TABLE OF CONTENTS In section 3.19 we discussed how the solution of boundary value (BV) problems involving ordinary differential equations (ODEs) can be obtained, directly and very conveniently, by the method of finite Fourier transforms (FFTs). Methods based on the Fourier transforms are developed in the present chapter to solve linear partial differential equations (PDEs). These methods are quite powerful and can be used for problems involving two or more independent variables and finite, semi-infinite, or infinite domains. Also, these methods are of general utility and can tackle second-order PDEs of all types-that is, hyperbolic, parabolic, and elliptic (see chapter 5 for this classification). As in the case of BV problems involving ODEs, the only restriction in applying the generalized Fourier transform methods is that the differential operator and the associated boundary conditions (BCs) in each spatial dimension should be self-adjoint. The books by Sneddon (1951) and Churchill (1972) are excellent sources for additional details about these methods.
A class of problems which can be solved by the method of FFT is that involving time and one space variable. An example is the parabolic heat conduction equation. In this context, let us consider the problem of transient cooling of a one-dimensional rod with internal heat generation. The corresponding steady-state problem was treated previously in section 3.19. The relevant PDE is
with BCs
and IC
where U 2 = hp/ kA, ? = k/ C p ?
