# Mathematical Physics with Partial Differential Equations

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Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field – the heat equation, the wave equation, and Laplace’s equation. The most common techniques of solving such equations are developed in this book, including Green’s functions, the Fourier transform, and the Laplace transform, which all have applications in mathematics and physics far beyond solving the above equations. The book’s focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book’s rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics.

## Key Features

- Examines in depth both the equations and their methods of solution
- Presents physical concepts in a mathematical framework
- Contains detailed mathematical derivations and solutions— reinforcing the material through repetition of both the equations and the techniques
- Includes several examples solved by multiple methods—highlighting the strengths and weaknesses of various techniques and providing additional practice

## Readership

Advanced Undergraduate and Graduate Students, Instructors, Academic Researchers in University Mathematics Departments

## Table of Contents

- Preface

1. Preliminaries

1-1. Self-Adjoint Operators

1-2. Curvilinear Coordinates

1-3. Approximate Identities and the Dirac-δ Function

1-4. The Issue of Convergence

1-5. Some Important Integration Formulas

2. Vector Calculus

2-1. Vector Integration

2-2. Divergence and Curl

2-3. Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem

3. Green’s Functions

3-1. Construction of Green’s Function using the Dirac-δ Function

3-2. Construction of Green’s Function using Variation of Parameters

3-3. Construction of Green’s Function from Eigenfunctions

3-4. More General Boundary Conditions

3-5. The Fredholm Alternative (Or, what if 0 is an Eigenvalue?)

3-6. Green’s function for the Laplacian in Higher Dimensions

4. Fourier Series

4-1. Basic Definitions

4-2. Methods of Convergence of Fourier Series

4-3. The Exponential Form of Fourier Series

4-4. Fourier Sine and Cosine Series

4-5. Double Fourier Series

5. Three Important Equations

5-1. Laplace’s Equation

5-2. Derivation of the Heat Equation in One Dimension

5-3. Derivation of the Wave equation in One Dimension

5-4. An Explicit Solution of the Wave Equation

5-5. Converting Second-Order PDEs to Standard Form

6. Sturm-Liouville Theory

6-1. The Self-Adjoint Property of a Sturm-Liouville Equation

6-2. Completeness of Eigenfunctions for Sturm-Liouville Equations

6-3. Uniform Convergence of Fourier Series

7. Separation of Variables in Cartesian Coordinates

7-1. Solving Laplace’s Equation on a Rectangle

7-2. Laplace’s Equation on a Cube

7-3. Solving the Wave Equation in One Dimension by Separation of Variables

7-4. Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables

7-5. Solving the Heat Equation in One Dimension using Separation of Variables

7-6. Steady State of the Heat equation

7-7. Checking the Validity of the Solution

8. Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables

8-1. The Solution to Bessel’s Equation in Cylindrical Coordinates

8-2. Solving Laplace’s Equation in Cylindrical Coordinates using Separation of Variables

8-3. The Wave Equation on a Disk (Drum Head Problem)

8-4. The Heat Equation on a Disk

9. Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables

9-1. An Example Where Legendre Equations Arise

9-2. The Solution to Bessel’s Equation in Spherical Coordinates

9-3. Legendre’s Equation and its Solutions

9-4. Associated Legendre Functions

9-5. Laplace’s Equation in Spherical Coordinates

10. The Fourier Transform

10-1. The Fourier Transform as a Decomposition

10-2. The Fourier Transform from the Fourier Series

10-3. Some Properties of the Fourier Transform

10-4. Solving Partial Differential Equations using the Fourier Transform

10-5. The Spectrum of the Negative Laplacian in One Dimension

10-6. The Fourier Transform in Three Dimensions

11. The Laplace Transform

11-1. Properties of the Laplace Transform

11-2. Solving Differential Equations using the Laplace Transform

11-3. Solving the Heat Equation using the Laplace Transform

11-4. The Wave Equation and the Laplace Transform

12. Solving PDEs with Green’s Functions

12-1. Solving the Heat Equation using Green’s Function

12-2. The Method of Images

12-3. Green’s Function for the Wave Equation

12-4. Green’s Function and Poisson’s Equation

Appendix. Computing the Laplacian with the Chain Rule References Index

## Product details

- No. of pages: 432
- Language: English
- Copyright: © Academic Press 2012
- Published: December 1, 2011
- Imprint: Academic Press
- eBook ISBN: 9780123869944

## About the Author

### James Kirkwood

Bringing over 25 years of teaching expertise, James Kirkwood is the author of ten mathematics books published in a range of areas from calculus to real analysis and mathematical biology. He has been awarded 4 awards for continuing research in the area of mathematical physics, including the 2016 ‘Outstanding Faculty Award of the State Council for Higher Education in Virginia’ – the highest award the state bestows.

#### Affiliations and Expertise

Professor of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA, USA

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