Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation

doi:10.1007/s42967-020-00067-5

Abstract

Abstract: The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles. The related governing equations are a series of partial differential equations with tempered fractional derivatives. Using the polynomial interpolation technique, in this paper, we present three efficient numerical formulas, namely the tempered L1 formula, the tempered L1-2 formula, and the tempered L2-1_σ formula, to approximate the Caputo-tempered fractional derivative of order α∈(0,1). The truncation error of the tempered L1 formula is of order 2-α, and the tempered L1-2 formula and L2-1_σ formula are of order 3-α. As an application, we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations, respectively. Furthermore, the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1_σ formulas are proved by the Fourier analysis method. Finally, we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.

Key words: Caputo-tempered fractional derivative, Polynomial interpolation, Implicit ADI schemes, Stability

CLC Number:

65M06

Le Zhao, Can Li, Fengqun Zhao. Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation[J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 1-40.

TrendMD

References

1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
2. Baeumera, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)
3. Cao, J.X., Li, C.P., Chen, Y.Q.: High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (II). Fract. Calc. Appl. Anal. 18, 735–761 (2015)
4. Chen, M.H., Deng, W.H.: High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. SIAM J. Sci. Comput. 37, A890–A917 (2015)
5. Chen, M.H., Deng, W.H.: High order algorithm for the time-tempered fractional Feynman–Kac equation. J. Sci. Comput. 76, 867–887 (2018)
6. Chen, C.M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction–subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)
7. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85, 1603–1638 (2016)
8. Dehghan, M., Abbaszadeh, M., Deng, W.H.: Fourth-order numerical method for the space time tempered fractional diffusion–wave equation. Appl. Math. Lett. 73, 120–127 (2017)
9. Deng, W.H., Zhang, Z.J.: Numerical schemes of the time tempered fractional Feynman–Kac equation. Comput. Math. Appl. 73, 1063–1076 (2017)
10. Dimitrov, Y.: A second order approximation for the Caputo fractional derivative. J. Frac. Cal. Appl. 7, 175–195 (2016)
11. Dimitrov, Y.: Three-point approximation for the Caputo fractional derivative. Comm. Appl. Math. Comput. 31, 413–442 (2017)
12. Ding, H.F., Li, C.P.: High-order numerical approximation formulas for Riemann–Liouville (Riesz) tempered fractional derivatives: construction and application (II). Appl. Math. Lett. 86, 208–214 (2018)
13. Gajda, J., Magdziarz, M.: Fractional Fokker–Planck equation with tempered ?-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 82, 011117 (2010)
14. Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
15. Gracia, J.L., Stynes, M.: Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. J. Comput. Appl. Math. 273, 103–115 (2015)
16. Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction–diffusion equations. Phys. Rev. E 74, 031116 (2006)
17. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650–678 (2017)
18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equation. Elsevier, Amsterdam (2006)
19. Li, C., Deng, W.H.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42, 543–572 (2016)
20. Li, C.P., Wu, R.F., Ding, H.F.: High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (I). Commun. Appl. Ind. Math. 6, 536 (2014)
21. Li, C., Deng, W.H., Zhao, L.J.: Well posedness and numerical algorithm for the tempered fractional ordinary differential equations. Discrete Contin. Dyn. Syst. Ser. B 24, 1989–2015 (2019)
22. Li, C., Sun, X.R., Zhao, F.Q.: LDG schemes with second order implicit time discretization for a fractional sub-diffusion equation. Results Appl. Math. 4, 100079 (2019)
23. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
24. Liu, F., Zhuang, P.H., Liu, Q.X.: The Applications and Numerical Methods of Fractional Differential Equations. Science Press, Beijing (2015)
25. Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)
26. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
27. Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2008)
28. Metzler, R., Barkai, E., Klafter, J.: Deriving fractional Fokker–Planck equations from generalised master equation. Europhys. Lett. 46, 431–436 (1999)
29. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
30. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
31. Sabzikar, F., Meerschaert, M.M., Chen, J.H.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)
32. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993)
33. Schmidt, M.G.W., Sagués, F., Sokolov, I.M.: Mesoscopic description of reactions for anomalous diffusion: a case study. J. Phys. Condens. Matter. 19, 065118 (2007)
34. Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville drivative. Appl. Numer. Math. 90, 22–37 (2015)
35. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion wave system. Appl. Numer. Math. 56, 193–209 (2006)
36. Sun, X.R., Li, C., Zhao, F.Q.: Local discontinuous Galerkin methods for the time tempered fractional diffusion equation. Appl. Math. Comput. 365, 124725 (2020)
37. Tatar, N.: The decay rate for a fractional differential equation. J. Math. Anal. Appl. 295, 303–314 (2004)
38. Wang, Z.B., Vong, S.W.: Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion–wave equation. J. Comput. Phys. 277, 1–15 (2014)
39. Zayernouri, M., Ainsworth, M., Karniadakis, G.: Tempered fractional Sturm–Liouville eigenproblems. SIAM J. Sci. Comput. 37, A1777–A1800 (2015)
40. Zhang, Y.: Moments for tempered fractional advection–diffusion equations. J. Stat. Phys. 139, 915–939 (2010)
41. Zhang, Z.J., Deng, W.H.: Numerical approaches to the functional distribution of anomalous diffusion with both traps and flights. Adv. Comput. Math. 43, 1–34 (2017)
42. Zhang, Y.N., Sun, Z.Z.: Alternating direction implicit schemes for the two-dimensional fractional subdiffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)
43. Zhang, H., Liu, F., Turner, I., Chen, S.: The numerical simulation of the tempered fractional Black– Scholes equation for European double barrier option. Appl. Math. Model. 40, 5819–5834 (2016)

[1]	Min Zhang, Yang Liu, Hong Li. High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 613-640.
[2]	Mostafa Abbaszadeh, Hanieh Amjadian. Second-Order Finite Diference/Spectral Element Formulation for Solving the Fractional Advection-Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 653-669.
[3]	Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven. A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 689-709.
[4]	Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.
[5]	Qiang Du, Lili Ju, Jianfang Lu, Xiaochuan Tian. A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 31-55.
[6]	Yubo Yang, Fanhai Zeng. Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 621-637.
[7]	Zachary Grant, Sigal Gottlieb, David C. Seal. A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions [J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 21-59.
[8]	Ying He, Jie Shen. Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model [J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 61-80.