Progr. Fract. Differ. Appl. 6, No. 1, 55-66 (2020) 55 Progress in Fractional Differentiation and Applications An International Journal http://dx.doi.org/10.18576/pfda/060106 Using Finite Volume-Element Method for Solving Space Fractional Advection-Dispersion Equation Allahbakhsh Yazdani 1,∗ , Navid Mojahed 1 , Afshin Babaei 1 and Elena Vazquez Cendon 2 1 Faculty of Mathematical Sciences,University of Mazandaran,Babolsar, Iran 2 Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain Received: 2 Sep. 2018, Revised: 27 May. 2019, Accepted: 27 Jul. 2019 Published online: 1 Jan. 2020 Abstract: In this paper, the numerical solution for space fractional advection-dispersion problem in one-dimension is proposed by B-spline finite volume element method. The fractional derivative is Grunwald-Letnikov in the proposed scheme. The stability and convergence of the proposed numerical method are studied, and the numerical results support the exact results. Keywords: Finite volume-element method, Advection-dispersion equation, Grunwald-Letnikov derivative, space fractional, fractional calculus. 1 Introduction Fractional calculus (FC) has been applied in different fields of engineering and science, including electro-magnetics, visco-elasticity, optics, electro-chemistry, fluid mechanics, and signals processing [1–9]. This method has been used in modeling contaminant flow as well [2, 3, 10–13]. Moreover, a wide range of physical phenomena can be modelled by FC to be described more precisely. Furthermore, the fractional derivative-based models are perfect in analysing the damping systems. Because of the wide range of FC applications, most of the analytical and numerical methods, which are recently proposed, are inapplicable [14–23]. Many people have recently worked on solving fractional partial differential equations (FPDEs). Some have addressed the analytical solution of FPDE [2, 24–32], others have explored numerical solutions [24, 33–79]. To solve the differential equations, the three following approaches are adopted: Finite Difference Methods (FDM), Finite Volume Methods (FVM), and Finite Element Methods (FEM) [80, 81]. Finite element volume methods (FEVM) are linked to finite element methods. Precisely, FVEMs are the Petrov- Galerkin form of FEMs, which are developed using two types of partitions; a primal partition and its dual, on a domain Ω . The primal mesh approximates the exact solution, and the equations are discretized over the control volumes by its dual. The two main advantages of FVEM are the accuracy of the method, dependent only on the degree of the approximation polynomial, and flexibility of the control volumes. It is advantageous to handle complicated domains. Badr et al. [82] investigated FVEM for solving a time-fractional advection-diffusion equation and proved the stability of this method. Transport activity enclosed by complex and non-homogenous conditions sometimes leads to non-classical diffusion that is not completely matched by Fick’s law or pedesis theory [2, 3, 10–13, 83]. Fractional calculus helps overcome such a challenge. If a random walk model takes place as continuous time, it results in a fractional advection-dispersion equation [6]. Space advection-dispersion equation is obtained by putting the fractional derivative term in classical diffusion equation [12]. In the present paper, we will work on space fractional homogenous advection-dispersion equation. The present paper is outlined as follows. In section two, we apply the FVEM to approximate the numerical solution of the initial value fractional advection-dispersion equation. Stability and convergence of this method are discussed in section three. Some numerical results are illustrated in section four. Section five is devoted to some conclusion. ∗ Corresponding author e-mail: yazdani@umz.ac.ir c  2020 NSP Natural Sciences Publishing Cor.