Citation: Kamran; Irfan, M.; Alotaibi, F.M.; Haque, S.; Mlaiki, N.; Shah, K. RBF-Based Local Meshless Method for Fractional Diffusion Equations. Fractal Fract. 2023, 7, 143. https:// doi.org/10.3390/fractalfract7020143 Academic Editors: Jordan Hristov and Denis N. Gerasimov Received: 3 January 2023 Revised: 21 January 2023 Accepted: 31 January 2023 Published: 2 February 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article RBF-Based Local Meshless Method for Fractional Diffusion Equations Kamran 1, *, Muhammad Irfan 1 , Fahad M. Alotaibi 2 , Salma Haque 3, *, Nabil Mlaiki 3, * and Kamal Shah 3,4 1 Department of Mathematics, Islamia College Peshawar, Jamrod Road, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan 2 Department of Information Systems, Faculty of Computing and Information Technology (FCIT), King Abdulaziz University, Jeddah 34025, Saudi Arabia 3 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia 4 Department of Mathematics, University of Malakand, Chakdara Dir(L) 18000, Khyber Pakhtunkhwa, Pakistan * Correspondence: kamran.maths@icp.edu.pk (K.); shaque@psu.edu.sa (S.H.); nmlaiki@psu.edu.sa or nmlaiki2012@gmail.com (N.M.) Abstract: 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. Keywords: RBF-based local meshless method; Laplace transform; Stehfest’s method; Liouville–Caputo fractional derivative; diffusion equations 1. Introduction Fractional calculus (FC) has recently received much attention among the research community. Applications of FC can be found in a large number of phenomena, such as traffic models, stochastic systems, diffusion processes, earthquake design, and control processing, etc. [1–3]. The time-fractional order diffusion equations (TFDEs) have been shown to be very efficient at describing diffusion in complex systems [4,5]. The diffusion processes have been observed in various areas, for example, seepage in porous media [6], electron transportation [7], magmatic plasma [8], dissipation [9], and turbulence [10], etc. Recently, TFDEs have attracted remarkable attention [11–17]. However, the analytic solution of TFDEs is rare, except for simple initial-boundary data [18]. Therefore, the numerical methods play a vital role in obtaining the solution of TFDEs. Numerous nu- merical techniques have been designed for the approximation of TFDEs. For example, Bayrak et al. [19] developed an efficient Chebyshev collocation technique for the simulations of TFDEs. In [20], the splines method was utilized for approximating the solution of TFDEs. Li et al. [21] utilized the Galerkin finite element method to study the solutions of TFDEs. In [22], the authors approximated the solution of variable order TFDEs. Bouchama et al. [23] proposed a finite difference method (FDM) for TFDEs. In [24], the others approximated Fractal Fract. 2023, 7, 143. https://doi.org/10.3390/fractalfract7020143 https://www.mdpi.com/journal/fractalfract