Skip to main content
SpringerLink
Log in

Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this paper, we develop an efficient finite difference/spectral method to solve a coupled system of nonlinear multi-term time-space fractional diffusion equations. In general, the solutions of such equations typically exhibit a weak singularity at the initial time. Based on the L1 formula on nonuniform meshes for time stepping and the Legendre–Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed. Taking into account the initial weak singularity of the solution, the convergence of the method is proved. The optimal error estimate is obtained by providing a generalized discrete form of the fractional Grönwall inequality which enables us to overcome the difficulties caused by the sum of Caputo time-fractional derivatives and and the positivity of the reaction term over the nonuniform time mesh. The error estimate reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. Furthermore, numerical experiments are presented to confirm the theoretical claims.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alikhanov AA (2015) Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation. Appl Math Comput 268:12–22

    MathSciNet  MATH  Google Scholar 

  2. Bhrawy A, Zaky M (2016) Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl Math Model 40(2):832–845

    MathSciNet  MATH  Google Scholar 

  3. Bhrawy A, Zaky MA (2015) A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J Comput Phys 281:876–895

    MathSciNet  MATH  Google Scholar 

  4. Chen J, Liu F, Anh V, Shen S, Liu Q, Liao C (2012) The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl Math Comput 219(4):1737–1748

    MathSciNet  MATH  Google Scholar 

  5. Daftardar-Gejji V, Bhalekar S (2008) Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method. Appl Math Comput 202(1):113–120

    MathSciNet  MATH  Google Scholar 

  6. Dehghan M, Abbaszadeh M (2018) An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional bloch-torrey equations. Appl Numer Math 131:190–206

    MathSciNet  MATH  Google Scholar 

  7. Dehghan M, Safarpoor M, Abbaszadeh M (2015) Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J Comput Appl Math 290:174–195

    MathSciNet  MATH  Google Scholar 

  8. Diethelm K, Garrappa R, Stynes M (2020) Good (and not so good) practices in computational methods for fractional calculus. Mathematics 8(3):324

    Google Scholar 

  9. Ding XL, Jiang YL (2013) Analytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions. Nonlinear Anal Real World Appl 14(2):1026–1033

    MathSciNet  MATH  Google Scholar 

  10. Ervin VJ, Roop JP (2007) Variational solution of fractional advection dispersion equations on bounded domains in \({\mathbb{R}}^d\). Numer Methods Partial Differ Equ IntJournal 23(2):256–281

    MATH  Google Scholar 

  11. Ezz-Eldien SS, Wang Y, Abdelkawy MA, Zaky MA, Aldraiweesh AA, Machado JTM (2020) Chebyshev spectral methods for multi-order fractional neutral pantograph equations. Nonlinear Dyn 100:3785–3797

    Google Scholar 

  12. Feng L, Liu F, Turner I, Zhuang P (2017) Numerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates. Int J Heat Mass Transf 115:1309–1320

    Google Scholar 

  13. Fu ZJ, Yang LW, Zhu HQ, Xu WZ (2019) A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations. Eng Anal Bound Elem 98:137–146

    MathSciNet  MATH  Google Scholar 

  14. Hendy AS (2020) Numerical treatment for after-effected multi-term time-space fractional advection–diffusion equations. Eng Comput pp 1–11

  15. Hendy AS, Zaky MA (2020) Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations. Appl Numer Math 156:276–302

    MathSciNet  MATH  Google Scholar 

  16. Jiang H, Liu F, Turner I, Burrage K (2012) Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J Math Anal Appl 389(2):1117–1127

    MathSciNet  MATH  Google Scholar 

  17. Jin B, Lazarov R, Liu Y, Zhou Z (2015) The Galerkin finite element method for a multi-term time-fractional diffusion equation. J Comput Phys 281:825–843

    MathSciNet  MATH  Google Scholar 

  18. Kelly JF, McGough RJ, Meerschaert MM (2008) Analytical time-domain Green’s functions for power-law media. J Acoust Soc Am 124(5):2861–2872

    Google Scholar 

  19. Kopteva N (2019) Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math Comput 88(319):2135–2155

    MathSciNet  MATH  Google Scholar 

  20. Li D, Liao HL, Sun W, Wang J, Zhang J (2018) Analysis of \(L1\)-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun Comput Phys 24(1):86–103

    MathSciNet  MATH  Google Scholar 

  21. Li D, Wang J, Zhang J (2017) Unconditionally convergent \(L1\)-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J Sci Comput 39(6):A3067–A3088

    MATH  Google Scholar 

  22. Li Z, Liu Y, Yamamoto M (2015) Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl Math Comput 257:381–397

    MathSciNet  MATH  Google Scholar 

  23. Liao Hl, Li D, Zhang J (2018) Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J Numer Anal 56(2):1112–1133

    MathSciNet  MATH  Google Scholar 

  24. Liao Hl, McLean W, Zhang J (2019) A discrete Gronwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J Numer Anal 57(1):218–237

    MathSciNet  MATH  Google Scholar 

  25. Liu F, Meerschaert M, McGough R, Zhuang P, Liu Q (2013) Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract Calc Appl Anal 16(1):9–25

    MathSciNet  MATH  Google Scholar 

  26. Liu Z, Liu F, Zeng F (2019) An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. Appl Numer Math 136:139–151

    MathSciNet  MATH  Google Scholar 

  27. Luchko Y (2011) Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J Math Anal Appl 374(2):538–548

    MathSciNet  MATH  Google Scholar 

  28. Ming C, Liu F, Zheng L, Turner I, Anh V (2016) Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid. Comput Math Appl 72(9):2084–2097

    MathSciNet  MATH  Google Scholar 

  29. Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order,  vol 111. Dover Publications, Inc., Mineola, New York

    MATH  Google Scholar 

  30. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198, 1st edn. Academic Press, San Diego, California

    Google Scholar 

  31. Shen J, Tang T, Wang LL (2011) Spectral methods: algorithms, analysis and applications, vol 41. Springer, Berlin, Heidelberg

    MATH  Google Scholar 

  32. Shi Z, Zhao Y, Liu F, Wang F, Tang Y (2018) Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshes. Appl Math Comput 338:290–304

    MathSciNet  MATH  Google Scholar 

  33. Sokolov IM, Klafter J, Blumen A (2002) Fractional kinetics. Phys Today 55(11):48–54

    Google Scholar 

  34. Stynes M (2016) Too much regularity may force too much uniqueness. Fract Calc Appl Anal 19(6):1554–1562

    MathSciNet  MATH  Google Scholar 

  35. Stynes M (2019) Singularities. In: Karniadakis GE (ed) Handbook of fractional calculus with applications, vol 3. De Gruyter, Berlin, pp 287–305

    Google Scholar 

  36. Van Bockstal K (2020) Existence and uniqueness of a weak solution to a non-automonous time-fractional diffusion equation (of distributed order). Appl Math Lett 109:106540

    MathSciNet  MATH  Google Scholar 

  37. Vieru D, Fetecau C, Fetecau C (2008) Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate. Appl Math Comput 200(1):459–464

    MathSciNet  MATH  Google Scholar 

  38. Wang Y, Liu F, Mei L, Anh VV (2020) A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions. Numer Algorithms. https://doi.org/10.1007/s11075-020-00940-7

    Article  MATH  Google Scholar 

  39. Wang YM, Wen X (2020) A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions. Appl Math Comput 381:125,316

    MathSciNet  Google Scholar 

  40. Wei L (2017) Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations. Numer Algorithms 76(3):695–707

    MathSciNet  MATH  Google Scholar 

  41. Zaky M, Doha E, Machado JT (2018) A spectral framework for fractional variational problems based on fractional Jacobi functions. Appl Numer Math 132:51–72

    MathSciNet  MATH  Google Scholar 

  42. Zaky MA (2018) A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput Appl Math 37(3):3525–3538

    MathSciNet  MATH  Google Scholar 

  43. Zaky MA (2020) An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205–222

    MathSciNet  MATH  Google Scholar 

  44. Zaky MA, Hendy AS, Macías-Díaz JE (2020) Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions. J Sci Comput 82(1):1–27

    MathSciNet  MATH  Google Scholar 

  45. Zeng F, Li C, Liu F, Turner I (2015) Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J Sci Comput 37(1):A55–A78

    MathSciNet  MATH  Google Scholar 

  46. Zeng F, Liu F, Li C, Burrage K, Turner I, Anh V (2014) A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J Numer Anal 52(6):2599–2622

    MathSciNet  MATH  Google Scholar 

  47. Zhang H, Jiang X, Wang C, Fan W (2018) Galerkin-Legendre spectral schemes for nonlinear space fractional schrödinger equation. Numer Algorithms 79(1):337–356

    MathSciNet  MATH  Google Scholar 

  48. Zhou Y, Suzuki JL, Zhang C, Zayernouri M (2020) Implicit-explicit time integration of nonlinear fractional differential equations. Appl Numer Math 156:555–583

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

ASH wishes to acknowledge the support of RFBR Grant 19-01-00019.

Author information

Authors and Affiliations

  1. Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, 620002, Russia

    Ahmed S. Hendy

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

    Ahmed S. Hendy

  3. Department of Applied Mathematics, National Research Centre, Dokki, 12622, Cairo, Egypt

    Mahmoud A. Zaky

Authors
  1. Ahmed S. Hendy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mahmoud A. Zaky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed S. Hendy.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hendy, A.S., Zaky, M.A. Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Engineering with Computers 38, 1351–1363 (2022). https://doi.org/10.1007/s00366-020-01095-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01095-8

Keywords

Search

Navigation