Generalized finite difference method for a class of multidimensional space-fractional diffusion equations
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ORIGINAL PAPER
Generalized finite difference method for a class of multidimensional space-fractional diffusion equations Hong Guang Sun1 · Zhaoyang Wang1 · Jiayi Nie1 · Yong Zhang2 · Rui Xiao3 Received: 11 March 2020 / Accepted: 22 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Fractional diffusion equations have been widely used to accurately describe anomalous solute transport in complex media. This paper proposes a local meshless method named the generalized finite difference method (GFDM), to solve a class of multidimensional space fractional diffusion equations (SFDEs) in a finite domain. In the GFDM, the spatial derivative terms are expressed as linear combinations of neighboring-node values with different weighting coefficients using the moving least-square approximation. An explicit formula for the SFDE is then obtained. The numerical solution is achieved by solving a sparse linear system. Four numerical examples are provided to verify the effectiveness of the proposed method. Numerical analysis indicates that the relative errors of prediction results are stable and less than 1% (0.001–1%). The method can also be applied for irregular grids with acceptable accuracy. Keywords Space fractional diffusion equations · Generalized finite difference method · Local meshless method · Moving least-square approximation
1 Introduction In 1695, Leibniz first introduced the d n y d x n notation to denote the fractional derivative [1]. After that, many researchers have made valuable contributions in this interesting and promising field. Nowadays, fractional calculus, especially fractional derivative has been widely used to describe complex physical systems with temporal memory or spatial nonlocality [2, 3]. Successful applications of fractional differential equation models include anomalous diffusion, non-Newtonian fluid mechanics, viscoelasticity, turbulence, hydrology and control, etc. [4, 5]. Especially, as an important research filed in both physics and engineering,
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Hong Guang Sun [email protected] Rui Xiao [email protected]
1
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, Jiangsu, China
2
Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
3
Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
the study of fractional differential equation models has been extensively investigated with fractional differential equation models, due to historical dependence and global correlation properties of anomalous diffusion [6–8]. At present, most literature focused on theoretical and numerical analysis of time fractional diffusion equation (TFDE) models which can well characterize power-law decay of anomalous diffusion [9–13]. However, numerical algorithms and application investigations on space fractional diffusion equation models (SFDE) attracted less at
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