Partial Differential Equations: An IntroductionOur 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 Laplace 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. |
Contents
Chapter 1Where PDEs Come From | 1 |
could be covered as desired A computational emphasis following | 10 |
Chapter 2Waves and Diffusions | 34 |
Chapter 3Reflections and Sources | 58 |
Chapter 4Boundary Problems | 86 |
Chapter 5Fourier Series | 104 |
Chapter 6Harmonic Functions | 152 |
Chapter 7Greens Identities and Greens Functions | 178 |
Chapter 10Boundaries in the Plane and in Space | 258 |
Chapter 11General Eigenvalue Problems | 299 |
Chapter 12Distributions and Transforms | 331 |
Chapter 13PDE Problems from Physics | 358 |
Chapter 14Nonlinear PDEs | 380 |
Appendix | 414 |
| 427 | |
33 | 443 |
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Common terms and phrases
approximation arbitrary assume Bessel boundary conditions c²uxx calculate coefficients compute Consider constant continuous function coordinates cosine cosine series curve deduce defined denote derivative differentiable diffusion equation Dirichlet problem domain dx dy eigenfunctions eigenvalues energy Example Exercise Figure Find finite formula Fourier series Fourier sine series full Fourier series function f(x Green's function harmonic function Hint infinite inhomogeneous initial condition u(x initial data integral interval kuxx Laplace Let f(x linear matrix maximum principle method Neumann odd function orthogonal piecewise continuous plane polynomial proof prove Robin boundary conditions satisfies scheme Section A.3 series converges Show solution Solve spherical surface term Theorem three dimensions three-dimensional trial functions u₁ unique vanishes variables vector wave equation write Απ ди дп дх