An Inverse Problem for A Time-Fractional Heat Equation: Determination of A Time-Dependent Coefficient Via The Laplace Homotopy Analysis Method

This study addresses an inverse problem for a time-fractional heat equation involving the Caputo derivative, where the objective is to determine an unknown time-dependent thermal coefficient. The governing equation incorporates a fractional order time derivative, a spatially ‎dependent diffusion term, and a known source term. To solve the inverse problem, we employ the Laplace Homotopy Analysis Method ‎‎(LHAM)—a semi-analytical approach that combines the Laplace transform with the Homotopy Analysis Method. This technique allows for ‎the construction of a convergent series solution for both the temperature distribution ‎ ‎ and the unknown coefficient ‎ ‎. By appropriately selecting the initial guess ‎ ‎, the convergence-control parameter h becomes embedded in the solution process, reducing com-‎computational effort. The proposed method is validated through an illustrative example, demonstrating the effectiveness and accuracy of LHAM ‎in solving time-fractional inverse problems with a source term‎.