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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 | |