The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse source problem of time-fractional diffusion equation in two dimensions. The missing solely time-dependent source is recovered from an additional integral measurement. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of SMRPI. Firstly, we use a difference scheme for the fractional derivative to discretize the governing equation, and we obtain a finite difference scheme with respect to time. Then we use the SMRPI approach to approximate the spatial derivatives. Also, it is proved that the scheme is unconditionally stable with respect to time in space H1. Consequently, when the input data is contaminated with noise, we use the Tikhonov regularization method in order to obtain a stable solution. Numerical results show that the solution is accurate for exact data and stable for noisy data.