Abstract
In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for space Riesz variable-order fractional diffusion equations. Based on weighted-shifted-Grünwald-difference (WSGD) operators proposed in Lin and Liu (J. Comput. Appl. Math. 363, 77–91 (2020)) for Riemann-Liouville fractional derivatives, we derive WSGD operators for variable-order ones by using the relation between variable-order fractional derivative and (constant-order) fractional derivative. We then apply Crank-Nicolson-weighted-shifted-Grünwald-difference (CN-WSGD) schemes to the initial-boundary problem for space Riesz variable-order diffusion equations. Theoretical results on the stability and convergence of CN-WSGD schemes are presented and proved. Moreover, we derive a problem-based method to choose suitable CN-WSGD schemes, which leads to unconditioned stable linear systems with optimal upper bound for accuracy. Numerical results show that the proposed schemes are very efficient.
Similar content being viewed by others
References
Abd-Elhameed, W.M., Youssri, Y.H.: Generalized Lucas polynomial sequence approach for fractional differential equations. Nonlinear Dyn. 89, 1341–1355 (2017)
Abd-Elhameed, W.M., Youssri, Y.H.: Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations. Comp. Appl. Math. 37, 2897–2921 (2018)
Abd-Elhameed, W.M., Youssri, Y.H.: Spectral tau algorithm for certain coupled system of fractional differential equations via generalized Fibonacci polynomial sequence. Iran. J. Sci. Technol. Trans. Sci. 43, 543–554 (2019)
Abd-Elhameed, W.M., Youssri, Y.H.: Sixth-kind Chebyshev spectral approach for solving fractional differential equations. IJNSNS 20, 191–203 (2019)
Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Physica A: Statistical Mechanics and its Applications 500, 40–49 (2018)
Bai, J., Feng, X.C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Proc. 16, 2492–2502 (2007)
Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)
Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)
Cao, J., Qiu, Y.: A high order numerical scheme for variable order fractional ordinary differential equation. Appl. Math. Let. 61, 88–94 (2016)
Cao, J., Qiu, Y., Song, G.: A compact finite difference scheme for variable order subdiffusion equation. Commun. Nonlinear Sci. Numer. Simulat. 48, 140–149 (2017)
Carreras, B.A., Lynch, V.E., Zaslavsky, G.M.: Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas 8, 5096–5103 (2001)
Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12, 692–703 (2003)
Daftardar-Gejji, V., Jafari, H.: Solving a multi-order fractional differential equation using Adomian decomposition. Appl. Math. Comput. 189, 541–548 (2007)
Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM. J. Numer. Anal. 47, 204–226 (2008)
Diethelm, K.: The Analysis of Fractional Differential Equations. Spring, Berlin (2010)
Ding, H., Li, C.: High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J. Sci. Comput. 71, 759–784 (2017)
Doha, E.H., Abdelkawy, M.A., Amin, A.Z.M., Baleanu, D.: Spectral technique for solving variable-order fractional Volterra integro-differential equations. Numer. Methods Partial Differ. Equ. 34, 1659–1677 (2017)
Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent,nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)
Feller, W.: On a generalization of Marcel Riesz’ potentials and the semi-groups generated by them. Medd. Lunds Univ. Mat. Sem. 1952, 72–81 (1952)
Hafez, R.M., Youssri, Y.H.: Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation. Comp. Appl. Math. 37, 5315–5333 (2018)
Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Physica A 276, 448–455 (2000)
Khalid, A., Naeem, M.N., Agarwal, P., Ghaffar, A., Ullah, Z., Jain, S.: Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline. Advances in Difference Equations 2019, 492 (2019)
Jin, X.Q.: Preconditioning Techniques for Toeplitz Systems. Higher Education Press, Beijing (2010)
Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84, 2665–2700 (2015)
Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)
Li, C., Deng, W.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42, 543–572 (2016)
Lin, F.R., Liu, W.D.: The accuracy and stability of CN-WSGD schemes for space fractional diffusion equation. J. Comput. Appl. Math. 363, 77–91 (2020)
Lin, F.R., Yang, S.W., Jin, X.Q.: Preconditioned iterative methods for fractional diffusion equation. J. Comput. Phys. 256, 109–117 (2014)
Lin, R., Liu, F., Anh, V., Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212, 435–445 (2009)
Lin, X.L., Ng, M.K., Sun, H.W.: Efficient preconditioner of one-sided space fractional diffusion equation. BIT Numer. Math. 58(3), 729–748 (2018)
Lorenzo, C.F., Hartley, T.T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers Inc., U.S. (2006)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314, 749–755 (2002)
Ruiz-Medina, M.D., Anh, V., Angulo, J.M.: Fractional generalized random fields of variable order. Stochastic Anal. Appl. 22, 775–799 (2004)
Salahshour, S., Ahmadian, A., Senu, N., Baleanu, D., Agarwal, P.: On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem. Entropy 17, 885–902 (2015)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Newark, NJ (1993)
Seki, K., Wojcik, M., Tachiya, M.: Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165–2174 (2003)
Shen, S., Liu, F., Chen, J., Turner, I., Anh, V.: Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 218, 10861–10870 (2012)
Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987)
Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today 55, 48–54 (2002)
Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)
Tariboon, J., Ntouyas, S.K, Agarwal, P.: New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Advances in Difference Equations 2015, 18 (2015)
Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)
Zeng, F.H., Zhang, Z.Q., EmKarniadakis, G.: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. SIAM J. Sci. Comput. 37, A2710–A2732 (2015)
Zhao, X., Sun, Z.Z., EmKarniadakis, G.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)
Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)
Acknowledgments
We thank the referees for providing valuable comments and suggestions, which are very helpful for improving our paper.
Funding
This work is supported by National Natural Science Foundation of China (11771265) and the research grants MYRG2016-00077-FST, MYRG2019-00042-FST from University of Macau.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lin, FR., Wang, QY. & Jin, XQ. Crank-Nicolson-weighted-shifted-Grünwald-difference schemes for space Riesz variable-order fractional diffusion equations. Numer Algor 87, 601–631 (2021). https://doi.org/10.1007/s11075-020-00980-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-020-00980-z