AIMS Mathematics, 2020, 5(3): 1729-1744. doi: 10.3934/math.2020117

Research article

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Fast algorithm based on the novel approximation formula for the Caputo-Fabrizio fractional derivative

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Abstract    Full Text(HTML)    Figure/Table    Related pages

In this study, we propose a novel second-order numerical formula that approximates the Caputo-Fabrizio (CF) fractional derivative at node $t_{k+\frac{1}{2}}$. The nonlocal property of the CF fractional operator requires $O(M^2)$ operations and $O(M)$ memory storage, where $M$ denotes the numbers of divided intervals. To improve the efficiency, we further develop a fast algorithm based on the novel approximation technique that reduces the computing complexity from $O(M^2)$ to $O(M)$, and the memory storage from $O(M)$ to $O(1)$. Rigorous arguments for convergence analyses of the direct method and fast method are provided, and two numerical examples are implemented to further confirm the theoretical results and efficiency of the fast algorithm.
Figure/Table
Supplementary
Article Metrics

References

1. M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophys. J. R. Astr. Soc., 13 (1967), 529-539.

2. J. Li, Y. Huang, Y. Lin, Developing finite element methods for Maxwell's equations in a cole-cole dispersive medium, SIAM J. Sci. Comput., 33 (2011), 3153-3174.

3. D. Shi, H. Yang, A new approach of superconvergence analysis for two-dimensional time fractional diffusion equation, Comput. Math. Appl., 75 (2018), 3012-3023.

4. M. Zheng, F. Liu, Q. Liu, et al., Numerical solution of the time fractional reaction-diffusion equation with a moving boundary, J. Comput. Phys., 338 (2017), 493-510.

5. Y. Liu, M. Zhang, H. Li, et al., High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation, Comput. Math. Appl., 73 (2017), 1298-1314.

6. Y. Liu, Y. Du, H. Li, et al., A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl., 70 (2015), 2474-2492.

7. A. Khan, J. F. Gómez-Aguilar, T. S. Khan, et al., Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos Soliton. Fract., 122 (2019), 119-128.

8. L. Liu, L. Zheng, Y. Chen, et al., Fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness, J. Heat Transfer, 140 (2018), 091701-1-9.

9. Y. M. Lin, C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.

10. Z. Z. Sun, X. N. Wu, A fully discrete scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193-209.

11. Z. J. Zhou, W. Gong, Finite element approximation of optimal control problems governed by time fractional diffusion equation, Comput. Math. Appl., 71 (2016), 301-318.

12. B. Jin, R. Lazarov, Y. K. Liu, et al., The Galerkin finite element method for a multi-term timefractional diffusion equation, J. Comput. Phys., 281 (2015), 825-843.

13. Y. J. Jiang, J. T. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), 3285-3290.

14. S. Jiang, J. Zhang, Q. Zhang, et al., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650-678.

15. Y. Zhao, W. Bu, J. Huang, et al., Finite element method for two-dimensional space-fractional advection-dispersion equations, Appl. Math. Comput., 257 (2015), 553-565.

16. N. Liu, Y. Liu, H. Li, et al., Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term, Comput. Math. Appl., 75 (2018), 3521-3536.

17. M. Abbaszadeh, M. Dehghan, A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method, Comput. Math. Appl., 70 (2015), 2493-2512.

18. C. Li, Z. Zhao, Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855-875.

19. B. L. Yin, Y. Liu, H. Li, A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations, Appl. Math. Comput., 368 (2020), 124799.

20. H. Z. Chen, H. Wang, Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation, J. Comput. Appl. Math., 296 (2016), 480-498.

21. C. Li, W. H. Deng, A new family of difference schemes for space fractional advection diffusion equation, Adv. Appl. Math. Mech., 9 (2017), 282-306.

22. L. Feng, F. Liu, I. Turner, et al, Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains, Appl. Math. Model., 59 (2018), 441-463.

23. F. Zeng, F. Liu, C. Li, et al., A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (2014), 2599-2622.

24. B. Yin, Y. Liu, H. Li, et al., Fast algorithm based on TT-M FE system for space fractional AllenCahn equations with smooth and non-smooth solutions, J. Comput. Phys., 379 (2019), 351-372.

25. G. Zhang, C. Huang, M. Li, A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations, Eur. Phys. J. Plus, 133 (2018), 155.

26. H. L. Liao, P. Lyu, S. W. Vong, Second-order BDF time approximation for Riesz space-fractional diffusion equations, Int. J. Comput. Math., 95 (2018), 144-158.

27. A. Atangana, B. S. T. Alkahtani, Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 871-877.

28. A. Atangana, J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1-7.

29. D. Baleanu, A. Mousalou, S. Rezapour, A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative, Adv. Differ. Equ., 2017 (2017), 51.

30. V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. Saad, et al., Application of the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect, Math. Meth. Appl. Sci., 42 (2019), 1167-1193.

31. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.

32. E. F. D. Goufo, An application of the Caputo-Fabrizio operator to replicator-mutator dynamics: Bifurcation, chaotic limit cycles and control, Eur. Phys. J. Plus, 80 (2018), 133.

33. E. Fan, Y. Liu, Y. Hou, et al., Finite element method with time second-order approximation scheme for a nonlinear time Caputo-Fabrizio fractional reaction-diffusion model, submitted, 2019.

34. T. Akman, B. Yildiz, D. Baleanu, New discretization of Caputo-Fabrizio derivative, Comput. Appl. Math., 37 (2018), 3307-3333.

35. Z. Liu, A. Cheng, X. Li, A second order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math., 95 (2018), 396-411.

36. M. Zhang, Y. Liu, H. Li, High-order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo-Fabrizio fractional derivative, Numer Methods Partial Differential Eq., 35 (2019), 1588-1612.

37. F. Yu, M. Chen, Finite difference/spectral approximations for the two-dimensional time CaputoFabrizio fractional diffusion equation, arXiv:1906.00328v1 [math.NA], 2019.

38. J. Y. Cao, Z. Q. Wang, C. J. Xu, A high-order scheme for fractional ordinary differential equations with the Caputo-Fabrizio derivative, Commun. Appl. Math. Comput., Available from: https://doi.org/10.1007/s42967-019-00043-8.

39. H. Liu, A. Cheng, H. Yan, et al., A fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel, Int. J. Comput. Math., 96 (2019), 1444-1460.

40. S. S. Roshan, H. Jafari, D. Baleanu, Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials, Math. Method Appl. Sci., 41 (2018), 9134-9141.

41. T. Kaczorek, Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition, Bulletin Polish Academy Sci. Tech. Sci., 65 (2017), 45-51.

42. S. Ullah, M. A. Khan, M. Farooq, A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 237.

43. V. E. Tarasov, No nonlocality. No fractional derivative, Commun. Nonlinear Sci. Numer. Simulat., 62 (2018), 157-163.

44. V. E. Tarasov, Caputo-Fabrizio operator in terms of integer derivatives: Memory or distributed lag?, Comput. Appl. Math., 38 (2019), 113.

45. M. Stynes, Fractional-order derivatives defined by continuous kernels are too restrictive, Appl. Math. Lett. 85 (2018), 22-26.

46. X. J. Yang, H. M. Srivastava, J. T. Machado, A new fractional derivative without singular kernel, Therm. Sci., 20 (2016), 753-756.

47. I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.

48. C. P. Li, M. Cai, Theory and numerical approximation of fractional integrals and derivatives, 2019.

49. C. Li, R. Wu, H. Ding, High-order approximation to Caputo derivatives and Caputotype advection-diffusion equations, Commun. Appl. Indust. Math., 536 (2015), DOI:10.1685/journal.caim.

50. Y. Liu, Y. Du, H. Li, et al., Some second-order θ schemes combined with finite element method for nonlinear fractional cable equation, Numer. Algor., 80 (2019), 533-555.

51. B. Yin, Y. Liu, H. Li, et al., Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations, arXiv preprint arXiv:1906.01242, (2019).

52. F. Zeng, I. Turner, K. Burrage, A stable fast time-stepping method for fractional integral and derivative operators, J. Sci. Comput., 77 (2018), 283-307.

53. A. Schädle, M. López-Fernández, C. Lubich, Fast and oblivious convolution quadrature, SIAM J. Sci. Comput., 28 (2006), 421-438.

54. M. Li, X. M. Gu, C. M. Huang, et al., A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys., 358 (2018), 256-282.

55. S. Chen, F. Liu, X. Jiang, et al., A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients, Appl. Math. Comput., 257 (2015), 591-601.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Article outline

Show full outline