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| Where PDEs Come From | |
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| What is a Partial Differential Equation? | |
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| First-Order Linear Equations | |
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| Flows, Vibrations, and Diffusions | |
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| Initial and Boundary Conditions | |
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| Well-Posed Problems | |
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| Types of Second-Order Equations | |
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| Waves and Diffusions | |
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| The Wave Equation | |
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| Causality and Energy | |
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| The Diffusion Equation | |
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| Diffusion on the Whole Line | |
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| Comparison of Waves and Diffusions | |
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| Reflections and Sources | |
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| Diffusion on the Half-Line | |
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| Reflections of Waves | |
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| Diffusion with a Source | |
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| Waves with a Source | |
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| Diffusion Revisited | |
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| Boundary Problems | |
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| Separation of Variables, The Dirichlet Condition | |
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| The Neumann Condition | |
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| The Robin Condition | |
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| Fourier Series | |
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| The Coefficients | |
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| Even, Odd, Periodic, and Complex Functions | |
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| Orthogonality and General Fourier Series | |
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| Completeness | |
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| Completeness and the Gibbs Phenomenon | |
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| Inhomogeneous Boundary Conditions | |
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| Harmonic Functions | |
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| Laplace's Equation | |
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| Rectangles and Cubes | |
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| Poisson's Formula | |
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| Circles, Wedges, and Annuli | |
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| Green's Identities and Green's Functions | |
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| Green's First Identity | |
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| Green's Second Identity | |
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| Green's Functions | |
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| Half-Space and Sphere | |
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| Computation of Solutions | |
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| Opportunities and Dangers | |
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| Approximations of Diffusions | |
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| Approximations of Waves | |
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| Approximations of Laplace's Equation | |
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| Finite Element Method | |
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| Waves in Space | |
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| Energy and Causality | |
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| The Wave Equation in Space-Time | |
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| Rays, Singularities, and Sources | |
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| The Diffusion and Schrodinger Equations | |
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| The Hydrogen Atom | |
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| Boundaries in the Plane and in Space | |
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| Fourier's Method, Revisited | |
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| Vibrations of a Drumhead | |
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| Solid Vibrations in a Ball | |
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| Nodes | |
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| Bessel Functions | |
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| Legendre Functions | |
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| Angular Momentum in Quantum Mechanics | |
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| General Eigenvalue Problems | |
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| The Eigenvalues Are Minima of the Potential Energy | |
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| Computation of Eigenvalues | |
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| Completeness | |
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| Symmetric Differential Operators | |
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| Completeness and Separation of Variables | |
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| Asymptotics of the Eigenvalues | |
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| Distributions and Transforms | |
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| Distributions | |
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| Green's Functions, Revisited | |
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| Fourier Transforms | |
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| Source Functions | |
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| Laplace Transform Techniques | |
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| PDE Problems from Physics | |
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| Electromagnetism | |
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| Fluids and Acoustics | |
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| Scattering | |
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| Continuous Spectrum | |
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| Equations of Elementary Particles | |
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| Nonlinear PDEs | |
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| Shock Waves | |
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| Solitons | |
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| Calculus of Variations | |
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| Bifurcation Theory | |
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| Water Waves | |
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| Appendix | |
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| Continuous and Differentiable Functions | |
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| Infinite Series of Functions | |
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| Differentiation and Integration | |
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| Differential Equations | |
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| The Gamma Function | |
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| References | |
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| Answers and Hints to Selected Exercises | |
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| Index | |