The numerical solution for the time-fractional inverse problem of diffusion equation

In this study, the spectral meshless radial point interpolation (SMRPI) technique is applied to the Cauchy problem of two-dimensional fractional diffusion equation. We obtain the unknown data on the inner boundary when overspecified boundary data is imposed on the outer boundary. The SMRPI is based on a combination of meshfree methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Here, similar to other meshless methods, localization in SMRPI can reduce the ill-posedness of the Cauchy problem. However, it does not require to use regularization algorithms and therefore reduces computational time. Two numerical examples, are tested to show that the SMRPI can overcome the ill-posedness of the Cauchy problem and has acceptable accuracy. Also, by adding some large perturbations, the proposed method is still stable.