Recovering the time dependent coefficient in a multidimensional multi-term fractional diffusion equation

Abstract

We consider the inverse problem of determining the time-dependent coefficient for a diffusion equation involving a fractional Laplacian operator in space and Hilfer multi-term fractional derivatives in time with Dirichlet zero boundary conditions. The solution of the initial-boundary value problem was investigated using the Fourier method. First, we will study the spectral problem related to the fractional Laplacian in a spatial variable. In addition, the existence of a solution to the Cauchy problem for a fractional time differential equation is considered. Using the generalized Gronwall inequality, we obtain an estimate of the solution in terms of the unknown coefficient. The inverse problem is to recover a time-dependent coefficient with an integral type over-determination condition. We consider the analytical solution of the inverse problem and prove the existence and uniqueness of the analytical solution. The inverse problem reduces to an equivalent Volterra-type integral equation. The local existence and global uniqueness of the solution of the inverse problem are proven according to the Banach fixed-point theorem.