Efficient Kansa-type MFS algorithm for time-fractional inverse diffusion problems

In this study we propose an efficient Kansa-type method of fundamental solutions (MFS-K) for the numerical solution of time-fractional inverse diffusion problems. By approximating the time-fractional derivative through a finite difference scheme, the time-fractional inverse diffusion problem is transformed into a sequence of Cauchy problems associated with inhomogeneous elliptic-type equations, which can be conveniently solved using the MFS-K. Since the matrix arising from the MFS-K discretization is highly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization method, while the choice of the regularization parameter is based on the generalized cross-validation criterion. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The stability of the method with respect to the noise in the data is investigated.