In this study, one explicit and one implicit finite difference scheme is introduced for the numerical solution of time-fractional Riesz space diffusion equation. The time derivative is approximated by the standard Gr¨unwald Letnikov formula of order one, while the Riesz space derivative is discretized by Fourier transform-based algorithm of order four. The stability and convergence of the proposed methods are studied. It is proved that the implicit scheme is unconditionally stable, while the explicit scheme is stable conditionally. Some examples are solved to illustrate the efficiency and accuracy of the proposed methods. Numerical results confirm that the accuracy of present schemes is of order one.
 J. Chen, F. Liu, I. Turner, and V. Anh, The fundamental and numerical solutions of the Riesz space-fractional reaction dispersion equation, The ANZIAM Journal, 50(1) (2008), 45-57.
 C. M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886-897.
 M. Dehghan, M. Abbaszadeh, and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klien-Gordon equations, Engineering Analysis with Boundary Elements, 50, 2015, 412-434.
 H. Ding, Ch. Li, and Y. Chen, High-Order Algorithms for Riesz Derivative and Their Applications (I), Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 653797, 17 pages http://dx.doi.org/10.1155/2014/653797
 C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comput., 58(198) (1992), 603-630.
 E. Hanert, On the numerical solution of space-time fractional diffusion models, Computers & Fluids, 46 (2011), 33-39.
 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of NorthHolland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
 C. Li, A. Chen, and J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, Journal of Computational Physics, 230(9) (2011), 3352-3368.
 L. Qiu, C. Hu, and Q. Qin, A novel homogenization function method for inverse source problem of nonlinear time-fractional wave equation, Applied Mathematics Letters, 109, 2020, 106554.
 J. Lin, W. Feng, S. Reutskiy, H. Xu, and Y. He, A new semi-analytical method for solving a class of time fractional partial differential equations with variable coefficients, Applied Mathematics Letters, 112, 2021, 106712.
 J. Lin, Y. Zhang, S. Reutskiy, and W. Feng, A novel meshless space-time backward substitution method and its application to nonhomogeneous advection-diffusion problems, Applied Mathematics and Computation, 398, 2021, 125964.
 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons Inc. New York 1993.
 K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, (1974).
 C. Piret and E. Hanert, A radial basis functions method for fractional diffusion equations, Journal of Computa- tional Physics, 238 (2013), 71-81.
 I. Podlubny, Fractional Differential Equations, Academic Press, SanDiego (1999).
 S. Shen, F. Liu, and V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numerical Algorithms, 56(3) (2011), 383-403.
 S. Shen, F. Liu, V. Anh, and I. Turner, A novel numerical approximation for the space fractional advection- dispersion equation, IMA Journal of Applied Mathematics, 79(3) (2014), 431444.
 Q. Yang, F. Liu, and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling, 34(1) (2010), 200-218.
 H. Zhang and F. Liu, The fundamental solutions of the space-time Riesz fractional partial differential equations with periodic conditions, Numerical Mathematics: A Journal of Chinese Universities, English Series, 16(2) (2007), 181-192.
Abdollahy, Z., Mahmoudi, Y., Salimi shamloo, A., Baghmisheh, M. (2022). Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation. Computational Methods for Differential Equations, 10(3), 799-815. doi: 10.22034/cmde.2021.45950.1927
Zeynab Abdollahy; Yaghoub Mahmoudi; Ali Salimi shamloo; Mahdi Baghmisheh. "Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation". Computational Methods for Differential Equations, 10, 3, 2022, 799-815. doi: 10.22034/cmde.2021.45950.1927
Abdollahy, Z., Mahmoudi, Y., Salimi shamloo, A., Baghmisheh, M. (2022). 'Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation', Computational Methods for Differential Equations, 10(3), pp. 799-815. doi: 10.22034/cmde.2021.45950.1927
Abdollahy, Z., Mahmoudi, Y., Salimi shamloo, A., Baghmisheh, M. Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation. Computational Methods for Differential Equations, 2022; 10(3): 799-815. doi: 10.22034/cmde.2021.45950.1927