Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A high-order compact difference scheme for the multi-term time-fractional Sobolev-type convection-diffusion equation

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper presents two high-order compact difference schemes to discuss the numerical solution of the one-dimensional and two-dimensional multi-term time-fractional convection-diffusion equation of the Sobolev type based on the Caputo fractional derivative. For this purpose, we employ the L2 formula for the temporal discretization of the Caputo fractional derivatives and introduce a new compact difference operator for the space discretization. The proposed schemes transform the original problem into a system of algebraic equation. We present a novel analysis of the convergence and theoretical stability of both methods. The difference schemes have fourth-order and \(3-\max \{\alpha _r, \beta _r\}\) order of accuracy in space and time respectively, where \(\alpha _r\) and \(\beta _r\) are the orders of the fractional derivatives in the multi-term convection-diffusion model. Additionally, we construct an L2-type numerical scheme on non-uniform graded meshes to address problems where the solution exhibits weak initial regularity. Some test functions are provided to demonstrate the accuracy and efficiency of the proposed schemes. The numerical results of inclusive examples confirm the proposed schemes’ theoretical results and illustrate their applicability and efficiency.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data availability

No data was used for the research described in the article.

References

  • Abro KA, Hussain M, Baig MM (2017) Slippage of magnetohydrodynamic fractionalized Oldroyd-B fluid in porous medium. Progress Fract Differ Appl 3(1):69–80

    Article  MATH  Google Scholar 

  • Alavi SA, Haghighi A, Yari A, Soltanian F (2022) A numerical method for solving fractional optimal control problems using the operational matrix of mott polynomials. Comput Methods Differ Equ 10(3):755–773

    MathSciNet  MATH  Google Scholar 

  • Alikhanov AA (2010) A priori estimates for solutions of boundary value problems for fractional-order equations. Differ Equ 46(5):660–666

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Alikhanov AA, Huang C (2021) A high-order L2 type difference scheme for the time-fractional diffusion equation. Appl Math Comput 411:126545

    MathSciNet  MATH  Google Scholar 

  • Alikhanov AA, Asl MS, Huang C (2024a) Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation. Fract Calculus Appl Anal 27(1):102–123

    Article  MathSciNet  MATH  Google Scholar 

  • Alikhanov AA, Asl MS, Huang C, Apekov AM (2024b) Temporal second-order difference schemes for the nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay. Phys D 464:134194

    Article  MathSciNet  MATH  Google Scholar 

  • Alikhanov AA, Asl MS, Li D (2024c) A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application. Comput Math Appl 175:330–344

    Article  MathSciNet  MATH  Google Scholar 

  • Asl MS, Javidi M (2018) Novel algorithms to estimate nonlinear FDEs: applied to fractional order nutrient-phytoplankton-zooplankton system. J Comput Appl Math 339:193–207

    Article  MathSciNet  MATH  Google Scholar 

  • Asl MS, Javidi M (2019) Numerical evaluation of order six for fractional differential equations: stability and convergency. Bull Belgian Math Soc-Simon Stevin 26(2):203–221

    Article  MathSciNet  MATH  Google Scholar 

  • Asl MS, Javidi M, Ahmad B (2019) New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability. J Appl Anal Comput 9(4):1527–1557

    MathSciNet  MATH  Google Scholar 

  • Asl MS, Javidi M, Yan Y (2021) High order algorithms for numerical solution of fractional differential equations. Adv Differ Equ 2021(1):1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Chang YK, Ponce R, Rueda S (2019) Fractional differential equations of Sobolev type with sectorial operators. In: Semigroup Forum, vol. 99, pp. 591–606. Springer

  • Chen H, Stynes M (2019) Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J Sci Comput 79:624–647

    Article  MathSciNet  MATH  Google Scholar 

  • Chen F, Xu Q, Hesthaven JS (2015) A multi-domain spectral method for time-fractional differential equations. J Comput Phys 293:157–172

    Article  MathSciNet  MATH  Google Scholar 

  • Chen R, Liu F, Anh V (2019) Numerical methods and analysis for a multi-term time-space variable-order fractional advection-diffusion equations and applications. J Comput Appl Math 352:437–452

    Article  MathSciNet  MATH  Google Scholar 

  • De Andrade M, Lenzi E, Evangelista L, Mendes R, Malacarne L (2005) Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces. Phys Lett A 347(4–6):160–169

    Article  MATH  Google Scholar 

  • Debbouche A, Nieto JJ (2014) Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls. Appl Math Comput 245:74–85

    MathSciNet  MATH  Google Scholar 

  • Gao GH, Alikhanov AA, Sun ZZ (2017) The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J Sci Comput 73:93–121

    Article  MathSciNet  MATH  Google Scholar 

  • Glöckle WG, Nonnenmacher TF (1995) A fractional calculus approach to self-similar protein dynamics. Biophys J 68(1):46–53

    Article  MATH  Google Scholar 

  • Heydari MH, Avazzadeh Z, Haromi MF (2019) A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation. Appl Math Comput 341:215–228

    MathSciNet  MATH  Google Scholar 

  • Kedia N, Alikhanov AA, Singh VK (2022) Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel. Appl Numer Math 172:546–565

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Kopteva N (2021) Error analysis of an L2-type method on graded meshes for a fractional-order parabolic problem. Math Comput 90(327):19–40

    Article  MathSciNet  MATH  Google Scholar 

  • Kopteva N (2025) Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations. Appl Math Lett 160:109306

    Article  MathSciNet  MATH  Google Scholar 

  • Kopteva N, Meng X (2020) Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions. SIAM J Numer Anal 58(2):1217–1238

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar Y, Singh S, Srivastava N, Singh A, Singh VK (2020) Wavelet approximation scheme for distributed order fractional differential equations. Comput Math Appl 80(8):1985–2017

    Article  MathSciNet  MATH  Google Scholar 

  • Lazopoulos K (2006) Non-local continuum mechanics and fractional calculus. Mech Res Commun 33(6):753–757

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Li S (2021) A fast element-free Galerkin method for the fractional diffusion-wave equation. Appl Math Lett 122:107529

    Article  MathSciNet  MATH  Google Scholar 

  • Liu X, Stynes M (2019) Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem. SIAM J Numer Anal 55:1057–1079

    MATH  Google Scholar 

  • Liu L, Feng L, Xu Q, Zheng L, Liu F (2020) Flow and heat transfer of generalized maxwell fluid over a moving plate with distributed order time fractional constitutive models. Int Commun Heat Mass Transfer 116:104679

    Article  MATH  Google Scholar 

  • Liu Y, Yin X, Liu F, Xin X, Shen Y, Feng L (2022a) An alternating direction implicit Legendre spectral method for simulating a 2D multi-term time-fractional Oldroyd-B fluid type diffusion equation. Comput Math Appl 113:160–173

    Article  MathSciNet  MATH  Google Scholar 

  • Liu F, Zhuang P, Liu Q, Zheng M, Anh VV (2022b) Fractional-order systems, numerical techniques, and applications: finite element and spectral methods for multiterm time-and time-space fractional dynamic systems and applications. In: Fractional order systems, pp. 179–256. Elsevier

  • Lyu P, Liang Y, Wang Z (2020) A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation. Appl Numer Math 151:448–471

    Article  MathSciNet  MATH  Google Scholar 

  • Mahmudov NI, Ahmadova A, Huseynov IT (2022) A novel technique for solving Sobolev-type fractional multi-order evolution equations. Comput Appl Math 41(2):1–35

    Article  MathSciNet  MATH  Google Scholar 

  • Maurya RK, Devi V, Singh VK (2020) Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation. J Comput Appl Math 380:112985

    Article  MathSciNet  MATH  Google Scholar 

  • Maurya RK, Devi V, Singh VK (2021) Stability and convergence of multistep schemes for 1D and 2D fractional model with nonlinear source term. Appl Math Model 89:1721–1746

    Article  MathSciNet  MATH  Google Scholar 

  • Nemati S, Anas AH (2018) Numerical solution of multi-order fractional differential equations using generalized sine-cosine wavelets. Univ J Math Appl 1(4):215–225

    Article  MATH  Google Scholar 

  • Nigmatullin R (1986) The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi (b) 133(1):425–430

    Article  MathSciNet  MATH  Google Scholar 

  • Niu Y, Liu Y, Li H, Liu F (2023) Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media. Math Comput Simul 203:387–407

    Article  MathSciNet  MATH  Google Scholar 

  • Patnaik S, Hollkamp JP, Semperlotti F (2020) Applications of variable-order fractional operators: a review. Proc R Soc A 476(2234):20190498

    Article  MathSciNet  MATH  Google Scholar 

  • Roohi M, Zhang C, Chen Y (2020) Adaptive model-free synchronization of different fractional-order neural networks with an application in cryptography. Nonlinear Dyn 100(4):3979–4001

    Article  MATH  Google Scholar 

  • Roohi M, Mirzajani S, Basse-O’Connor A (2023) A no-chatter single-input finite-time PID sliding mode control technique for stabilization of a class of 4D chaotic fractional-order laser systems. Mathematics 11(21):4463

    Article  MATH  Google Scholar 

  • Roohi M, Zhang C, Taheri M, Basse-O’Connor A (2023) Synchronization of fractional-order delayed neural networks using dynamic-free adaptive sliding mode control. Fractal Fract 7(9):682

    Article  MATH  Google Scholar 

  • Roohi M, Mirzajani S, Haghighi AR, Basse-O’Connor A (2024a) Robust design of two-level non-integer SMC based on deep soft actor-critic for synchronization of chaotic fractional order memristive neural networks. Fractal Fract 8(9):548

    Article  MATH  Google Scholar 

  • Roohi M, Mirzajani S, Haghighi AR, Basse-O’Connor A (2024b) Robust stabilization of fractional-order hybrid optical system using a single-input TS-fuzzy sliding mode control strategy with input nonlinearities. AIMS Math 9:25879–25907

    Article  MathSciNet  MATH  Google Scholar 

  • Shen S, Liu F, Anh VV (2019) The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation. J Comput Appl Math 345:515–534

    Article  MathSciNet  MATH  Google Scholar 

  • Shiralashetti S, Deshi A (2016) An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn 83(1):293–303

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava V, Rai K (2010) A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math Comput Model 51(5–6):616–624

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava N, Singh A, Kumar Y, Singh VK (2021) Efficient numerical algorithms for Riesz-space fractional partial differential equations based on finite difference/operational matrix. Appl Numer Math 161:244–274

    Article  MathSciNet  MATH  Google Scholar 

  • Stynes M, O’Riordan E, Gracia JL (2017) Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J Numer Anal 55(2):1057–1079

    Article  MathSciNet  MATH  Google Scholar 

  • Vijayakumar V, Udhayakumar R, Zhou Y, Sakthivel N (2023) Approximate controllability results for Sobolev-type delay differential system of fractional order without uniqueness. Numer Methods Partial Differ Equ 39(5):3479–3498

    Article  MathSciNet  MATH  Google Scholar 

  • Wang YM, Ren L (2019) A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients. Appl Math Comput 342:71–93

    MathSciNet  MATH  Google Scholar 

  • Zeng F, Li C (2017) A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation. Appl Numer Math 121:82–95

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. North-Caucasus Center for Mathematical Research, North-Caucasus Federal University, Pushkin str. 1, 355017, Stavropol, Russia

    Anatoly A. Alikhanov & Mohammad Shahbazi Asl

  2. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, India

    Poonam Yadav & Vineet Kumar Singh

Authors
  1. Anatoly A. Alikhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Poonam Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vineet Kumar Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohammad Shahbazi Asl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatoly A. Alikhanov.

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

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alikhanov, A.A., Yadav, P., Singh, V.K. et al. A high-order compact difference scheme for the multi-term time-fractional Sobolev-type convection-diffusion equation. Comp. Appl. Math. 44, 115 (2025). https://doi.org/10.1007/s40314-024-03077-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-024-03077-8

Keywords

Mathematics Subject Classification

Search

Navigation