An efficient approximate solution of Riesz fractional advection-diffusion equation

Document Type : Research Paper


1 Department of Mathematics, College of Science, Yadegar-e-Imam Khomeini (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran.

2 Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.


The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It’s of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [−1, 1] × [−1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.


  • [1]          Y. E. Aghdam, H. Safdari, Y. Azari, H. Jafari, and D. Baleanu, Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact  finite  difference  scheme,  Discrete Continuous Dynamical Systems-S, (2018), 1-15.
  • [2]          A. Baseri , E. Babolian, and S. Abbasbandy, Normalized Bernstein polynomials in solving space-time fractional diffusion equation, Advances in Difference Equations, 346 (2017), 1-25.
  • [3]          Y. Chen, Y. Sun, and L. Liu, Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional-order Legendre functions. Applied Mathematics and Computation, 244 (2014), 847- 858.
  • [4]          L. Feng, P. Zhuang, F. Liu, I. Turner, and Q. Yang, Second-order approximation for the space fractional diffusion equation with variable coefficient, Progr. Fract. Differ. Appl, 1(1)(2015), 23-35.
  • [5]          H. Jafari, and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Applied Mathematics and Computation, 180(2) (2006), 488-497.
  • [6]          H. Jafari, M. Nazari, D. Baleanu, and C. M. Khalique, A new approach for solving a system of fractional partial differential equations, Computers & Mathematics with Applications, 66(5) (2013), 838-843.
  • [7]          H. Jafari, and S. Seifi, Solving a system of nonlinear fractional partial differential equations  using  homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14(5) (2009), 1962-1969.
  • [8]          A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, North-Holland Mathematics Studies , Theory and applications of fractional differential equations, (204) (2006).
  • [9]          J. C. Mason and C. David Handscomb, Chebyshev polynomials, CRC Press, 2002.
  • [10]        H. Mesgarani, A. Beiranvand, and Y. E. Aghdam, The impact of the Chebyshev collocation method on solutions   of the time-fractional Black–Scholes, Mathematical Sciences, (2020), 1-7.
  • [11]        S. Mockary, E. Babolian, and A. R. Vahidi, A fast numerical method for fractional partial differential equations, Advances in Difference Equations, 452 (2019), 1–20.
  • [12]        Z. Odibat, and S. Momani,The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers & Mathematics with Applications, 58(11) (2009), 2199-2208.
  • [13]        M. Parvizi, M. R. Eslahchi, and M. Dehghan, Numerical solution of fractional advection-diffusion equation with    a nonlinear source term, Numerical Algorithms, 68(3) (2015), 601-629.
  • [14]        I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Academic press, 1998.
  • [15]        M. Pourbabaee and A. Saadatmandi, Cllocation method based o chebyshev polynomials for solving  distributed  order fractional differential equations, Computational Methods for Differential Equations, (2020).
  • [16]        H. Safdari, Y. E. Aghdam, and J. F. Gómez-Aguilar, Shifted Chebyshev collocation of the fourth kind with convergence analysis for the space–time fractional advection-diffusion equation, Engineering with Computers, (2020), 1-12.
  • [17]        H. Safdari, H. Mesgarani, M. Javidi, and Y. E. Aghdam, Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme, Computational and Applied Mathematics, 39(2) (2020),1-15.
  • [18]        A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos: An Interdis- ciplinary Journal of Nonlinear Science, 7(4) (1997), 753-764.
  • [19]        S. Shen, F. Liu, V. Anh, and I. Turner, The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation, IMA Journal of Applied Mathematics, 73(6) (2008), 850-872.
  • [20]        B. Shiri, I. Perfilieva, and Z. Alijani, Classical approximation for fuzzy Fredholm integral equation, Fuzzy Sets and Systems, 404 (2021),159-177.
  • [21]        N. H. Sweilam, A. M. Nagy, and A. A. El-Sayed, Sinc-Chebyshev collocation method for time-fractional order telegraph equation, Appl. Comput. Math, 19(2) (2020), 162-174.
  • [22]        L. Wang, Y. Ma, and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 227 (2014), 66-76.
  • [23]        J. L. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 214(1) (2009), 31-40.
  • [24]        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.
  • [25]        G. M. Zaslavsky, Chaos: fractional kinetics, and anomalous transport, Physics Reports, 371(6) (2002), 461-580.
Volume 10, Issue 2
April 2022
Pages 307-319
  • Receive Date: 25 September 2020
  • Revise Date: 04 April 2021
  • Accept Date: 11 April 2021
  • First Publish Date: 11 April 2021