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

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

H. Zeidabadi, R. Pourgholi, A. Hosseini

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Abstract

In this paper, two numerical methods are presented to solve a nonlinear in- verse parabolic problem of determining the unknown reaction term in the scalar reaction- di usion equation. In the  rst method, the  nite element method will be used to dis- cretize the variational form of the problem and in the second method, we use the radial basis functions (RBFs) method for spatial discretization and  nite-di erence for time discretization. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such di cul- ties, we use Tikhonov regularization method. In fact, this work considers a comparative study between the  nite 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 e ectiveness of the proposed methods are illustrated by numerical examples.

Keywords

Nonlinear inverse problem, Parabolic equations, Finite element method, Ra- dial basis functions method, Least square method, Tikhonov regularization method, Stability analysis.