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Chapter 7: Higher dimensional PDE

7.1 Introduction

7.2 Separation of the Time Variable

7.2.1 Vibrating Membrane: Any Shape

Displacement satisfies the two dimensional wave equation

Initial conditions

Product Solution

Separate

Choose separation constant because the time dependent differential equation has oscillatorysolutions

7.2.2 Heat Conduction: Any Region

Flow of thermal energy in any two dimensional region Seek product solutions of the form

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Separate

Heat flow in 3d region

Separate

7.2.3 Summary

7.3 Vibrating Rectangular Membrane

Satisfies 2d wave equation

Boundary conditions

Initial position & velocity

Product solutions

Separate

Separation constant must be positive

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Homogenous boundary conditions imply that the eigenvalue problem is

Product solution Substitute into PDE

Separate

Get a second separation constant Two ODEs result from separation of variables of PDE with two independent variables Product form implies the following homogenous boundary conditions

Regular Sturm Liouville eigenvalue problems

Eigenvalues

Corresponding eigenfunctions

Combine eigenvalues

Corresponding eigenfunctions

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Separation constant

Corresponding 2d eigenfunctions

Explicitly shown that eigenvalues are positive U(x,y,t)

Use initial conditions to determine coefficient Amn

Use initial conditions to determine coefficient Bmn

IN general for PDE in N variables that completely separate, there will be N ODEs, N-1 of whichare 1D eigenvalue problems

7.4 Statements and Illustration of Theorems for the Eigenvalue Problem

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Helmholtz equation

Boundary Condition

Generalized version

Generalized 2d version of SL problemo All eigenvalues are realo Infinite numbero Corresponding to an eigenvalue, there may be many eigenfunctionso Eigenfunctions form a complete set

o Orthogonality

o Rayleigh quotient

Exampleo PDE

o Eigenvalues

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o Eigenfunctions

o Real eigenvalues

o Ordering of eigenvalues

o Multiple eigenvalues

o Series of eigenfunctions

o Orthogonality of eigenfunctions

Convergence

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Chapter 8: Nonhomogenous Problems

8.1 Introduction

8.2 Heat flow with Sources and Nonhomogenous Boundary Conditions

Time independent boundary conditionso PDE:

o BC1: u(0, t) = Ao BC2: u(L, t) = Bo IC: u(x, 0) = f(x)

Equilibrium Temperatureo Satisfies the steady state heat equation

o Time independent boundary conditions

o Steady state implies linear temperature distribution

o Usually uEwill not be the desired time dependent solution since it satisfies tehintiailconditions only if f(x) = uE

Displacement from equilibriumoo First derivative & second derivatives are equal

o v(x,t) satisfies the heat equation

o u and uE agree at endpointso v(x,t) at endpoints

v(0,t) = 0 v(L, t) = 0

o Plug in the initial value

o v(x,t) is a linear homogenous partial differential equation with linear homogenousboundary conditions

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o Determine rest of v(x,t) by separation of variables

o Plug in initial conditions to find Fourier sine coefficients of f(x) - uE

o Explicit u(x,t)

o The temperature approaches its equilibrium distribution for all initial conditions

Steady nonhomogenous terms (i.e. steady sources of thermal energy)o PDE

o BC1: u(0,t) = Ao BC2: u(L,t) = Bo IC = u(x,0) = f(x)o Determine again the displacement form equilibrium

Time dependent nonhomogenous termso PDE

o BC: u(0,t) = A(t) & u(L,t) = B(t)o IC: u(x, 0) = f(x)

Related homogenous boundary conditionso We can always transform our problem into one with homogenous boundary conditions,

although in general the partial differential equation will remain nonhomogenous

o Consider any reference temperature distribution r(x,t)o Must satisfy given nonhomogenous boundary conditions

r(0,t) = A(t) r(L,t) = B(t)

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o r(x,t)

o v(x,t)

o v(0,t) = 0 & v(L,t) = 0o PDE satisfied by v(x,t) is derived by substituting u(x,t)

o Into the heat equation with sources. Thus

o In general the PDE for v(x,t) I sof the same type as for u(x,t) but with a differentnonhomogenous term, since r(x,t) usually does not satisfy the homogenous heat

equation

o Initial condition is also altered

o In general, only the boundary conditions have been made homogenous8.3 Method of Eigenfunction Expansion with Homogenous Boundary Conditions (Differentiating Series

of Eigenfunctions)

PDE:

BC: v(0,t) = v(L,t ) = 0

IC: v(x,0) = g(x)

Steps:

1. The related homogenous problema. PDE

b. BC: u(0,t) = u(L,t) = 02. Eigenfunctions of this related homogenous problem satisfy

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3. The method of eigenfunction expansion, employed to solve the nonhomogenous problem withhomogenous boundary conditions, consists of expanding the unknown solution v(x,t) in a series

of the related homogenous eigenfunctions

4. an(t) are just the generalized Fourier coefficients for v(x,t)

5. v(x,t) automatically satisfies the homogenous boundary conditions6. Term by term differentiation of v(x,t)

First derivative

Second derivative

7. Substitute results back into the original PDE

8. By orthogonality of eigenfunctions, we get first order differential equation for an(t)

9. Fourier coefficient of Q(x,t)

10.Multiply by integrating factor

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11. Integrate from 0 to t

12.Solve for an(t)a. an(t) is in the form of a constant, an(0) * homogenous solution + a particular solution

13.an(t) if the problem were homogenousElementary Example

PDE:

BC

Displacement from equilibrium v(x,t)

PDE of v(x,t)

BC of v(x,t)

Eigenfunctions

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Eigenfunction expansion substituted into the PDE

Solve for the unknown Fourier coefficients

Chapter 9: Greens Functions for Time-Independent Problems

9.1: Introduction

9.2: One-dimensional Heat equation

PDE:

BC

By method of separation of variables, we obtained u(x,t)

an

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Eliminate Fourier sine coefficients from u(x,t)

Interchange the order of operations of the infinite summation and integration, we obtain u(x,t)

Influence function- expresses the fact that the temperature at position x at time t is due to theinitial temperature at x0. To obtain the temperature u(x,t), we sum the influences of all possible

initial positions

General heat equation including sources (PDE)

BC

IC

Method of eigenvalue expansion

Differentiate term by term to get a first order differential equation for an(t)

Coefficients of the Fourier sine series of Q(x,t)

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Integrating factor to get an

u(x,t)

interchange order to get new u(x,t)

GreenS function

Rewrite u(x,t) in terms of Greens function

Causality principle- temperature at time t is only due to the thermal sources that acted beforetime t. Any future sources of heat energy cannot influence the temperature now

9.3 Greens Functions for Boundary Value Problems for ODEs

9.3.1 One dimensional Steady-State Heat equation

Steady state heat equation with homogenous boundary conditions, arising in situations in whichthe source term Q(x,t) = Q(x) is independent of time

BCs: u(0) = u(L) = 0

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Limit of Time-dependent problemo u(x,t)

o G(x, t; x0, t0)

o As t gets large

o Limit as t goes to infinity, steady source is still important

o As t gets large, u(x,t).

o G(x,x0) =

o Influence of Greens function for the steady state problem

9.3.3 The Method of Eigenfunction Expansion for Greens Functions

Generalization of L(u) = f(x) subject to 2 homogenous boundary conditions, related eigenvalueproblem

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Solve L(u) = f(x) to get u(x) as a generalized Fourier series of the eigenfunctions

Differentiate term by term twice to solve for coefficients

Plug back into u(x)

Greens function has representation in terms of the eigenfunctions

Greens function does not exist if one of the eigenvalues is 0.Example:

PDE:

BC: u(0) = u(L) = 0 Related eigenvalue problem

BC:

Solve for eigenvalues and corresponding eigenfunctions:

Fourier sine series of u(x)

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Fourier sine series of the Greens function

9.3.4 The Dirac Delta Function and Its Relationship to Greens Functions

Dirac Delta Functiono Isolate the effect of each individual point by decomposing f(x) into a linear combination

of unit pulses

o f(x) = sum of f(xi)(unit pulse starting at x = x i)o Similar to definition of integral

o Infinitely concentrated pulse, delta, called the concentrated source or impulsive force atx = xi

o Defn. Dirac delta function- so concentrated that in integrating it with any continuousfunction f(xi), it sifts out the value at x i=x.o Properties of Dirac delta function

Unit area

Even function

Heaviside unit step function

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Scaling property

o Greens function Solution of the nonhomogenous problem, L(u) = f(x) subject to two homogenous

boundary conditions is

Example: if f(x) is a concentrated source at x = xs, f(x)

then the response at x, u(x), satisfies

Greens function G(x, xs) is the response at x due to a concentrated source at xs,where G also satisfies the same homogenous boundary conditions at endpoints

Often with two homogenous boundary conditions is thought of as anindependent definition of Greens function. In this case we want to derive the

representation of the solution of the nonhomogenous problem in terms of the

Greens function satisfying L*G(x, xs)].

Greens formula

Definition of Dirac delta function

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Interchange variables

Maxwells reciprocity- symmetry of Greens functiono The response at x due to a concentrated source at x0is the same as the response at x0

due to a concentrated source at x.

Jump conditionso Greens function is continuous at x = xsbut its derivative is not

9.3.5 Nonhomogeneous Boundary Conditions

PDE

u(0) = a & u(L) = b Greens function PDE

BCs:

The Greens function always satisfies the related homogeneous boundary conditions Greens formula

Apply Greens function to endpoint

Use Diracs delta function

Representation of the solution of our nonhomogeneous problem (including thenonhomogeneous boundary conditions) in terms of the standard Greens function

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Evaluate Derivative with respect to the source position of the Greens function

Evaluate at endpoints and get new u(x,t)

The solution is the sum of a particular solution satisfying the homogeneous boundary conditionsand a homogeneous solution satisfying the two required nonhomogeneous boundary conditions

9.3.6 Summary

We have described 2 fundamental methods to obtain Greens function

1. Method of eigenfunction expansion2. Using the defining differential equation for Greens function3. Steady state Greens functions can be obtained as the limit as t goes to infinity of the solution

with steady sources.

9.4 Fredholm Alternative and Generalized Greens Functions

9.4.1 Introduction

Nonhomogeneous problem: L(u) = f(x) subject to homogeneous boundary conditions Used eigenfunction expansion to get u(x,t)

Where by substitution:

Zero eigenvalue

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Nontrivial homogeneous solutions of L(u) = f(x) solving the same homogeneous boundaryconditions are equivalent to eigenfunctions corresponding to the zero eigenvalue.

o If there are no nontrivial homogeneous solutions solving the same homogeneousboundary conditions, then 0 is not an eigenvalue

o If there are nontrivial homogeneous solutions solving the same homogeneous boundaryconditions, then 0 is an eigenvalue

o Example

9.4.2 Fredholm Alternative

The fredholm alternative summarizes the results of eigenfunction expansion fornonhomogeneous problems

L(u) = f(x) Subject to homogeneous boundary conditions (of self-adjoint type). Either

o U = 0 is the only homogeneous solution (0 is not an eigenvalue), in which case thenonhomogeneous problem has a unique solution or

o There are nontrivial homogeneous solutions (0 is an eigenvalue), in which case thenonhomogeneous problem has no solutions or an infinite number of solutions

There is an infinite number of solutions of L(u) = f(x) if

o Corresponds to the forcing function being orthogonal to the homogeneous solution Because the corresponding anis arbitrary These nonunique solutions correspond to an arbitrary additive multiple of a homogeneous

solution

If

o Then the nonhomogeneous problem with homogeneous BC has no solutions

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o Table 9.4.1: Number of Solutions of L(u) = f(x) subject to Homogeneous BoundaryConditions

o For the nonhomogeneous problem with homogeneous boundary conditions, solutionsexist only if the forcing function is orthogonal to all homogeneous solutions

If u = 0 is the only homogeneous solution, then f(x) is automatically orthogonalto it and there is a solution.

o Examples

9.5 Greens Functions for Poissons Equation

9.5.1 Introduction

o Greens formula for the Laplacian

9.5.2 Mutlidimensional Dirac Delta Function and Greens Functions

o Dirac delta function

o Fundamental operator property of multidimensional Dirac delta function

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o Greens functiono Nonhomogeneous PDE:

Homogeneous conditions along the boundaryo Greens function for the Poissons equation subject to the same homogeneous

boundary conditions

o Representation formula using Greens functiono Greens formula

o

o Reverse the roles

oo Symmetry

9.5.6 Infinite Space Greens Functions

o Greens function

o Since this is a model of steady state heat flow with a concentrated source located at x = x0withno boundaries, there should be a solution that is symmetric around the source point x = x0.

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o Radial distances

o Assume G only depends on radial distance

o Circularly/spherically symmetric solutions

o General solution

o Obtain singularity by integrating around a small circle/sphere

o Divergence theorem

o Singularity Condition

o Constants

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o Greens function for Poissons equation

o Symmetrico These infinite space Greens functions are themselves singular at the concentrated

source

o Solution of Poissons equation in infinite space, using the infinite space Greens function, weneed to utilize Greens formula

o Take the limit as radius approaches infinity

o Integrate with Dirac Delta function, reversing roles

o Conditions that must be satisfied at infinity in order for the boundary terms to vanish there

Chapter 10: Infinite Domain Problems: Fourier Transform Solutions of PDEs

10.1 Introduction

o Extend indefinitely in at least 1 direction10.2 Heat Equation on an Infinite Domain

o PDE

o Initial condition: u(x,0) = f(x)

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o Usually there are physical conditions at infinity

o Separation of Variables

o Integrate w over the continuous spectrum instead of summing over discrete spectrumcorresponding to Fourier series

o Separate variables

o Determining the separation constant

o Eigenvalue Problem

o Superposition Principle

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o Complex Exponentials

10.3 Fourier Transform Pair

10.3.1 Motivation from Fourier Series Identity

Complex form of a Courier Series

Complex Fourier coefficients

Extend to domain

The Fourier Series Identityo Eliminate cn

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o Wave number

o Integral partitions

10.3.2 Fourier Transform

Functions defined on the infinite interval may be thought of in some sense as periodic functionswith an infinite period

F(x) should be represented by a sum of waves of all possible wave lengths

New Fourier Series Identity

Fourier Integral Identity

Fourier Transform

Fourier integral representation of f(x) or Fourier integral

o f(x) is composed of waves e-iwxof all wave numbers w (and all wave lengths)o F(w) the fourier transform of f(x), represents the amplitude of the wave with wave

number w

Inverse Fourier transform of F(w)- If Fourier transform F(w) is known, f(x) as determined by theFourier integral is called the inverse fourier transform.

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10.3.3 Inverse Fourier Transform of a Gaussian

Gaussian

The function g(x), whose Fourier transform is G(w), is given by

The inverse Fourier transform of a Gaussian is itself a Gaussian Table 10.3.1 Fourier Transform of a Gaussian

Broadly spread

Sharply peaked

10.4 Fourier Transform and the Heat Equation

10.4.1 Heat Equation

Heat equation is solved by

Initial condition

o Fourier Integral representation of f(x)

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c(w) is the Fourier transform of the initial temperature distribution f(x)

Substitute c(w) into a solution

Interchange the order

g(x) is the inverse Fourier transform of

o Thus the integrand of u(x, t) contains Influence Function

o Determine the function g(x) whose Fourier transform iso And then make it a function ofo Is Gaussiano Let a = kt and we obtain the Gaussian g(x)

o The solution of the heat equation is

o Solution depends on the entire initial temperature distribution u(x, 0) = f(X)o

Each initial temperature influences the temperature at time t

Measures the effect of the initial temperature at position x on the temperatureat time t at location x.

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o As t goes to 0, the influence function becomes more concentrated Fundamental Solution of the heat equation

o Initial condition concentrated at x = 0, u(x,0) = f(x) = dirac deltao Solution of the heat equation with this initial condition

o Infinite domain Example

o Initial value problem:

o Solution of heat equation

o Change of variables

o Evenness

o As shown earlier

o Solution

o Similarity variable- temperature is constant whenever the similarity variable is constant

o Thermal energy spreads at an infinite propagation speed.

o Error function

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o Complementary error function

o Solution

Similarity solutiono Solutions remain the same under the elementary scaling x = Lxo PDE remain the same if time is scaledo Similarity variableo u(x,t)

o Derivatives of u

o F solves the following linear ODE

o General solution following form separation

o Integrating yields a similarity solution of the diffusion equation

o Self similar solutions must have very special self similar intial conditions, which have astep at x = 0

10.4.2 Fourier Transforming the Heat Equation: Transforms of Derivvatives

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Simpler method to solve the heat equation on an infinite interval: Fourier transform in thespatial variable the entire problem

Calculate Fourier transforms of derivatives of u

Define the spatial Fourier transform of u(x,t)

Multiply by eiwxand integrate

The spatial Fourier transform of a time derivative = the time derivative of the Fourier transform Simplify using integration by parts

IF u goes to 0 as x goes to infinity, the endpoint contributions of integration by parts vanish

Fourier transofmrs of higher derivatives may be obtained

In general, the Fourier transform of the nth derivative of a function with respect to x equalstimes the Fourier transform of the function, assuming that

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The Fourier transform operation converts a linear PDE with constant coefficients into an ODEsince spatial derivatives are transformed into algebraic multiples of the transform

General solution

d/dt is an ODE keeping w fixed and thus c is constant if w is fixed. C depends on w . Therefore:

c(s) = initial value of the transform (obtained by transforming the initial condition, f(x)

10.4.3 Convolution Theorem

Problem with inverting a transform is that a product of transforms of known functions occursvery frequently

Let F and G be the Fourier transforms of f and g respectively

we will determine h whose Fourier transform H equals the product of two transforms

Substitute one function

Interchange orders of integration

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Recognize inner integral as

h is

The inverse Fourier transform of the product of two Fourier transforms is 1/2pi times theconvolution of the two functions

Alternative form:

Heat Equationo Apply convolution theorem

o Solution

Summary procedureo Fourier thransform the PDEo Solve the ODEo Apply initial conditions to determine the initial Fourier transformo Use convolution theorem

Parsevals identityo Restatement of convolution theorem

o At x = 0

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o If we pick g(x) such that

o g(x) is the reflection of f(x) around x = 0. In general their Fourier transforms are related

o Let s = -xo Parsevals Identity

o The Fourier transform G(w) of a function g(x) is a complex quantity whose magnitudesquared is the spectral energy density (the amount of energy per unit wave number)