Mathematical Physics with Partial Differential Equations

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