Reconstruction of a right-hand side of parabolic equation by radial basis functions method

The inverse problem of reconstructing the right-hand side (RHS) of a parabolic equation using the radial basis functions (RBF) method from a solution specified at internal points is investigated. In this paper, the RHS is unknown about time, and the method we use is the meshless method. Some numerical experiments are presented to illustrate the accuracy, stability and effectiveness.

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 reactiondiffusion 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.Publisher's Versio