The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. The proposed scheme combines the local meshless method based on a radial basis function (RBF) with Laplace transform. This scheme first implements the Laplace transform to reduce the given problem to a time-independent inhomogeneous problem in the Laplace domain, and then the RBF-based local meshless method is utilized to obtain the solution of the reduced problem in the Laplace domain. Finally, Stehfest’s method is utilized to convert the solution from the Laplace domain into the real domain. The proposed method uses Laplace transform to handle the fractional order derivative, which avoids the computation of a convolution integral in a fractional order derivative and overcomes the effect of time-stepping on stability and accuracy. The method is tested using four numerical examples. All the results demonstrate that the proposed method is easy to implement, accurate, efficient and has low computational costs.

In this article, we propose a localized transform based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo’s sense with a weakly singular kernel. The purpose of coupling the localized meshless method with the Laplace transform is to avoid the time stepping procedure by eliminating the time variable. Then, we utilize the local meshless method for spatial discretization. The solution of the original problem is obtained as a contour integral in the complex plane. In the literature, numerous contours are available; in our work, we will use the recently introduced improved Talbot contour. We approximate the contour integral using the midpoint rule. The bounds of stability for the differentiation matrix of the scheme are derived, and the convergence is discussed. The accuracy, efficiency, and stability of the scheme are validated by numerical experiments.