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** An Introduction to Partial Differential Equations with MATLAB **exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB^{®} software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the “Big Three” PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.

Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.

Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green’s functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.

### Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs--Definitions

Linear PDEs--The Principle of Superposition

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

### The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation--The Potential Equation

Using Separation of Variables to Solve the Big Three PDEs

### Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

The Fourier Series---Proof of Pointwise Convergence

Fourier Sine and Cosine Series

Completeness

**Solving the Big Three PDE****s **

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace's Equation on a Rectangular

Domain

Nonhomogeneous Problems

**Characteristicsfor Linear PDE****s **

First-Order PDEs with Constant Coefficients

First-Order PDEs with Variable Coefficients

D'Alembert's Solution for the Wave Equation--The Infinite

String

Characteristics for Semi-Infinite and Finite String Problems

General Second-Order Linear PDEs and Characteristics

### Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Distributions, the Dirac Delta Function and Generalized Fourier

Transforms

Proof of the Fourier Integral Formula

Bessel Functions and Orthogonal Polynomials

The Special Functions and Their Differential Equations

Ordinary Points and Power Series Solutions; Chebyshev, Hermite

and Legendre Polynomials

The Method of Frobenius; Laguerre Polynomials

Interlude: The Gamma Function

Bessel Functions

Recap: A List of Properties of Bessel Functions and Orthogonal

Polynomials

Sturm-Liouville Theory and Generalized Fourier Series

Sturm-Liouville Problems

Regular and Periodic Sturm-Liouville Problems

Singular Sturm-Liouville Problems; Self-Adjoint Problems

The Mean-Square or *L*^{2} Norm and Convergence in the Mean

Generalized Fourier Series; Parseval's Equality and Completeness

**PDEs in Higher Dimensions **

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier

Series

Laplace's Equation in Polar Coordinates; Poisson's Integral

Formula

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;

Green's Identities for the Laplacian

Nonhomogeneous Problems and Green's Functions

Green's Functions for ODEs

Green's Function and the Dirac Delta Function

Green's Functions for Elliptic PDEs (I): Poisson's Equation in

Two Dimensions

Green's Functions for Elliptic PDEs (II): Poisson's Equation in

Three Dimensions; the Helmholtz Equation

Green's Function's for Equations of Evolution

### Numerical Methods

Finite Difference Approximations for ODEs

Finite Difference Approximations for PDEs

Spectral Methods and the Finite Element Method

References

Uniform Convergence; Differentiation and Integration of Fourier Series

Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations

Existence and Uniqueness Theorems

A Menagerie of PDEs

MATLAB Code for Figures and Exercises

Answers to Selected Exercises

** An Introduction to Partial Differential Equations with MATLAB **exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB^{®} software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the “Big Three” PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.

Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.

Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green’s functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.

### Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs--Definitions

Linear PDEs--The Principle of Superposition

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

### The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation--The Potential Equation

Using Separation of Variables to Solve the Big Three PDEs

### Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

The Fourier Series---Proof of Pointwise Convergence

Fourier Sine and Cosine Series

Completeness

**Solving the Big Three PDE****s **

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace's Equation on a Rectangular

Domain

Nonhomogeneous Problems

**Characteristicsfor Linear PDE****s **

First-Order PDEs with Constant Coefficients

First-Order PDEs with Variable Coefficients

D'Alembert's Solution for the Wave Equation--The Infinite

String

Characteristics for Semi-Infinite and Finite String Problems

General Second-Order Linear PDEs and Characteristics

### Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Distributions, the Dirac Delta Function and Generalized Fourier

Transforms

Proof of the Fourier Integral Formula

Bessel Functions and Orthogonal Polynomials

The Special Functions and Their Differential Equations

Ordinary Points and Power Series Solutions; Chebyshev, Hermite

and Legendre Polynomials

The Method of Frobenius; Laguerre Polynomials

Interlude: The Gamma Function

Bessel Functions

Recap: A List of Properties of Bessel Functions and Orthogonal

Polynomials

Sturm-Liouville Theory and Generalized Fourier Series

Sturm-Liouville Problems

Regular and Periodic Sturm-Liouville Problems

Singular Sturm-Liouville Problems; Self-Adjoint Problems

The Mean-Square or *L*^{2} Norm and Convergence in the Mean

Generalized Fourier Series; Parseval's Equality and Completeness

**PDEs in Higher Dimensions **

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier

Series

Laplace's Equation in Polar Coordinates; Poisson's Integral

Formula

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;

Green's Identities for the Laplacian

Nonhomogeneous Problems and Green's Functions

Green's Functions for ODEs

Green's Function and the Dirac Delta Function

Green's Functions for Elliptic PDEs (I): Poisson's Equation in

Two Dimensions

Green's Functions for Elliptic PDEs (II): Poisson's Equation in

Three Dimensions; the Helmholtz Equation

Green's Function's for Equations of Evolution

### Numerical Methods

Finite Difference Approximations for ODEs

Finite Difference Approximations for PDEs

Spectral Methods and the Finite Element Method

References

Uniform Convergence; Differentiation and Integration of Fourier Series

Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations

Existence and Uniqueness Theorems

A Menagerie of PDEs

MATLAB Code for Figures and Exercises

Answers to Selected Exercises

** An Introduction to Partial Differential Equations with MATLAB **exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB^{®} software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the “Big Three” PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.

Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.

Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green’s functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.

### Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs--Definitions

Linear PDEs--The Principle of Superposition

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

### The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation--The Potential Equation

Using Separation of Variables to Solve the Big Three PDEs

### Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

The Fourier Series---Proof of Pointwise Convergence

Fourier Sine and Cosine Series

Completeness

**Solving the Big Three PDE****s **

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace's Equation on a Rectangular

Domain

Nonhomogeneous Problems

**Characteristicsfor Linear PDE****s **

First-Order PDEs with Constant Coefficients

First-Order PDEs with Variable Coefficients

D'Alembert's Solution for the Wave Equation--The Infinite

String

Characteristics for Semi-Infinite and Finite String Problems

General Second-Order Linear PDEs and Characteristics

### Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Distributions, the Dirac Delta Function and Generalized Fourier

Transforms

Proof of the Fourier Integral Formula

Bessel Functions and Orthogonal Polynomials

The Special Functions and Their Differential Equations

Ordinary Points and Power Series Solutions; Chebyshev, Hermite

and Legendre Polynomials

The Method of Frobenius; Laguerre Polynomials

Interlude: The Gamma Function

Bessel Functions

Recap: A List of Properties of Bessel Functions and Orthogonal

Polynomials

Sturm-Liouville Theory and Generalized Fourier Series

Sturm-Liouville Problems

Regular and Periodic Sturm-Liouville Problems

Singular Sturm-Liouville Problems; Self-Adjoint Problems

The Mean-Square or *L*^{2} Norm and Convergence in the Mean

Generalized Fourier Series; Parseval's Equality and Completeness

**PDEs in Higher Dimensions **

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier

Series

Laplace's Equation in Polar Coordinates; Poisson's Integral

Formula

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;

Green's Identities for the Laplacian

Nonhomogeneous Problems and Green's Functions

Green's Functions for ODEs

Green's Function and the Dirac Delta Function

Green's Functions for Elliptic PDEs (I): Poisson's Equation in

Two Dimensions

Green's Functions for Elliptic PDEs (II): Poisson's Equation in

Three Dimensions; the Helmholtz Equation

Green's Function's for Equations of Evolution

### Numerical Methods

Finite Difference Approximations for ODEs

Finite Difference Approximations for PDEs

Spectral Methods and the Finite Element Method

References

Uniform Convergence; Differentiation and Integration of Fourier Series

Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations

Existence and Uniqueness Theorems

A Menagerie of PDEs

MATLAB Code for Figures and Exercises

Answers to Selected Exercises

** An Introduction to Partial Differential Equations with MATLAB **exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB^{®} software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

### Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs--Definitions

Linear PDEs--The Principle of Superposition

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

### The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation--The Potential Equation

Using Separation of Variables to Solve the Big Three PDEs

### Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

The Fourier Series---Proof of Pointwise Convergence

Fourier Sine and Cosine Series

Completeness

**Solving the Big Three PDE****s **

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace's Equation on a Rectangular

Domain

Nonhomogeneous Problems

**Characteristicsfor Linear PDE****s **

First-Order PDEs with Constant Coefficients

First-Order PDEs with Variable Coefficients

D'Alembert's Solution for the Wave Equation--The Infinite

String

Characteristics for Semi-Infinite and Finite String Problems

General Second-Order Linear PDEs and Characteristics

### Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Distributions, the Dirac Delta Function and Generalized Fourier

Transforms

Proof of the Fourier Integral Formula

Bessel Functions and Orthogonal Polynomials

The Special Functions and Their Differential Equations

Ordinary Points and Power Series Solutions; Chebyshev, Hermite

and Legendre Polynomials

The Method of Frobenius; Laguerre Polynomials

Interlude: The Gamma Function

Bessel Functions

Recap: A List of Properties of Bessel Functions and Orthogonal

Polynomials

Sturm-Liouville Theory and Generalized Fourier Series

Sturm-Liouville Problems

Regular and Periodic Sturm-Liouville Problems

Singular Sturm-Liouville Problems; Self-Adjoint Problems

The Mean-Square or *L*^{2} Norm and Convergence in the Mean

Generalized Fourier Series; Parseval's Equality and Completeness

**PDEs in Higher Dimensions **

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier

Series

Laplace's Equation in Polar Coordinates; Poisson's Integral

Formula

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;

Green's Identities for the Laplacian

Nonhomogeneous Problems and Green's Functions

Green's Functions for ODEs

Green's Function and the Dirac Delta Function

Green's Functions for Elliptic PDEs (I): Poisson's Equation in

Two Dimensions

Green's Functions for Elliptic PDEs (II): Poisson's Equation in

Three Dimensions; the Helmholtz Equation

Green's Function's for Equations of Evolution

### Numerical Methods

Finite Difference Approximations for ODEs

Finite Difference Approximations for PDEs

Spectral Methods and the Finite Element Method

References

Uniform Convergence; Differentiation and Integration of Fourier Series

Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations

Existence and Uniqueness Theorems

A Menagerie of PDEs

MATLAB Code for Figures and Exercises

Answers to Selected Exercises

** An Introduction to Partial Differential Equations with MATLAB **exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB^{®} software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

### Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs--Definitions

Linear PDEs--The Principle of Superposition

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

### The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation--The Potential Equation

Using Separation of Variables to Solve the Big Three PDEs

### Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

The Fourier Series---Proof of Pointwise Convergence

Fourier Sine and Cosine Series

Completeness

**Solving the Big Three PDE****s **

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace's Equation on a Rectangular

Domain

Nonhomogeneous Problems

**Characteristicsfor Linear PDE****s **

First-Order PDEs with Constant Coefficients

First-Order PDEs with Variable Coefficients

D'Alembert's Solution for the Wave Equation--The Infinite

String

Characteristics for Semi-Infinite and Finite String Problems

General Second-Order Linear PDEs and Characteristics

### Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Distributions, the Dirac Delta Function and Generalized Fourier

Transforms

Proof of the Fourier Integral Formula

Bessel Functions and Orthogonal Polynomials

The Special Functions and Their Differential Equations

Ordinary Points and Power Series Solutions; Chebyshev, Hermite

and Legendre Polynomials

The Method of Frobenius; Laguerre Polynomials

Interlude: The Gamma Function

Bessel Functions

Recap: A List of Properties of Bessel Functions and Orthogonal

Polynomials

Sturm-Liouville Theory and Generalized Fourier Series

Sturm-Liouville Problems

Regular and Periodic Sturm-Liouville Problems

Singular Sturm-Liouville Problems; Self-Adjoint Problems

The Mean-Square or *L*^{2} Norm and Convergence in the Mean

Generalized Fourier Series; Parseval's Equality and Completeness

**PDEs in Higher Dimensions **

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier

Series

Laplace's Equation in Polar Coordinates; Poisson's Integral

Formula

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;

Green's Identities for the Laplacian

Nonhomogeneous Problems and Green's Functions

Green's Functions for ODEs

Green's Function and the Dirac Delta Function

Green's Functions for Elliptic PDEs (I): Poisson's Equation in

Two Dimensions

Green's Functions for Elliptic PDEs (II): Poisson's Equation in

Three Dimensions; the Helmholtz Equation

Green's Function's for Equations of Evolution

### Numerical Methods

Finite Difference Approximations for ODEs

Finite Difference Approximations for PDEs

Spectral Methods and the Finite Element Method

References

Uniform Convergence; Differentiation and Integration of Fourier Series

Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations

Existence and Uniqueness Theorems

A Menagerie of PDEs

MATLAB Code for Figures and Exercises

Answers to Selected Exercises

** An Introduction to Partial Differential Equations with MATLAB **exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB^{®} software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

### Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs--Definitions

Linear PDEs--The Principle of Superposition

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

### The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation--The Potential Equation

Using Separation of Variables to Solve the Big Three PDEs

### Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

The Fourier Series---Proof of Pointwise Convergence

Fourier Sine and Cosine Series

Completeness

**Solving the Big Three PDE****s **

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace's Equation on a Rectangular

Domain

Nonhomogeneous Problems

**Characteristicsfor Linear PDE****s **

First-Order PDEs with Constant Coefficients

First-Order PDEs with Variable Coefficients

D'Alembert's Solution for the Wave Equation--The Infinite

String

Characteristics for Semi-Infinite and Finite String Problems

General Second-Order Linear PDEs and Characteristics

### Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Distributions, the Dirac Delta Function and Generalized Fourier

Transforms

Proof of the Fourier Integral Formula

Bessel Functions and Orthogonal Polynomials

The Special Functions and Their Differential Equations

Ordinary Points and Power Series Solutions; Chebyshev, Hermite

and Legendre Polynomials

The Method of Frobenius; Laguerre Polynomials

Interlude: The Gamma Function

Bessel Functions

Recap: A List of Properties of Bessel Functions and Orthogonal

Polynomials

Sturm-Liouville Theory and Generalized Fourier Series

Sturm-Liouville Problems

Regular and Periodic Sturm-Liouville Problems

Singular Sturm-Liouville Problems; Self-Adjoint Problems

The Mean-Square or *L*^{2} Norm and Convergence in the Mean

Generalized Fourier Series; Parseval's Equality and Completeness

**PDEs in Higher Dimensions **

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier

Series

Laplace's Equation in Polar Coordinates; Poisson's Integral

Formula

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;

Green's Identities for the Laplacian

Nonhomogeneous Problems and Green's Functions

Green's Functions for ODEs

Green's Function and the Dirac Delta Function

Green's Functions for Elliptic PDEs (I): Poisson's Equation in

Two Dimensions

Green's Functions for Elliptic PDEs (II): Poisson's Equation in

Three Dimensions; the Helmholtz Equation

Green's Function's for Equations of Evolution

### Numerical Methods

Finite Difference Approximations for ODEs

Finite Difference Approximations for PDEs

Spectral Methods and the Finite Element Method

References

Uniform Convergence; Differentiation and Integration of Fourier Series

Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations

Existence and Uniqueness Theorems

A Menagerie of PDEs

MATLAB Code for Figures and Exercises

Answers to Selected Exercises