Partial Differential Equations and Boundary Value Problems With Applications

Now available from Waveland Press, this corrected version of a classic text offers a level of rigor and completeness that rivals any book of its kind. Building on the basic techniques of separation-of-variables and Fourier series-integral methods, the book contains the solution of boundary-value problems for the heat equation, wave equation, and Laplace's equation in the standard coordinate systems--rectangular, cylindrical, and spherical. Each of the basic equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system rather than by the type of the equation, which makes the mathematics especially clear. Bessel and Legendre functions are developed in their own right, and their use is specifically indicated, where appropriate. The notion of steady-state solution and the closely related stationary solutions are developed for the heat equation and applied to the problem of heat flow in the earth. The problem of the vibrating string is also studied in detail, both from the Fourier viewpoint and the viewpoint of the explicit representation (d'Alembert's formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. Special features include more than 200 worked examples and 700 exercises, with nearly 450 answers; asymptotic methods (Laplace and stationary phase); and over 300 typographical cleanups from the previous publisher's versions!