The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data

Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation u(t) = u(xx) + p(t)u + phi, in [0, 1] x (0, T], where it is unknown while the initial condition and boundary conditions are given. Also an additional condition integral(1)(0) k(x)u (x, t)dx = E(t), 0 <= t <= T, for known functions E(t)k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form integral(s(t))(0) u(x, t)dx = E(t), 0 < t <= T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. (c) 2007 Wiley Periodicals, Inc.