A numerical algorithm for determining time-dependent coefficient in a parabolic inverse problem using Legendre multiwavelet base (in persian)

Abstract

In this paper, we introduce Legendre multiwavelet functions and use them as a set of base functions to approximate the solution of a parabolic differential equation with an unknown time-dependent coefficient in an inverse problem. Using the expansion formula of a known function in terms of the Legendre multiwavelet base, we define integral and product operational matrices from a general point of view. With the help of these matrices, we transform the problem into a system of algebraic equations. By solving the obtained system of algebraic equations using the existing optimization algorithms, we provide an approximation for the solution of the problem in the form of its expansion in terms of the Legendre multiwavelet base. In addition to expressing the algorithm of the proposed numerical method, we perform the proposed method on two examples and report its numerical results. We also compare the results of the proposed method with the results reported from other methods.