A Numerical Method for Solving of a Nonlinear Inverse Diffusion Problem

In this paper, we will first study the existence and uniqueness of the solution for a one-dimensional nonlinear inverse diffusion problem via an auxiliary problem and the Schauder fixed-point theorem. Furthermore, a numerical algorithm based on the finite-difference method and the least-squares scheme for solving a nonlinear inverse problem is proposed. At the beginning of the numerical algorithm, Taylor's series expansion is employed to linearize nonlinear terms and then the finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate the unknown diffusion coefficient. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study, and thus, it is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the diffusion coefficient can be obtained within a couple of minutes of CPU time on a Pentium IV-2.4 GHz PC.