Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions

doi:10.1007/s42967-022-00244-8

Abstract

Abstract: In this paper, finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed. The temporal derivative is in the Caputo-Hadamard sense for both cases. The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian. The schemes are proved to be unconditionally stable and convergent. The numerical results are in line with the theoretical analysis.

Key words: Time-space fractional diffusion equation, Caputo-Hadamard derivative, Riesz derivative, Fractional Laplacian, Numerical analysis

CLC Number:

35R11
65M06

Yu Wang, Min Cai. Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1674-1696.

TrendMD

References

1. Agrawal, O.P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynam. 29, 145-155 (2002)
2. Arshad, S., Huang, J.F., Khaliq, A.Q.M., Tang, Y.F.: Trapezoidal scheme for time-space fractional diffusion equation with Riesz derivative. J. Comput. Phys. 350, 1-15 (2017)
3. Cai, M., Karniadakis, G.E., Li, C.P.: Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant. Chaos 32(7), 071101 (2022)
4. Cao, J.X., Li, C.P.: Finite difference scheme for the time-space fractional diffusion equations. Cent. Eur. J. Phys. 11(10), 1440-1456 (2013)
5. Denisov, S.I., Kantz, H.: Continuous-time random walk theory of super-slow diffusion. Europhys. Lett. 92(3), 30001 (2010)
6. Ding, H.F., Li, C.P., Chen, Y.Q.: High-order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218-237 (2015)
7. Ding, H.F., Li, C.P.: High-order algorithms for Riesz derivative and their applications (III). Fract. Calc. Appl. Anal. 19, 19-55 (2016)
8. Ding, H.F., Li, C.P.: High-order algorithms for Riesz derivative and their applications (V). Numer. Methods Partial Differ. Equ. 33, 1754-1794 (2017)
9. Ding, H.F., Li, C.P.: High-order algorithms for Riesz derivative and their applications (IV). Fract. Calc. Appl. Anal. 22, 1537-1560 (2019)
10. E, W.N., Ma, C., Wu, L.: The Barron space and the flow-induced function spaces for neural network models. Constr. Approx. 55, 369-406 (2022)
11. Fan, E.Y., Li, C.P., Li, Z.Q.: Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems. Commun. Nonlinear Sci. Numer. Simul. 106, 106096 (2022)
12. Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Solitons Fractals 102, 333-338 (2017)
13. Gohar, M., Li, C.P., Li, Z.Q.: Finite difference methods for Caputo-Hadamard fractional differential equations. Mediterr. J. Math. 17(6), 194 (2020)
14. Hao, Z.P., Zhang, Z.Q., Du, R.: Fractional centered difference scheme for high-dimensional integral fractional Laplacian. J. Comput. Phys. 424, 109851 (2021)
15. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier B.V, Amsterdam (2006)
16. Li, C.P., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)
17. Li, C.P., Li, Z.Q.: Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation. J. Nonlinear Sci. 31, 31 (2021)
18. Li, C.P., Li, Z.Q., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85, 41 (2020)
19. Li, C.P., Wang, Z.: The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis. Appl. Numer. Math. 140, 1-22 (2019)
20. Liu, F.W., Zhuang, P.H., Anh, V., Turner, I.: A fractional-order implicit difference approximation for the space-time fractional diffusion equation. ANZIAM J. 47, 203-235 (2005)
21. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65-77 (2004)
22. Ou, C.X., Cen, D.K., Vong, S., Wang, Z.B.: Mathematical analysis and numerical methods for CaputoHadamard fractional diffusion-wave equations. Appl. Numer. Math. 177, 34-57 (2022)
23. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
24. Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Phys. A 284(1/2/3/4), 376-384 (2000)
25. Sousa, E.: A second order explicit finite difference method for the fractional advection diffusion equation. Comput. Math. Appl. 64, 3141-3152 (2012)
26. Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comp. 84, 1703-1727 (2015)
27. Wang, Y.Y., Hao, Z.P., Du, R.: A linear finite difference scheme for the two-dimensional nonlinear Schrödinger equation with fractional Laplacian. J. Sci. Comput. 90, 24 (2022)
28. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)
29. Xie, C.P., Fang, S.M.: Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions. Math. Methods Appl. Sci. 43, 3473-3487 (2020)
30. Yang, Q.Q., Liu, F.W., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34, 200-218 (2010)
31. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461-580 (2002)

[1]	Junhong Tian, Hengfei Ding. A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 571-584.
[2]	Yuxin Zhang, Hengfei Ding. Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 57-72.
[3]	Prashant K. Jha, Robert Lipton. Finite Element Convergence for State-Based Peridynamic Fracture Models [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 93-128.
[4]	Yaoqiang Yan, Weihua Deng, Daxin Nie. A Finite-Diference Approximation for the One- and Two-Dimensional Tempered Fractional Laplacian [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 129-145.
[5]	Kamran Kazmi, Abdul Khaliq. A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 525-544.
[6]	Changpin Li, Qian Yi. Modeling and Computing of Fractional Convection Equation [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 565-595.
[7]	Zhiping Mao, Zhen Li, George Em Karniadakis. Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 597-619.