Abstract
This paper deals with two fractional Crank–Nicolson–Galerkin finite element schemes for coupled time-fractional nonlinear diffusion system. The first scheme is iterative and is based on Newton’s method, while the other one is a linearized scheme. Existence-uniqueness results of the fully discrete solution for both schemes are discussed. In addition, a priori bounds and a priori error estimates are derived for proposed schemes using a new discrete fractional Grönwall-type inequality. Both the schemes yield \(O({\varDelta } t^2)\) accuracy in time and hence, superior to \(O({\varDelta } t^{2-\alpha })\) accurate L1 scheme existing in the literature. Moreover, three different numerical examples are provided to illustrate the theoretical estimates .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chaudhary S (2018) Crank-Nicolson-Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton’s method. Math Methods Appl Sci 41(2):724–749
Dimitrov Y (2014) Numerical approximations for fractional differential equations. J Fract Calc Appl 5(22):1–45
Dumitru B, Kai D, Enrico S (2012) Fractional calculus: models and numerical methods, vol 3. World Scientific, Singapore
Gao GH, Sun HW, Sun ZZ (2015) Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J Comput Phys 280:510–528
Govaerts W, Pryce J (1990) Block elimination with one refinement solves bordered linear systems accurately. BIT Numer Math 30(3):490–507
Hajipour M, Jajarmi A, Baleanu D, Sun H (2019) On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun Nonlinear Sci Num Simul 69:119–133
Jin B, Li B, Zhou Z (2017) An analysis of the Crank–Nicolson method for subdiffusion. IMA J Numer Anal 38(1):518–541
Jin B, Li B, Zhou Z (2018) Numerical analysis of nonlinear subdiffusion equations. SIAM J Numer Anal 56(1):1–23
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam
Kou SC (2008) Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. Ann Appl Stat 2(2):501–535
Kumar D, Chaudhary S, Kumar V (2018) Fractional Crank–Nicolson–Galerkin finite element scheme for the time-fractional nonlinear diffusion equation. arXiv preprint arXiv:1811.08485
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
Li L, Jin L, Fang S (2015) Existence and uniqueness of the solution to a coupled fractional diffusion system. Adv Diff Equ 2015(1):370
Li M, Gu XM, Huang C, Fei M, Zhang G (2018) A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations. J Comput Phys 358:256–282
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
Liao Hl, Yan Y, Zhang J (2018) Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations. J Sci Comput. https://doi.org/10.1007/s10915-019-00927-0
Lin Y, Xu C (2007) finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225(2):1533–1552
Liu N, Liu Y, Li H, Wang J (2018) Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term. Comput Math Appl 75(10):3521–3536
Podlubny I (1999) Fractional differential equations. Academic press, Cambridge
Rannacher R, Scott R (1982) Some optimal error estimates for piecewise linear finite element approximations. Math Comput 38(158):437–445
Sun W, Wang J (2017) Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D. J Comput Appl Math 317:685–699
Thomée V (1984) Galerkin finite element methods for parabolic problems, vol 1054. Springer, Berlin
Wang Y, Liu Y, Li H, Wang J (2016) Finite element method combined with second-order time discrete scheme for nonlinear fractional cable equation. Eur Phys J Plus 131(3):61
West BJ (2007) Fractional calculus in bioengineering. J Stat Phys 126(6):1285–1286
Acknowledgements
We would like to express our gratitude to the editor and anonymous reviewers for their constructive comments and suggestions. In addition, the first author is grateful to the University Grants Commission, India, for the financial grant through Senior Research Fellowship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dileep Kumar.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kumar, D., Chaudhary, S. & Kumar, V.V.K.S. Galerkin finite element schemes with fractional Crank–Nicolson method for the coupled time-fractional nonlinear diffusion system. Comp. Appl. Math. 38, 123 (2019). https://doi.org/10.1007/s40314-019-0889-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-019-0889-2
Keywords
- Time-fractional diffusion system
- Fractional Crank–Nicolson method
- Error estimates
- Grönwall-type inequality
- Newton’s method
- Linearized schemes