A Meshless Method to Determine a Source Term in Heat Equation with Radial Basis Functions

This paper considers a numerical method based on the radial basis functions for the inverse problem of heat equation; the inverse problem is determining an unknown source term subject to the overdetermination along with the usual initial boundary conditions, and the unknown source term is only time-dependent. The radial basis functions method is a meshless method with high accuracy for the inverse problem. Some numerical experiments using this method are presented and discussed. 1. Introduction The inverse problem of parabolic equations appears naturally in a wide variety of physical and engineering settings, such as elasticity, hydrology, material sciences, heat transfer, medical imaging, transport problems, and control theory. In this paper, we consider an inverse problem for determining an unknown source term in a heat equation. There are many numerical methods which have been developed to solve the inverse problem of parabolic equations. One approach is the method of output least squares, which assumes that the unknown source function is a specific functional form depending on some parameters and then seeks to determine optimal parameter values which minimized an error function based on the overdetermined data. Other effective and broadly applicable techniques such as finite difference method, finite elements method, boundary elements method, and finite volume method are widely used in inverse problems; however, the finite elements method approximates the solution by using low-order piecewise polynomials, and in the finite difference method, the derivatives of the solution are approximated by difference quotients. These methods depend on a suitable generation of meshes, which is difficult for problems with very complicated and irregular geometries. The method we used in this paper is the meshless method based on the radial basis functions. Meshless methods are a class of numerical methods for solving partial differential equations. In these methods, mesh generation on the spatial domain of the problem is not needed, and this property is the main advantage of these techniques over the mesh-dependent methods such as finite difference methods and finite element methods. In addition, the radial basis functions method is a technique for interpolation on scattered data; this method uses the distributed nodal points to approximate the unknown function; the distribution of nodes could be selected regularly or randomly in the analyzed domain, and there is no demand for the geometry of the domain. The meshless method based on the radial basis functions