Fourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term

Document Type : Research Paper


Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran.


‎This paper aims to propose a high-order and accurate numerical scheme for the solution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we first discretize the Riesz fractional derivative with a fourth-order finite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those provided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an efficient algorithm for solving such problems and can arrive at the high-precision.


[1] M. Abbaszadeh, Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation, Appl. Math. Let., 88 (2019), 179-185.
[2] P. Amodio, F. Mazzia, and D. Trigiante, Stability of some boundary value methods for the solution of initial value problems, BIT 33 (1993), 434-451.
[3] M. R. Azizi and A. Khani, Sinc operational matrix method for solving the Bagley-Torvik equation, Comput. Methods Differ. Eq. 5 (2017), 56-66.
[4] L. Brugnano and D. Trigiante, Solving differential problems by multistep initial and boundary value methods, Gordon and Beach Science Publishers, Amsterdam, 1998.
[5] L. Brugnano and D. Trigiante, Stability properties of some BVM methods, Appl. Numer. Math. 13 (1993) 201-304.
[6] L. Brugnano and D. Trigiante,Boundary value methods: the third way between linear multistep and Runge-Kutta methods, Computers Math. Applic. 36 (1998), 269-284.
[7] C. Celik and M. Duman,Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phy. 231 (2012), 1743-1750.
[8] J. Chen, F. Liu, I. Turner, and V. Anh, The fundamental and numerical solutions of the Riesz space-fractional reaction dispersion equation, ANZIAM J.,50 (2008), 45-57.
[9] M. Dehghan and A. Mohebbi,High-order compact boundary value method for the solution of unsteady convection diffusion problems, Math. Comput. Simul. 79 (2008), 683-699.
[10] M. Dehghan and A. Mohebbi, The use of compact boundary value method for the solution of two-dimensional Schrodinger equation, J. Comput. Appli. Math.225 (2009), 124-134.
[11] M. Dehghan and M. Abbaszadeh, An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch-Torrey equations, Appl. Numer. Math.,131 (2018) 190-206.
[12] M. Dehghan, M. Abbaszadeh, and W. Deng, Fourth-order numerical method for the space-time tempered fractional diffusion-wave equation, Appl. Math. Let.,73 (2017), 120-127.
[13] Z. P. Hao, Z. Z. Sun, and W. R. Cao,A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), 787-805.
[14] Z. Hao, K. Fan, W. Cao, and Z. Sun, A finite difference scheme for semilinear space-fractional diffusion equations with time delay, Appl. Math. Comput. 275 (2016), 238-254.
[15] F. Iavernaro and F. Mazzia, Convergence and stability of multistep methods solving nonlinear initial value problems, SIAM J. Sci. Comput. 18 (1997), 270-285.
[16] H. L. Liao, P. Lyu, and S. Vong, Second-order BDF time approximation for Riesz spacefractional diffusion equations, Int. J. Comput. Math., 95 (2018), 144-158.
[17] F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math. 166 (2004), 209-219.
[18] A. Mohebbi, On the split-step method for the solution of nonlinear Schrodinger equation with the Riesz spacefractional derivative, Comput. Methods Differ. Eq. 4 (2016), 54-69.
[19] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, 1974.
[20] I. Podulbny, Fractional differential equations, New York: Academic Press; 1999.
[21] A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (2011), 1135-1142.
[22] A. Saadatmandi and M. A. Darani, The operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications, Comput. Methods Differ. Eq. 5 (2017), 67-87.
[23] H. Wang, C. Zhang, and Y. Zhou, A class of compact boundary value methods applied to semilinear reactiondiffusion equations, Appl. Math. Comput. 325 (2018), 69-81.
[24] H. Zhang and F. Liu, Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term, J. Appli. Math. Informatics, 26 (2008), 1-14.
[25] Y. X. Zhang and H. F. Ding, Improved matrix transform method for the Riesz space fractional reaction dispersion equation, J. Comput. Appl. Math. 260 (2014), 266-280.
[26] Y. Zhou and Z. Luo, A Crank-Nicolson finite difference scheme for the Riesz space fractionalorder parabolic-type sine-Gordon equation, Adv. Di. Equ. (2018), 2018:216.
[27] S. Yang, Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term, J. Nonlinear Sci. Appl., 11 (2018), 17-25.