Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs, the wave, heat and Lapace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
Chapter 1: Where PDEs Come From 1.1 What is a Partial Differential Equation? 1.2 First-Order Linear Equations 1.3 Flows, Vibrations, and Diffusions 1.4 Initial and Boundary Conditions 1.5 Well-Posed Problems 1.6 Types of Second-Order Equations Chapter 2: Waves and Diffusions 2.1 The Wave Equation 2.2 Causality and Energy 2.3 The Diffusion Equation 2.4 Diffusion on the Whole Line 2.5 Comparison of Waves and Diffusions Chapter 3: Reflections and Sources 3.1 Diffusion on the Half-Line 3.2 Reflections of Waves 3.3 Diffusion with a Source 3.4 Waves with a Source 3.5 Diffusion Revisited Chapter 4: Boundary Problems 4.1 Separation of Variables, The Dirichlet Condition 4.2 The Neumann Condition 4.3 The Robin Condition Chapter 5: Fourier Series 5.1 The Coefficients 5.2 Even, Odd, Periodic, and Complex Functions 5.3 Orthogonality and the General Fourier Series 5.4 Completeness 5.5 Completeness and the Gibbs Phenomenon 5.6 Inhomogeneous Boundary Conditions Chapter 6: Harmonic Functions 6.1 Laplace's Equation 6.2 Rectangles and Cubes 6.3 Poisson's Formula 6.4 Circles, Wedges, and Annuli Chapter 7: Green's Identities and Green's Functions 7.1 Green's First Identity 7.2 Green's Second Identity 7.3 Green's Functions 7.4 Half-Space and Sphere Chapter 8: Computation of Solutions 8.1 Opportunities and Dangers 8.2 Approximations of Diffusions 8.3 Approximations of Waves 8.4 Approximations of Laplace's Equation 8.5 Finite Element Method Chapter 9: Waves in Space 9.1 Energy and Causality 9.2 The Wave Equation in Space-Time 9.3 Rays, Singularities, and Sources 9.4 The Diffusion and Schrodinger Equations 9.5 The Hydrogen Atom Chapter 10: Boundaries in the Plane and in Space 10.1 Fourier's Method, Revisited 10.2 Vibrations of a Drumhead 10.3 Solid Vibrations in a Ball 10.4 Nodes 10.5 Bessel Functions 10.6 Legendre Functions 10.7 Angular Momentum in Quantum Mechanics Chapter 11: General Eigenvalue Problems 11.1 The Eigenvalues Are Minima of the Potential Energy 11.2 Computation of Eigenvalues 11.3 Completeness 11.4 Symmetric Differential Operators 11.5 Completeness and Separation of Variables 11.6 Asymptotics of the Eigenvalues Chapter 12: Distributions and Transforms 12.1 Distributions 12.2 Green's Functions, Revisited 12.3 Fourier Transforms 12.4 Source Functions 12.5 Laplace Transform Techniques Chapter 13: PDE Problems for Physics 13.1 Electromagnetism 13.2 Fluids and Acoustics 13.3 Scattering 13.4 Continuous Spectrum 13.5 Equations of Elementary Particles Chapter 14: Nonlinear PDEs 14.1 Shock Waves 14.2 Solitions 14.3 Calculus of Variations 14.4 Bifurcation Theory 14.5 Water Waves Appendix A.1 Continuous and Differentiable Functions A.2 Infinite Sets of Functions A.3 Differentiation and Integration A.4 Differential Equations A.5 The Gamma Function References Answers and Hints to Selected Exercises Index