Gegenbauer Cardinal Functions for the Inverse Source Parabolic Problem with a Time-Fractional Diffusion Equation

In this paper, we study a time-fractional inverse source problem. We introduce a new variable and transform inverse problem to an equivalent direct problem. By using maximum principle approach, the existence, uniqueness and stability of the inverse problem are displayed, then a numerical method is proposed to solve the problem. The main idea of the proposed method is based on expanding the approximate solution as the elements of Gegenbauer cardinal function. By using derivative and fractional derivative matrixes, the problem is reduced to the solution of a system of algebraic equations thus greatly simplifying the problem. This study concerns both theoretical and numerical aspects, where we deal with the construction and convergence analysis of the discretization schemes.