A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

Abstract

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.