Partial Differential Equations and Boundary-value Problems with Applications

✍ Scribed by Mark A. Pinsky

• Publisher: American Mathematical Society
• Year: 2011
• Tongue: English
• Leaves: 534
• Series: Pure and Applied Undergraduate Texts 15
• Edition: 3rd
• Category: Library

✦ Synopsis

Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.

✦ Table of Contents

1. contents......Page 1
   2. chap 0, preliminaries......Page 13
   3a. chap 1, fourier series......Page 47
   3b. chap 1, fourier series......Page 83
   4a. chap 2, boundary value problems in rectangular coordinates......Page 110
   4b. chap 2, boundary value problems in rectangular coordinates......Page 152
   5a. chap 3, boundary value problems in cylindrical coordinates......Page 182
   5b. chap 3, boundary value problems in cylindrical coordinates......Page 220
   6a. chap 4, boundary value problems in spherical coordinates......Page 246
   6b. chap 4, boundary value problems in spherical coordinates......Page 266
   7a. chap 5, fourier transforms and applications......Page 288
   7b. chap 5, fourier transforms and applications......Page 326
   8. chap 6, asymptotic analysis......Page 355
   9a. chap 7, numerical analysis......Page 388
   9b. chap 7, numerical analysis......Page 412
   10. chap 8, green's functions......Page 436
   11. appendixes......Page 473
   12. answers to problems. index......Page 510