Home Error estimates of high-order numerical methods for solving time fractional partial differential equations
Article
Licensed
Unlicensed Requires Authentication

Error estimates of high-order numerical methods for solving time fractional partial differential equations

  • Zhiqiang Li EMAIL logo and Yubin Yan
Published/Copyright: July 12, 2018
Become an author with De Gruyter Brill
Author Information Explore this Subject
Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 3

Abstract

Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(τ3−α) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. [21]. We show that this new method also has the convergence order O(τ3−α) for all α ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(τ3−α + h2) are proved in the fully discrete case, where h is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.

MSC 2010: Primary 65M06; Secondary 65M12, 65M15, 26A33, 35R11
Key Words and Phrases: time fractional partial differential equations; finite difference method; stability; error estimates

References

[1] E.E. Adams, L.W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis. Water Resources Res. 28 (1992), 3293–3307.10.1029/92WR01757Search in Google Scholar

[2] J. Cao, C. Li, and Y. Chen, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II). Fract. Calc. Appl. Anal. 18, No 3 (2015), 735–761; 10.1515/fca-2015-0045; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.Search in Google Scholar

[3] F. Chen, Q. Xu, and J.S. Hesthaven, A multi-domain spectral method for time-fractional differential equations. J. Comput. Phys. 293 (2015), 157–172.10.1016/j.jcp.2014.10.016Search in Google Scholar

[4] S. Chen, J. Shen, and L.-L. Wang, Generalized Jacobi functions and their applications to fractional differential equations. Math. Comp. 85 (2016), 1603–1638.10.1090/mcom3035Search in Google Scholar

[5] N.J. Ford, M.L. Morgado, and M. Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16, No 4 (2013), 874–891; 10.2478/s13540-013-0054-3; https://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml.Search in Google Scholar

[6] N.J. Ford, K. Pal, and Y. Yan, An algorithm for the numerical solution of two-sided spacefractional partial differential equations. Comput. Methods Appl. Math. 15 (2015), 497–514.10.1515/cmam-2015-0022Search in Google Scholar

[7] N.J. Ford, M.M. Rodrigues, J. Xiao, and Y. Yan, Numerical analysis of a two-parameter fractional telegraph equation. J. Comput. Appl. Math. 249 (2013), 95–106.10.1016/j.cam.2013.02.009Search in Google Scholar

[8] N.J. Ford, J. Xiao, and Y. Yan, Stability of a numerical method for a space-time-fractional telegraph equation. Comput. Methods Appl. Math. 12 (2012), 1–16.10.2478/cmam-2012-0009Search in Google Scholar

[9] N.J. Ford, J. Xiao, and Y. Yan, A finite element method for time-fractional partial differential equations. Fract. Calc. Appl. Anal. 14, No 3 (2011), 454–474; 10.2478/s13540-011-0028-2; https://www.degruyter.com/view/j/fca.2011.14.issue-3/issue-files/fca.2011.14.issue-3.xml.Search in Google Scholar

[10] N.J. Ford, Y. Yan, An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data. Fract. Calc. Appl. Anal. 20, No 5 (2017), 1076–1105; 10.1515/fca-2017-0058; https://www.degruyter.com/view/j/fca.2017.20.issue-5/issue-files/fca.2017.20.issue-5.xml.Search in Google Scholar

[11] G.-H. Gao, Z.-Z. Sun, and H.-W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259 (2014), 33–50.10.1016/j.jcp.2013.11.017Search in Google Scholar

[12] R. Gorenflo, F. Mainardi, Random walk models for space fractional diffusion processes. Fract. Calc. Appl. Anal. 1, No 2 (1998), 167–191.Search in Google Scholar

[13] Y. Hatano, N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles. Water Resources Res. 34 (1998), 1027–1033.10.1029/98WR00214Search in Google Scholar

[14] B. Jin, R. Lazarov, D. Sheen, and Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fract. Calc. Appl. Anal. 19, No 1 (2016), 69–93; 10.1515/fca-2016-0005; https://www.degruyter.com/view/j/fca.2016.19.issue-1/issue-files/fca.2016.19.issue-1.xml.Search in Google Scholar

[15] B. Jin, R. Lazarov, and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38 (2016), A146–A170.10.1137/140979563Search in Google Scholar

[16] B. Jin, R. Lazarov, and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. of Numer. Anal. 36 (2016), 197–221.10.1093/imanum/dru063Search in Google Scholar

[17] T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205 (2005), 719–736.10.1016/j.jcp.2004.11.025Search in Google Scholar

[18] H. Li, J. Cao, and C. Li, Higher-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III). J. Comput. Appl. Math. 299 (2016), 159–175.10.1016/j.cam.2015.11.037Search in Google Scholar

[19] C. Li, H. Ding, Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38 (2014), 3802–3821.10.1016/j.apm.2013.12.002Search in Google Scholar

[20] Z. Li, Z. Liang, and Y. Yan, High-order numerical methods for solving time fractional partial differential equations. J. Sci. Comput. 71 (2017), 785–803.10.1007/s10915-016-0319-1Search in Google Scholar

[21] C. Li, R. Wu, and H. Ding, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations. Commun. Appl. Ind. Math. 6 (2014), e–536.Search in Google Scholar

[22] X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8 (2010), 1016–1051.10.4208/cicp.020709.221209aSearch in Google Scholar

[23] Z. Li, Y. Yan, and N.J. Ford, Error estimates of a high order numerical method for solving linear fractional differential equation. Appl. Numer. Math. 114 (2017), 201–220.10.1016/j.apnum.2016.04.010Search in Google Scholar

[24] H. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56 (2018), 1112–1133.10.1137/17M1131829Search in Google Scholar

[25] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007), 1533–1552.10.1016/j.jcp.2007.02.001Search in Google Scholar

[26] C. Lv, C. Xu, Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38 (2016), A2699–A2724.10.1137/15M102664XSearch in Google Scholar

[27] W. McLean, K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data. J. Comput. Phys. 293 (2015), 201–217.10.1016/j.jcp.2014.08.050Search in Google Scholar

[28] R. Metzler, J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), 161–208.10.1088/0305-4470/37/31/R01Search in Google Scholar

[29] R.R. Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B133 (1986), 425–430.10.1515/9783112495483-049Search in Google Scholar

[30] K. Pal, Y. Yan, and G. Roberts, Numerical Solutions of Fractional Differential Equations by Extrapolation. In: Finite Difference Methods, Theory and Applications. FDM 2014 (I. Dimov, I. Faragó, L. Vulkov, Eds.), Lecture Notes in Computer Science # 9045 (2015), 291–299.Search in Google Scholar

[31] E. Sousa, A second order explicit finite difference method for the fractional advection diffusion equation. Comput. Math. Appl. 64 (2012), 3143–3152.10.1016/j.camwa.2012.03.002Search in Google Scholar

[32] Y. Yan, K. Pal, and N.J. Ford, Higher order numerical methods for solving fractional differential equations. BIT Numer. Math. 54 (2014), 555–584.10.1007/s10543-013-0443-3Search in Google Scholar

[33] S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216 (2006), 264–274.10.1016/j.jcp.2005.12.006Search in Google Scholar

[34] S. B. Yuste, L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42 (2005), 1862–1874.10.1137/030602666Search in Google Scholar

[35] F. Zeng, C. Li, F. Liu, and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35 (2013), A2976–A3000.10.1137/130910865Search in Google Scholar

[36] F. Zeng, Z. Zhang, and G.E. Karniadakis, Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. J. Comput. Phys. 307 (2016), 15–33.10.1016/j.jcp.2015.11.058Search in Google Scholar

[37] Y.-N. Zhang, Z.-Z. Sun, and H.-L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265 (2014), 195–210.10.1016/j.jcp.2014.02.008Search in Google Scholar

[38] M. Zheng, F. Liu, V. Anh, and I. Turner, A high order spectral method for the multi-term time-fractional diffusion equations. Appl. Math. Model. 40 (2016), 4970–4985.10.1016/j.apm.2015.12.011Search in Google Scholar

Received: 2017-12-03
Published Online: 2018-07-12
Published in Print: 2018-06-26

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–Volume 21–3–2018)
  4. Research Paper
  5. Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces
  6. Novel polarization index evaluation formula and fractional-order dynamics in electric motor insulation resistance
  7. The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces
  8. Parallel algorithms for modelling two-dimensional non-equilibrium salt transfer processes on the base of fractional derivative model
  9. Some iterated fractional q-integrals and their applications
  10. Lebesgue regularity for nonlocal time-discrete equations with delays
  11. Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions
  12. Error estimates of high-order numerical methods for solving time fractional partial differential equations
  13. The Laplace transform induced by the deformed exponential function of two variables
  14. Attractivity for fractional evolution equations with almost sectorial operators
  15. The multiplicity solutions for nonlinear fractional differential equations of Riemann-Liouville type
  16. Well-posedness of general Caputo-type fractional differential equations
  17. Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions
  18. Inverse source problem for a space-time fractional diffusion equation
  19. Erratum
  20. Erratum to: Invariant subspace method: A tool for solving fractional partial differential equations, in: FCAA-20-2-2017
https://doi.org/10.1515/fca-2018-0039
Keywords for this article
time fractional partial differential equations; finite difference method; stability; error estimates

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–Volume 21–3–2018)
  4. Research Paper
  5. Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces
  6. Novel polarization index evaluation formula and fractional-order dynamics in electric motor insulation resistance
  7. The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces
  8. Parallel algorithms for modelling two-dimensional non-equilibrium salt transfer processes on the base of fractional derivative model
  9. Some iterated fractional q-integrals and their applications
  10. Lebesgue regularity for nonlocal time-discrete equations with delays
  11. Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions
  12. Error estimates of high-order numerical methods for solving time fractional partial differential equations
  13. The Laplace transform induced by the deformed exponential function of two variables
  14. Attractivity for fractional evolution equations with almost sectorial operators
  15. The multiplicity solutions for nonlinear fractional differential equations of Riemann-Liouville type
  16. Well-posedness of general Caputo-type fractional differential equations
  17. Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions
  18. Inverse source problem for a space-time fractional diffusion equation
  19. Erratum
  20. Erratum to: Invariant subspace method: A tool for solving fractional partial differential equations, in: FCAA-20-2-2017
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 8.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2018-0039/html
Scroll to top button