Numerical solution of the problem of finding two unknowns in time-fractional diffusion equations

We study an inverse problem for a time-fractional diffusion equation with initial-boundary and overdetermination conditions.This inverse problem aims to determine a time varying coefficient and source in the equation with overdetermination integral conditions. First, we establish the unique existence of the classical solution using the Fourier method, Gronwall inequality for direct problem. Second, by using the fixed point theorem in Banach space, the local existence and uniqueness of this inverse problem are obtained. To verify the theoretical results, a numerical solution to the problem was constructed using the finite difference method. Finally, a numerical example is presented to show the effectiveness of the proposed method.

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