ORIGINAL PAPER

A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation

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  • School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, Hunan Province, China

Received date: 2019-11-19

  Revised date: 2020-03-10

  Online published: 2021-03-15

Abstract

By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation, and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative, the fractional central difference operator to discritize the space Riesz fractional partial derivative, and the classical central difference formula to discretize the advection term, a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation. The stability and the convergence analysis of the numerical method are given. Numerical experiments show that the numerical method is effective.

Cite this article

Wenhui Guan, Xuenian Cao . A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation[J]. Communications on Applied Mathematics and Computation, 2021 , 3(1) : 41 -59 . DOI: 10.1007/s42967-020-00080-8

References

1. Boris, B., Mark, M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)
2. Cartea, A., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A. 374, 749–763 (2007)
3. Çelik, C., Duman, M.: Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)
4. Chen, M.H., Deng, W.H.: Fourth order difference approximations for space Riemann–Liouville derivatives based on weighted and shifted Lubich difference operators. Commun. Comput. Phys. 16, 516– 540 (2014)
5. Chen, M.H., Deng, W.H.: A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation. Appl. Math. Lett. 68, 87–93 (2017)
6. Deng, W.H., Chen, M.H., Barkai, E.: Numerical algorithms for the forward and backward fractional Feynman–Kac equations. J. Sci. Comput. 62, 718–746 (2015)
7. 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)
8. Ding, H.F., Li, C.P.: A high-order algorithm for time-Caputo-tempered partial differential equation with Riesz derivatives in two spatial dimensions. J. Sci. Comput. 80, 81–109 (2019)
9. Ding, H.F.: A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation. Appl. Numer. Math. 135, 30–46 (2019)
10. Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional dreivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
11. Guan, W.H., Cao, X.N.: The implicit midpoint method for Riesz tempered fractional advection-diffusion equation. J. Numer. Methods Comput. 41, 51–65 (2020)
12. He, J.Q., Dong, Y., Li, S.T., Liu, H.L., Yu, Y.J., Jin, G.Y., Liu, L.D.: Study on force distribution of the tempered glass based on laser interference technology. Optik 126, 5276–5279 (2015)
13. Hu, D.D., Cao, X.N.: The implicit midpoint method for Riesz tempered fractional diffusion equation with a nonlinear source term. Adv. Differ. Equ. 2019, 66 (2019)
14. Li, C., Deng, W.H., Wu, Y.J.: Numerical analysis and physical simulations for the time fractional radial diffusion equation. Comput. Math. Appl. 62, 1024–1037 (2011)
15. Li, C., Deng, W.H.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42, 543–572 (2016)
16. Magin, R.: Fractional Calculus in Bioengneering. Begell House Publishers, Danbury (2006)
17. Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35(17), L17403 (2008)
18. Moghaddam, B.P., Tenreiro Machado, J.A., Babaei, A.: A computationally efficient method for tempered fractional differential equations with application. Comput. Appl. Math. 37, 3657–3671 (2017)
19. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1998)
20. Qiu, Z.S., Cao, X.N.: Second-order numerical methods for the tempered fractional diffusion equations. Adv. Differ. Equ. 2019, 485 (2019)
21. Rosenau, P.: Tempered diffusion: a transport process with propagating fronts and inertial delay. Phys. Rev. A. 46, 7371–7374 (1992)
22. Shen, S.J., Liu, F.W., Anh, V., Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. Math. Comput. 73, 850–872 (2008)
23. Shen, S.J., Liu, F.W., Anh, V.: Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation. Numer. Algor. 56, 383–403 (2011)
24. Sabzikar, F., Meerschaert, M.M., Chen, J.H.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)
25. Sun, X.R., Zhao, F.Q., Chen, S.P.: Numerical algorithms for the time-space tempered fractional Fokker–Planck equation. Adv. Differ. Equ. 2017, 259 (2017)
26. Wu, X.C., Deng, W.H., Barkai, E.: Tempered fractional Feynman–Kac equation: theory and examples. Phys. Rev. E. 93, 032151 (2016)
27. Yu, Y.Y., Deng, W.H., Wu, Y.J.: Third order difference schemes (without using points outside of the domain) for one side space tempered fractional partial differential equations. Appl. Numer. Math. 112, 126–145 (2017)
28. Zhuang, P., Liu, F.W., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)
29. Zhang, Y.: Moments for tempered fractional advection-diffusion equations. J. Stat. Phys. 139, 915–939 (2010)
30. Zhang, Y.X., Li, Q., Ding, H.F.: High-order numerical approximation formulas for Riemann–Liouville (Riesz) tempered fractional derivative: construction and application (I). Appl. Math. Comput. 329, 432–443 (2018)
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