Numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm

This paper deals with an inverse problem of identifying a nonlinear source term g = g(u) in the heat equation u, - uxx = a(x)g(u). By data compatibility analysis, the forward problem is proved to have a unique positive solution with a maximum of M > 0, with which an optimal perturbation algorithm is applied to determine the source function g(u) on u ∈ [0, M]. Numerical inversions are carried out for g(u) with functional forms of polynomial, trigonometric and index functions. The inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here. By the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources. The inversion algorithm seems to be very sharp if the solution's maximum M of the forward problem is relatively small; otherwise, the deviations in the source solutions become large especially near the endpoint of u = M.