Partial Differential Equations

Dr. Walter A. Strauss is a professor of mathematics at Brown University. He has published numerous journal articles and papers. Not only is he is a member of the Division of Applied Mathematics and the Lefschetz Center for Dynamical Systems, but he is currently serving as the Editor in Chief of the SIAM Journal on Mathematical Analysis. Dr. Strauss' research interests include Partial Differential Equations, Mathematical Physics, Stability Theory, Solitary Waves, Kinetic Theory of Plasmas, Scattering Theory, Water Waves, Dispersive Waves.

• Chapter 1/Where PDEs Come From1.1* What is a Partial Differential Equation? 11.2* First-Order Linear Equations 61.3* Flows, Vibrations, and Diffusions 101.4* Initial and Boundary Conditions 201.5 Well-Posed Problems 251.6 Types of Second-Order Equations 28Chapter 2/Waves and Diffusions2.1* The Wave Equation 332.2* Causality and Energy 392.3* The Diffusion Equation 422.4* Diffusion on the Whole Line 462.5* Comparison of Waves and Diffusions 54Chapter 3/Reflections and Sources3.1 Diffusion on the Half-Line 573.2 Reflections of Waves 613.3 Diffusion with a Source 673.4 Waves with a Source 713.5 Diffusion Revisited 80Chapter 4/Boundary Problems4.1* Separation of Variables, The Dirichlet Condition 844.2* The Neumann Condition 894.3* The Robin Condition 92Chapter 5/Fourier Series5.1* The Coefficients 1045.2* Even, Odd, Periodic, and Complex Functions 1135.3* Orthogonality and General Fourier Series 1185.4* Completeness 1245.5 Completeness and the Gibbs Phenomenon 1365.6 Inhomogeneous Boundary Conditions 147Chapter 6/Harmonic Functions6.1* Laplace’s Equation 1526.2* Rectangles and Cubes 1616.3* Poisson’s Formula 1656.4 Circles, Wedges, and Annuli 172Chapter 7/Green’s Identities and Green’s Functions7.1 Green’s First Identity 1787.2 Green’s Second Identity 1857.3 Green’s Functions 1887.4 Half-Space and Sphere 191Chapter 8/Computation of Solutions8.1 Opportunities and Dangers 1998.2 Approximations of Diffusions 2038.3 Approximations of Waves 2118.4 Approximations of Laplace’s Equation 2188.5 Finite Element Method 222Chapter 9/Waves in Space9.1 Energy and Causality 2289.2 The Wave Equation in Space-Time 2349.3 Rays, Singularities, and Sources 2429.4 The Diffusion and Schrodinger Equations 248 ¨9.5 The Hydrogen Atom 254Chapter 10/Boundaries in the Plane and in Space10.1 Fourier’s Method, Revisited 25810.2 Vibrations of a Drumhead 26410.3 Solid Vibrations in a Ball 27010.4 Nodes 27810.5 Bessel Functions 28210.6 Legendre Functions 28910.7 Angular Momentum in Quantum Mechanics 294Chapter 11/General Eigenvalue Problems11.1 The Eigenvalues Are Minima of the Potential Energy 29911.2 Computation of Eigenvalues 30411.3 Completeness 31011.4 Symmetric Differential Operators 31411.5 Completeness and Separation of Variables 31811.6 Asymptotics of the Eigenvalues 322Chapter 12/Distributions and Transforms12.1 Distributions 33112.2 Green’s Functions, Revisited 33812.3 Fourier Transforms 34312.4 Source Functions 34912.5 Laplace Transform Techniques 353Chapter 13/PDE Problems from Physics13.1 Electromagnetism 35813.2 Fluids and Acoustics 36113.3 Scattering 36613.4 Continuous Spectrum 37013.5 Equations of Elementary Particles 373Chapter 14/Nonlinear PDEs14.1 Shock Waves 38014.2 Solitons 39014.3 Calculus of Variations 39714.4 Bifurcation Theory 40114.5 Water Waves 406AppendixA.1 Continuous and Differentiable Functions 414A.2 Infinite Series of Functions 418A.3 Differentiation and Integration 420A.4 Differential Equations 423A.5 The Gamma Function 425References 427Answers and Hints to Selected Exercises 431Index 446