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1st Edition - December 1, 2011

**Author:** James Kirkwood

eBook ISBN:

9 7 8 - 0 - 1 2 - 3 8 6 9 9 4 - 4

Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math… Read more

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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.

- 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

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

Preface1. Preliminaries1-1. Self-Adjoint Operators1-2. Curvilinear Coordinates1-3. Approximate Identities and the Dirac-δ Function1-4. The Issue of Convergence1-5. Some Important Integration Formulas2. Vector Calculus2-1. Vector Integration2-2. Divergence and Curl2-3. Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem3. Green’s Functions3-1. Construction of Green’s Function using the Dirac-δ Function3-2. Construction of Green’s Function using Variation of Parameters3-3. Construction of Green’s Function from Eigenfunctions3-4. More General Boundary Conditions3-5. The Fredholm Alternative (Or, what if 0 is an Eigenvalue?)3-6. Green’s function for the Laplacian in Higher Dimensions4. Fourier Series4-1. Basic Definitions4-2. Methods of Convergence of Fourier Series4-3. The Exponential Form of Fourier Series4-4. Fourier Sine and Cosine Series4-5. Double Fourier Series5. Three Important Equations5-1. Laplace’s Equation5-2. Derivation of the Heat Equation in One Dimension5-3. Derivation of the Wave equation in One Dimension5-4. An Explicit Solution of the Wave Equation5-5. Converting Second-Order PDEs to Standard Form6. Sturm-Liouville Theory6-1. The Self-Adjoint Property of a Sturm-Liouville Equation6-2. Completeness of Eigenfunctions for Sturm-Liouville Equations6-3. Uniform Convergence of Fourier Series7. Separation of Variables in Cartesian Coordinates7-1. Solving Laplace’s Equation on a Rectangle7-2. Laplace’s Equation on a Cube7-3. Solving the Wave Equation in One Dimension by Separation of Variables7-4. Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables7-5. Solving the Heat Equation in One Dimension using Separation of Variables7-6. Steady State of the Heat equation7-7. Checking the Validity of the Solution8. Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables8-1. The Solution to Bessel’s Equation in Cylindrical Coordinates8-2. Solving Laplace’s Equation in Cylindrical Coordinates using Separation of Variables8-3. The Wave Equation on a Disk (Drum Head Problem)8-4. The Heat Equation on a Disk9. Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables9-1. An Example Where Legendre Equations Arise9-2. The Solution to Bessel’s Equation in Spherical Coordinates9-3. Legendre’s Equation and its Solutions9-4. Associated Legendre Functions9-5. Laplace’s Equation in Spherical Coordinates10. The Fourier Transform10-1. The Fourier Transform as a Decomposition10-2. The Fourier Transform from the Fourier Series10-3. Some Properties of the Fourier Transform10-4. Solving Partial Differential Equations using the Fourier Transform10-5. The Spectrum of the Negative Laplacian in One Dimension10-6. The Fourier Transform in Three Dimensions11. The Laplace Transform11-1. Properties of the Laplace Transform11-2. Solving Differential Equations using the Laplace Transform11-3. Solving the Heat Equation using the Laplace Transform11-4. The Wave Equation and the Laplace Transform12. Solving PDEs with Green’s Functions12-1. Solving the Heat Equation using Green’s Function12-2. The Method of Images12-3. Green’s Function for the Wave Equation12-4. Green’s Function and Poisson’s EquationAppendix. Computing the Laplacian with the Chain Rule References Index

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

JK

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