RBF-Based Local Meshless Method for Fractional Diffusion Equations

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.

Automated summary

This article presents a method for numerical simulation of time-fractional diffusion equations, which combines the local meshless method based on a radial basis function (RBF) with Laplace transform. The proposed method first transforms the given problem into a time-independent inhomogeneous problem in the Laplace domain using Laplace transform, and then utilizes RBF-based local meshless method to solve the reduced problem in the Laplace domain. Finally, Stehfest's method is used to convert the solution back to the real domain. The method handles the fractional order derivative using Laplace transform, avoiding the computation of convolution integrals and overcoming the effects of time-stepping on stability and accuracy. The method is tested using four numerical examples, demonstrating its ease of implementation, accuracy, efficiency, and low computational costs.