL,∂u∂t

(x , 0) = g(x)

I solution to inhomogeneous 1D wave equation with fixed ends via eigenfunctionexpansion (not in the textbook).

Oct. 9th lecture: (tentative, 4.3, 4.4, 4.5)I normal modes (harmonics, overtones): sin nπx

L (An cos nπctL + Bn sin nπct

L )I natural frequencies: nπc/L, n = 1, 2, · · · ,I standing waves: motion of the normal modes

1mode standing wave 2 mode standing wave 3 mode standing wave

I nodes: zeros of the harmonicsI traveling waves (right and left with speed c):

sinnπx

Lsin

nπctL

=12

cosnπL

(x − ct)−12

cosnπL

(x + ct)

travelling wave

I differentiability of the solution to the wave equation;I Derivation of 2D wave equation (vibrating membrane)

I vertical displacement z = u(x, y, t)I mass density ρ0(x, y)I tensile force ~FT = T0~t × ~nI Newton’s law and Stokes theorem (

∫∫∇× ~B · ~ndA =

∮~B ·~tds)∫∫

ρ0∂2u∂t2

dA =

∮T0~t × ~n · ~kds =

∮T0(~n × ~k) ·~tds

ρ0∂2u∂t2

= T0∇× (~n × ~k)

I ~n ≈ −ux~i − uy~j + ~kI

∂2u∂t2

= c2(∂2u∂x2

+∂2u∂y2

)

I Conservation of total energy, uniqueness of solutions and continuous dependenceon data (not in the textbook)

Oct. 14 lecture (tentative, 4.6, Reflection and refraction ofelectromagnetic/light and acoustic/sound wave)

I plane traveling wave u = Aei(~k·~x−ωt), wave vector ~k , wave number k def= |~k |, wave

length (def= 2π

k ), wave front (~k · ~x − ωt = constant), phase velocity (def= ω

k = ±c)

I dispersive relationship for wave equation: ω2 = c2k2

I incident wave ei(~kI ·~x−ω+(kI )t), reflected wave Rei(~kR ·~x−ω+(kR )t), refracted(transmitted) wave Tei(~kT ·~x−ω−(kT )t). Continuity at z = 0:

ei(~kI ·~x−ω+(kI )t) + Rei(~kR ·~x−ω+(kR )t) = Tei(~kT ·~x−ω−(kT )t)

I ConsequencesI ω+(kI ) = ω+(kR) = ω−(kT )⇒ kI = kR ⇒ kI,z = −kR,z ⇒ θI = θRI Snell’s law of refraction (from horizontal frequency consideration kI sin θI = kT sin θT )

sin θT

sin θI=

kI

kT=

c−c+

I evanescent wave for large incident angle and c+ < c−

Oct. 23 lecture (examples and statements of S-L problems, 5.2, 5.3)I Examples of Sturm-Liouville problem

I heat flow in a non-uniform rod with a source term depending on solution Q = αuddx

(K0dφdx

) + αφ + λcρφ = 0

I circularly symmetric heat flowddr

(rdφdr

) + λrφ = 0

I Regular ( continuous plus p > 0, σ > 0 on [a, b]) Sturm-Liouville eigenvalueproblem

ddx (p(x) dφ

dx ) + q(x)φ+ λσ(x)φ = 0, a < x < b,

β1φ(a) + β2dφdx (a) = 0, β3φ(b) + β4

dφdx (b) = 0

I Statement of theorems for regular S-L problem1. All eigenvalues are real.2. There exist an infinite number of eigenvalues: λ1 < λ2 < · · · < λn < λn+1 < · · ·3. Corresponding eigenfunction φn(x) (unique up to an arbitrary multiplicative constant)

with n − 1 zeros in (a, b).4. {φn, n = 1, 2, · · · } form a "basis" or a "complete" set in the sense of the generalized

Fourier series

f (x) ∼∞∑

n=1

anφn(x)

for any piecewise smooth function (convergence to average at each jump discontinuity).5. Eigenfunctions are mutually orthogonal relative to the weight function σ(x), i.e.,∫ b

aφn(x)φm(x)σ(x)dx = 0, if λn 6= λm

6. Rayleigh quotient λ = (−pφdφ/dx|ba +∫ b

a [p(dφ/dx)2 − qφ2]dx)/∫ b

a φ2σdx

Oct. 28 lecture, (Examples of Sturm-Liouville problem and applications,5.3, 5.4)

I Example of Sturm-Liouville problem: heat conduction in a uniform rod without heatsource with zero temperature at the end. 5.3.3

I Application of Sturm-Liouville problem to heat conduction in a non-uniform rod(variable conductivity and density, formal manipulation). 5.4.Eigenfunction expansion!

Oct. 28 and 30 lecture, (Self-adjoint operators and S-L eigenvalueproblems, 5.5.)

Goal: motivate and prove a few properties for regular (p > 0, σ > 0) S-L problems:

ddx (p(x) dφ

dx ) + q(x)φ+ λσ(x)φ = 0, a < x < b,

β1φ(a) + β2dφdx (a) = 0, β3φ(b) + β4

dφdx (b) = 0

I L(u) = ddx (p(x) du

dx ) + q(x)u is self-adjoint , i.e.,∫ b

a [uL(v)− vL(u)]dx = 0(Lagrange identity uL(v)− vL(u) = d

dx [p(x)(u dvdx − v du

dx )])I eigenfunctions corresponding to different eigenvalues are mutually orthogonal

with weight σ(x)

I all eigenvalues are realI unique eigenfunctionI non-unique eigenfunction in the periodic caseI Gram-Schmidt orthogonalization

Oct. 30 lecture, ( S-L eigenvalue problem with Robin b.c., 5.8.)

−d2

dx2φ = λφ, φ(0) = 0, hφ(L) +

dφdx

(L) = 0

I positive eigenvalues: tan√λL = −

√λ

hI physical case h > 0

I nonphysical case h < 0 (two major subcases: −1/hL > 1 and 0 < −1/hL < 1, and aminor subcase: −1/hL = 1)

I zero eigenvalue: φ(x) = x if hL = −1, not an eigenvalue otherwise.I negative eigenvalues: tanh

√sL = −

√s

h , s = −λ

I special case h = 0: λn = (n − 12 )2π2/L2

Oct. 30 lecture, ( S-L eigenvalue problem with Robin b.c. summary andapp, 5.8.)

λ > 0 λ = 0 λ < 0Physical h > 0 sin

√λx

physical h = 0 sin√λx

nonphysical −1 < hL < 0 sin√λx

nonphysical hL = −1 sin√λx x

nonphysical hL < −1 sin√λx sinh

√s1x

Application: heat flow with a nonphysical boundary condition

∂u∂t

= κ∂2u∂x2

, u(0, t) = 0,∂u∂x

(L, t) = −hu(L, t), u(x , 0) = f (x), hL < −1

φn(x) =

{sinh√

s1x , n = 1,sin√λnx , n > 1

u(x , t) =∞∑

n=1

anφn(x)e−λnκt , f (x) =∞∑

n=1

anφn(x)

an =

∫ L0 f (x)φn(x)dx∫ L

0 φ2n dx

=

{ ∫ L0 f (x) sinh

√s1xdx/

∫ L0 sinh2√s1xdx , n = 1∫ L

0 f (x) sin√λnxdx/

∫ L0 sin2√λnxdx , n ≥ 2

Nov. 4th lecture (Approximation properties, connection to L2 theory,5.10)

"best approximation" of a function f with finitely many terms from the basis{φn, n = 1, 2, · · · }

I Mean-square deviation

EM (α1, · · · , αM ) =

∫ b

a[f (x)−

M∑n=1

αnφn(x)]2σ(x) dx .

I Minimizing mean-square deviation leads to

αn = an =

∫ ba f (x)φn(x)σ(x) dx∫ b

a φ2n(x)σ(x) dx

I Approximation error

EM =

∫ b

af 2σ dx −

M∑n=1

α2n

∫ b

aφ2

nσ dx

I Bessel’s inequalityEM (a1, · · · , an) ≥ 0

I Parseval’s identity (equality)(generalization of Pythagorean)∫ b

af 2σ dx =

∞∑n=1

a2n

∫ b

aφ2

nσ dx

Nov. 6th lecture, (separation of time variable, 7.1, 7.2)

I separation of time variable (vibrating membrane and heat conduction in arbitraryregion)

I eigenvalue-eigenfunction problem for the Laplace operator

−∆φ = λφ

aφ+ b∇φ · ~n = 0

Special case of a rectangular domain with homogeneous Dirichlet boundarycondition

I vibrating rectangular membrane with fixed boundary andu(x , y , 0) = α(x , y), ∂u

∂t (x , y , 0) = β(x , y)I separation of variables: u(x, y, t) = φ(x, y)h(t) = f (x)g(y)h(t)

φnm(x, y) = sinnπx

Lsin

mπyH

, λnm = (nπL

)2 + (mπH

)2

I general solution (and 2D F-series)

u(x, y, t) =∞∑

m=1

∞∑n=1

(Anm cos c√λnm t + Bnm sin c

√λnm t) sin

nπxL

sinmπy

H,

α(x, y) =∞∑

m=1

∞∑n=1

Anm sinnπx

Lsin

mπyH

,Anm =2L

∫ L

0

2H

∫ H

0α(x, y) sin

mπyH

dy sinnπx

Ldx,

c√λnmBnm =

4LH

∫ L

0

∫ H

0β(x, y) sin

nπxL

sinmπy

Hdydx

Nov. 13th lecture ( statement of Theorems for eigenvalue problem forLaplace operator, 7.4)

I statement of the eigenvalue problem for the Laplace operator

−∆φ = λφ

aφ+ b∇φ · ~n = 0

1. All the eigenvalues are real.2. There exists an infinite number of eigenvalues. There is a smallest one, but no largest

one.3. Dimension of eigenspace may be more than one.4. The eigenfunctions φ form a "complete" set (one can have a "basis" that consists of

eigenfunctions only), i.e, we have generalized F-series

f (x, y) ∼∑λ

aλφλ(x, y)

5. Eigenfunctions corresponding to different eigenvalues are mutually orthogonal.6. Rayleigh quotient

λ =−∮φ∇φ · ~n +

∫∫R |∇φ|

2 dxdy∫∫R φ

2 dxdyI example of Laplace operator on a rectangle with homogeneous Dirichlet boundary

condition

φnm(x , y) = sinnπx

Lsin

mπyH

, λnm = (nπL

)2 + (mπH

)2, n,m = 1, 2, . . .

I application to vibrating rectangular membrane

u =∑n,m

anm(t)φnm(x , y)

Nov. 13th lecture ( property of Laplace operator, 7.5)Goal: prove or motivate the results for eigenvalue problems for the Laplace operator

−∆φ = λφ, (x , y) ∈ R, aφ+ b∇φ · ~n = 0, (x , y) ∈ ∂R

I Green’s formula (2nd identity)∫∫R

(u∆v − v∆u) dxdy =

∮(u∇v − v∇u) · ~n ds

I L = −∆ plus b.c is self-adjoint.I All eigenvalues are real.I Eigenfunctions corresponding to different eigenvalues are mutually orthogonal.I Rayleigh quotient.

Appendix: Gram-Schmidt orthogonalizationI φ1, . . . , φn: independent eigenfunctions corresponding to the same eigenvalueI G-S procedure product: ψ1, . . . , ψn mutually orthogonal eigenfunctions for the

same eigenvalueI Description:

1.ψ1 = φ1

2. For j = 2, . . . , n,

ψj = φj −j−1∑i=1

∫∫R φjψi dxdy∫∫R ψ

2i dxdy

ψi

I Example: φ1 = 1, φ2 = x , φ3 = x2 on [0, 1]

Nov. 18th lecture ( Bessel’s DE and function, 7.7)I Motivation via separation of variable for eigenvalue problem (−∆φ = λφ) on a

disk: φ(r , θ) = f (r)g(θ) (a singular S-L problem for f )

d2gdθ2

= −µg,1r

ddr

(rdfdr

) + (λ− µ1r2

)f = 0, (|f (0)| <∞, f (a) = 0)

I Bessel’s DE of order m

z2 d2fdz2

+ zdfdz

+ (z2 −m2)f = 0 (r2 d2fdr2

+ rdfdr

+ (λr2 −m2)f = 0, z =√λr)

I Bessel function 1st and 2nd kind of order m (asymptotic properties near z = 0)

Jm(z) ∼{

1 m = 01

2mm!zm m > 0

, Ym(z) ∼{

2π

ln z m = 0− 2m(m−1)!

πz−m m > 0.

I Large z asymptotics: f (z) ∼ sin z, cos z ⇒ infinitely many zeros (denoted zmn forthe 1st kind ) (more precise result in 7.8.1)

I Eigenvalue-eigenfunction problems (1D S-L):

fmn(r) = Jm(zmnra

), (f = Jm(√λr), Jm(

√λa) = 0, λmn =

z2mn

a2)

α(r) =∞∑=1

anJm(√λmnr), an =

∫ a0 α(r)Jm(

√λmnr)r dr∫ 2

0 J2m(√λmnr)r dr

,

I Eigenvalue- eigenfunction for the Laplace operator on a disk (with Dirichlet bc)

φmn(r , θ) = Jm(√λmnr)

{cos mθsin mθ

}

Nov. 18th lecture (Bessel’s function applications to vibrating circularmembrane, 7.7)

I Application to vibrating circular membrane ∂2u∂t2 = c2∆u: u(r , θ, t) = f (r)g(θ)h(t)

(or just eigenfunction expansion u(r , θ, t) =∑λ αλφλ(r , θ)hλ(t))

Jm(√λmnr)

{cos mθsin mθ

}{cos c

√λmnt

sin c√λmnt

}I Special case of ∂u

∂t (r , θ, 0) = β(r , θ) = 0

u(r , θ, t) =∞∑

m=0

∞∑n=1

AmnJm(√λmnr) cos mθ cos c

√λmnt +

∞∑m=1

∞∑n=1

BmnJm(√λmnr) sin mθ cos c

√λmnt

=∞∑

m=0

∞∑n=1

Amnφmnc(r , θ) cos c√λmnt +

∞∑m=1

∞∑n=1

Bmnφmns(r , θ) cos c√λmnt

α(r , θ) =∑λ

Aλφλ(r , θ),Aλ =

∫∫α(r , θ)φλ(r , θ) dA∫∫

φ2λ(r , θ) dA

I circular symmetric case (u(r , 0) = α(r), ∂u∂t (r , 0) = β(r)) (λn = λ0n)

u(r , t) =∞∑

n=1

anJ0(√λnr) cos c

√λnt +

∞∑n=1

bnJ0(√λnr) sin c

√λnt

an =

∫ a0 α(r)J0(

√λnr)r dr∫ a

0 J20 (√λnr)r dr

, c√λnbn =

∫ a0 β(r)J0(

√λnr)r dr∫ a

0 J20 (√λnr)r dr

Eigenvalue-eigenfunction problem for the Laplace operator,Sturm-Liouville problems, special functions

I higher dimensional PDEs (ut = κ∆u, utt = c2∆u plus bc and ic, Laplace eq.)I reduction idea and relationship to the eigenvalue-eigenfunction problem for the

L-operatorI separation of variables (temp and spat) u = h(t)φ(~x)⇒ eigenvalue-eigenfunction prob

for L-operator⇒ λ, φλ(~x)⇒(superposition) u =∑λ αλhλ(t)φλ(~x) (why?)

I freeze time variable, expand in terms of eigenfunctions (why?) u =∑λ uλ(t)φλ(~x)

(uλ(t) = αλhλ(t))⇒ (for the heat eq) ddt uλ(t) = −λκuλ(t) (why?) ⇒

uλ(t) = e−λκt uλ(0)⇒ u =∑λ uλ(0)e−λκtφλ(~x)⇒ (IC)

α(~x)(= u(~x, 0)) =∑λ uλ(0)φλ(~x)⇒ (orthogonality) uλ(0) =

∫α(~x)φλ(~x) d~x∫φ2λ

(~x) d~x

I eigenvalue-eigenfunction problem for the L-operatorI 1D case: (standard) regular S-L problem with trigonometric functions as eigenfunctions⇒ solution = Fourier series

I 2D case (further reduction needed). 1. case rectangular domain (separation of variableor using 1D F-series): 2D eigenfunctions are double Fourier series. 2. case circular disk(separation of variables in polar coord or using 1D F-series in θ)⇒ Bessel’s equation (asingular S-L problem) in the radial direction and Bessel functions⇒ (bc) Bessel’sfunction of the 1st kind Jm and eigenvalues λmn = z2

mn/a2 ⇒ eigenfunctionsJm(√λmnr) cos mθ, Jm(

√λmnr) sin mθ.

I 3D case (additional reduction needed) 1. case of a rectangular box⇒ triple Fourierseries. 2. case of a sphere⇒ Legendre functions and polynomials. Pm

n and

eigenfunctions ρ−1/2Jn+ 12

(√λρ)

{cos mθsin mθ

}Pm

n (cosφ), λn+ 12 ,l

=z2n+1/2,l

a2

Nov. 25th lecture ( Legendre polynomials and functions, 7.10)I Laplace operator eigenvalue-eigenfunction problem (−∆w = λw) in a sphere in

spherical coordinates (x = ρ sinφ cos θ, y = ρ sinφ sin θ, z = ρ cosφ)

1ρ2

∂

∂ρ(ρ2 ∂w

∂ρ) +

1ρ2 sinφ

∂

∂φ(sinφ

∂w∂φ

) +1

ρ2 sin2 φ

∂2w∂θ2

+ λw = 0

I Use separation of variables (w = w̃(ρ, φ)q(θ) = f (ρ)q(θ)g(φ)) and 2π periodicityin θ

ddρ

(ρ2 dfdρ

) + (λρ2 − µ)f = 0,d

dφ(sinφ

dgdφ

) + (µ sinφ−m2

sinφ)g = 0

I Associated Legendre functions and Legendre polynomials(z = cosφ, dz = − sinφdφ)

ddz

[(1− z2)dgdz

] + (µ−m2

1− z2)g = 0, (singular S − L prob)

µ = n(n + 1) in order to ensure boundedness at z = ±1 (mysterious)⇒g(z) = Pm

n (z)I m = 0: Legendre polynomials:

P0(z) = 1,P1(z) = z, p2(z) = 12 (3z2 − 1),Pn(z) = P0

n (z) = 12nn!

dndzn (z2 − 1)n

I m > 0: associated Legendre functions (spherical harmonics)

Pmn (z) = (z2 − 1)m/2 dm

dzmPn(z),

∫ 1

−1[Pm

n (x)]2 dx = (n +12

)−1 (n + m)!

(n − m)!

Nov. 25th lecture ( eigenvalue and eigenfunctions for L-operator onsphere, 7.10)

I The radial S-L problem

ddρ

(ρ2 dfdρ

) + (λρ2 − n(n + 1))f = 0, f (a) = 0, |f (0)| <∞

I Relationship to Bessel’s function Zp(x) (HW)

f (ρ) = ρ−1/2Zn+ 1

2(√λρ) (spherical Bessel functions)

I boundedness⇒f (ρ) = ρ

−1/2Jn+ 12

(√λρ)

I bc⇒λ = λn+ 1

2 ,l, Jn+ 1

2(√λn+ 1

2 ,la) = 0, l = 1, 2, . . .

I relation to trigonometric functions

x−1/2Jn+ 12

(x) = xn(−1x

ddx

)n(sin x

x)

I Eigenvalue and eigenfunction for the Laplace operator on the sphere withhomogeneous bc

λ = λn+ 12 ,l, n = 1, 2, . . . , l = 1, 2, . . .

wλ(ρ, φ, θ) = ρ−1/2Jn+ 12

(√λρ)

{cos mθsin mθ

}Pm

n (cosφ), (form a basis)

I surface harmonics{cos mθsin mθ

}Pm

n (cosφ), n = 1, 2, . . . ,m = 0, 1, 2, . . .

Nov. 25th lecture ( application to heat conduction in a sphere)I The model

∂u∂t− κ∆u = 0, u

∣∣ρ=a = 0, u(ρ, φ, θ, 0) = α(ρ, φ, θ)

I Use eigenfunction expansion (why?)

u =∑λ

uλ(t)wλ(ρ, φ, θ)

λ = λn+ 12 ,l, n = 1, 2, . . . , l = 1, 2, . . . , (−∆wλ = λwλ)

wλ(ρ, φ, θ) = ρ−1/2Jn+ 12

(√λρ)

{cos mθsin mθ

}Pm

n (cosφ)

I reduced equationu′λ + κλuλ = 0

I initial condition

α =∑λ

uλ(0)wλ(ρ, φ, θ) =∑λ

αλ(t)wλ(ρ, φ, θ)

⇒uλ(t) = αλe−κλt , αλ =

∫∫∫αwλ dV∫∫∫w2λ dV

, (dV = ρ2 sinφ dρdφdθ)

I Final answeru(ρ, φ, θ, t) =

∑λ

αλe−κλt wλ(ρ, φ, θ)

Laplace equation in a cylinderI Laplace operator in cylindrical coordinates

∆ =1r∂

∂r(r∂

∂r) +

1r2

∂2

∂θ2+

∂2

∂z2

I Divide and conquer strategy u = u1(top bc) + u2(bottom bc) + u3(lateral bc)I strategy for u1 and u2: eigenfunction expansion for the 2D Laplace operator in a

disk

u1 =∑m,n

(AmnJm(√λmnr) cos mθ+BmnJm(

√λmnr) sin mθ)hmn(z), hmn = sinh

√λmnz

I strategy for u3:I eigenfunction expansion for the 1D Laplace operator in the vertical direction first

(hn(z) = sin nπzH , λ = ( nπ

H )2

I separation of var in the horizontal variables (r , θ) and use 2π periodicity in θ

rddr

(rdfdr

) + (−(nπH

)2r2 − m2)f = 0

w2 d2fdw2

+ wdfdw

+ (−w2 − m2)f = 0,w =nπH

r

I modified Bessel’s functions: 1st kind (well-defined at w = 0) Im(w), 2nd kind (singular atthe origin) Km(w)

I general solution (taking into consideration boundedness requirement)

u3 =∞∑

m=0

∞∑n=1

EmnIm(nπH

r) sinnπzH

cos mθ +∞∑

m=1

∞∑n=1

FmnIm(nπH

r) sinnπzH

sin mθ

I Properties of the modified Bessel functions (check Abramowitz and Stegun formore)

December 2nd lecture (non-homogeneous problem, 8.1, 8.2, 8.3)

How to deal with non-homogeneous problem? Reduction!I Source term with homogeneous boundary condition: eigenfunction expansion

(8.3). e.g. ut = uxx + sin 3x e−t , u(0, t) = u(π, t) = 0, u(x , 0) = f (x).I Nonhomogeneous boundary condition: convert to a problem with homogeneous

bc after subtracting a background profile b that satisfies the bc (sect.8.2). e.g.previous one with u(0, t) = 0, u(π, t) = 1⇒ b(x) = x/π.

I Eigenfunction expansion without converting to a problem with homogeneous bc:example of 1D heat equationut = κuxx + Q(x , t), u(0, t) = A(t), u(L, t) = B(t), u(x , 0) = f (x).

I eigenfunction expansion u =∑∞

n=1 bn(t)φn(x),Q(x, t) =∑∞

n=1 qn(t)φn(x)I equation in terms of expansion

∑∞n=1

dbndt φn(x) = κuxx +

∑∞n=1 qn(t)φn(x)

I equation in terms of the coefficients dbndt = κ

∫ L0 φn(x)uxx dx∫ L

0 φ2n dx

+ qn(t)

I equation in terms of the coefficients after utilizing Green’s formula and property of theeigenfunctions dbn

dt + κλnbn = κκ(nπ/L)[A(t)−(−1)nB(t)]∫ L

0 φ2n dx

+ qn(t)

I potential downside with this approach?

I Multi-dimensional case? Background profile could use Poisson equation. Sourceterm could use eigenfunction expansion. Read section 8.4.e.g. ut = κ∆u + Q(~x , t), u|t=0 = f plus inhomogeneous bc. Let b be abackground profile, say ∆b = 0 with the same b.c. as u (how?). Considerv = u − b⇒ vt = κ∆v + Q(~x , t)− bt plus homogeneous bc. Apply eigenfunctionexpansion method to find v . u = v + b.