DETERMINATION OF A NONLINEAR SOURCE TERM IN A REACTION-DIFFUSION EQUATION BY USING FINITE ELEMENT METHOD AND RADIAL BASIS FUNCTIONS METHOD

In this paper, two numerical methods are presented to solve a nonlinear inverse parabolic problem of determining the unknown reaction term in the scalar reaction-diffusion equation. In the first method, the finite element method will be used to discretize the variational form of the problem and in the second method, we use the radial basis functions (RBFs) method for spatial discretization and finite-difference for time discretization. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. In fact, this work considers a comparative study between the finite element method and radial basis functions method. As we will see, these methods are very useful and convenient tools for approximation problems and they are stable with respect to small perturbation in the input data. The effectiveness of the proposed methods are illustrated by numerical examples. Keywords: Nonlinear inverse problem, Parabolic equations, Finite element method, Radial basis functions method, Least square method, Tikhonov regularization method, Stability analysis. AMS Subject Classification: 65M32, 35K05.