A meshless method for solving an inverse spacewise-dependent heat source problem

Keywords: Meshless method Method of fundamental solutions Heat source Ill-posed problem a b s t r a c t In this paper an effective meshless and integration-free numerical scheme for solving an inverse spacewise-dependent heat source problem is proposed. Due to the use of the fundamental solution as basis functions, the method leads to a global approximation scheme in both spatial and time domains. The standard Tikhonov regularization technique with the generalized cross-validation criterion for choosing the regularization parameter is adopted for solving the resulting ill-conditioned system of linear algebraic equations. The effectiveness of the algorithm is illustrated by several numerical examples. In the process of transportation, diffusion and conduction of natural materials, the following heat equation is a suitable approximation: where u represents the state variable, X is a bounded domain in R d , and the right hand side f denotes physical laws, in our case source terms. Unfortunately, the characteristics of sources in actual problems are always unknown. These are inverse problems, and it is well-known that they are generally ill-posed, i.e. the existence, uniqueness and stability of their solutions are not always guaranteed [1]. In general, a complete recovery of the unknown source is not attainable from practically restricted boundary measurements. If no a priori information is available on the functional form of the unknown variable, the solution of the estimation problem becomes difficult. Inverse problems are unstable in nature because the unknown solutions have to be determined from indirect observable data which contain measurement errors. The major difficulty in establishing any numerical algorithm for approximating the solution is the ill-posedness of the problem and the ill-conditioning of the resulting discretized matrix. For instance, uniqueness and conditional stability results can be found in [2,3]. A number of techniques have been proposed for solving the inverse source problem, including the boundary element method (BEM) [4], iterative regularization methods [5–7] and mollification methods [8,9]. Besides, a sequential method [10] and linear least-squares error method [11] have also been used in solving the inverse source problem. In all of these methods, the partial differential equation must be discretized. The traditional mesh-dependent finite difference method (FDM) and finite element method (FEM) require a mesh on the domain to support the solution process, and hence tedious computational time. The BEM reduces the dimensionality of the problem by one, thus reduces the computational time. However, the main drawback 0021-9991/$-see front matter Ó …