A local meshless procedure to determine the unknown control parameter in the multi-dimensional inverse problems

Abstract

This article is devoted to applying a local meshless method for specifying an unknown control parameter in one- and multi-dimensional inverse problems which are considered with a temperature overspecification condition at a specific point or an energy overspecification condition over the computational domain. Finding the unknowns in inverse problems is a challenge because these problems are modeled as non-classical parabolic problems and also have a significant role in describing physical phenomena of the real world. In this study, a combination of the meshless method of radial basis functions and finite difference method (called radial basis function-finite difference method) is used to solve inverse problems because this method has two important features. First it does not require any mesh generation. Consequently, it can be exerted to handle the high-dimensional inverse problems. Secondly, since this method is local, at each time step, a system with a sparse coefficient matrix is solved. Hence, the computational time and cost will be much low. Various numerical examples are examined, and also the accuracy and computational time required are presented. The numerical results indicate that the mentioned procedure is appropriate for the identification of the unknown parameter of inverse problems.