Skip to main content
SpringerLink
Log in

Theta-type convolution quadrature OSC method for nonlocal evolution equations arising in heat conduction with memory

  • Original Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this paper, we propose a robust and simple technique with efficient algorithmic implementation for numerically solving the nonlocal evolution problems. A theta-type (\(\theta \)-type) convolution quadrature rule is derived to approximate the nonlocal integral term in the problem under consideration, such that \(\theta \in (\frac{1}{2},1)\), which remains untreated in the literature. The proposed approaches are based on the \(\theta \) method (\(\frac{1}{2}\le \theta \le 1\)) for the time derivative and the constructed \(\theta \)-type convolution quadrature rule for the fractional integral term. A detailed error analysis of the proposed scheme is provided with respect to the usual convolution kernel and the tempered one. In order to fully discretize our problem, we implement the orthogonal spline collocation (OSC) method with piecewise Hermite bicubic for spatial operators. Stability and error estimates of the proposed \(\theta \)-OSC schemes are discussed. Finally, some numerical experiments are introduced to demonstrate the efficiency of our theoretical findings.

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. Carillo, S., Valente, V., Caffarelli, G.V.: Heat conduction with memory: a singular kernel problem. Evol. Equ. Control. Theory. 3(3), 399 (2014)

    Article  MathSciNet  Google Scholar 

  2. Carillo, S., Giorgi, C.: Non-classical memory kernels in linear viscoelasticity. In: El-Amin, M. (ed.) Viscoelastic and Viscoplastic Materials. IntechOpen, London (2016)

    Google Scholar 

  3. Liemert, A., Sandev, T., Kantz, H.: Generalized Langevin equation with tempered memory kernel. Phys. A: Stat. Mech. 466, 356–369 (2017)

    Article  MathSciNet  Google Scholar 

  4. Stanislavsky, A., Weron, K., Weron, A.: Diffusion and relaxation controlled by tempered \(\alpha \)-stable processes. Phys. Rev. E. 78(5), 051106 (2008)

    Article  MathSciNet  Google Scholar 

  5. Chan, R.H.F., Jin, X.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)

    Book  Google Scholar 

  6. Cuesta, E., Palencia, C.: A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces. Appl. Numer. Math. 45, 139–159 (2003)

    Article  MathSciNet  Google Scholar 

  7. Friedman, A., Shinbrot, M.: Volterra integral equations in Banach space. Trans. Amer. Math. Soc. 26, 131–179 (1967)

    Article  MathSciNet  Google Scholar 

  8. Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. California Monographs in Mathematical Sciences. University of California Press, Berkeley (1958)

    Book  Google Scholar 

  9. Guo, L., Zeng, F., Turner, I., Burrage, K., Karniadakis, G.E.: Efficient multistep methods for tempered fractional calculus: Algorithms and simulations. SIAM J. Sci. Comput. 41, A2510–A2535 (2019)

    Article  MathSciNet  Google Scholar 

  10. Heard, M.L.: An abstract parabolic Volterra integrodifferential equation. SIAM J. Math. Anal. 13, 81–105 (1982)

    Article  MathSciNet  Google Scholar 

  11. López-Marcos, J.C.: A difference scheme for a nonlinear partial integro-differential equation. SIAM J. Numer. Anal. 27, 20–31 (1990)

    Article  MathSciNet  Google Scholar 

  12. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MathSciNet  Google Scholar 

  13. Lubich C., Convolution quadrature and discretized operational calculus, I., Numer. Math. 52, 129–145 (1988)

  14. Nohel, J.A., Shea, D.F.: Frequency domain methods for Volterra equations. Adv. Math. 22, 278–304 (1976)

    Article  MathSciNet  Google Scholar 

  15. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  16. Xu D.: The global behavior of time discretization for an abstract Volterra equation in Hilbert space., Calcolo 34, 71–104 (1997)

  17. Xu, D.: The long-time global behavior of time discretization for fractional order Volterra equations. Calcolo 35, 93–116 (1998)

    Article  MathSciNet  Google Scholar 

  18. Chen, M., Deng W.: Discretized fractional substantial calculus. ESAIM: Math. Mod. Numer. Anal. 49, 373-394 (2015)

  19. Chen, M., Deng, W.: A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation. Appl. Math. Lett. 68, 87–93 (2017)

    Article  MathSciNet  Google Scholar 

  20. Li, C., Deng, W., Zhao, L.: Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. Disc. Contin. Dyn. Syst. Ser. B 24, 1989–2015 (2019)

    Google Scholar 

  21. Zaky, M.A.: Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl. Numer. Math. 145, 429–457 (2019)

    Article  MathSciNet  Google Scholar 

  22. Guo, L., Zeng, F., Turner, I., Burrage, K., Karniadakis, G.E.: Efficient multistep methods for tempered fractional calculus: Algorithms and simulations. SIAM J. Sci. Comput. 41, A2510–A2535 (2019)

    Article  MathSciNet  Google Scholar 

  23. Sultana, F., Singh, D., Pandey, R.K., Zeidan, D.: Numerical schemes for a class of tempered fractional integro-differential equations. Appl. Numer. Math. 157, 110–134 (2020)

    Article  MathSciNet  Google Scholar 

  24. Fernandez, A., Ustaoǧlu, C.: On some analytic properties of tempered fractional calculus. J. Comput. Appl. Math. 366, 112400 (2020)

    Article  MathSciNet  Google Scholar 

  25. Pani, A., Fairweather, G., Fernandes, R.: Orthogonal spline collocation methods for partial integro-differential equations. SIAM. J. Numer. Anal. 30, 248–276 (2010)

    Article  MathSciNet  Google Scholar 

  26. Pani, A., Fairweather, G., Fernandes, R.: Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term. SIAM. J. Numer. Anal. 46, 344–364 (2008)

    Article  MathSciNet  Google Scholar 

  27. Qiu, W.: Optimal error estimate of accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels. Adv. Comput. Math. 49, 43 (2023)

    Article  MathSciNet  Google Scholar 

  28. Qiu, W., Nikan, O., Avazzadeh, Z.: Numerical investigation of generalized tempered-type integrodifferential equations with respect to another function. Fract. Calc. Appl. Anal. 26, 2580–2601 (2023). https://doi.org/10.1007/s13540-023-00198-5

    Article  MathSciNet  Google Scholar 

  29. Qiu, W., Fairweather, G., Yang, X., Zhang, H.: ADI finite element Galerkin methods for two-dimensional tempered fractional integro-differential equations. Calcolo 60, 41 (2023)

    Article  MathSciNet  Google Scholar 

  30. Qiao, L., Xu, D.: A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation. Adv. Comput. Math. 47, 64 (2021)

    Article  MathSciNet  Google Scholar 

  31. Van Bockstal, K., Zaky, M.A., Hendy, A.: On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations. Fract. Calc. Appl. Anal. 26, 2175–2201 (2023). https://doi.org/10.1007/s13540-023-00184-x

    Article  MathSciNet  Google Scholar 

  32. Bialecki, B., Fernandes, R.: An alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations. SIAM J. Numer. Anal. 47, 3429–50 (2009)

    Article  MathSciNet  Google Scholar 

  33. Bialecki, B., Fairweather, G.: Orthogonal spline collocation methods for partial differential equations. J. Comput. Appl. Math. 128, 55–82 (2001)

    Article  MathSciNet  Google Scholar 

  34. Fernandes, R., Fairweather, G.: Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables. Numer. Methods Part. Differ. Equa. 9, 191–211 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. M. A. Zaky

    Present address: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia

  2. A. S. Hendy

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

Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Shanxi, 030006, Taiyuan, People’s Republic of China

    Leijie Qiao

  2. School of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

    Wenlin Qiu

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

    M. A. Zaky

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

    A. S. Hendy

Authors
  1. Leijie Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenlin Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. Zaky
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. S. Hendy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. 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.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, L., Qiu, W., Zaky, M.A. et al. Theta-type convolution quadrature OSC method for nonlocal evolution equations arising in heat conduction with memory. Fract Calc Appl Anal (2024). https://doi.org/10.1007/s13540-024-00265-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13540-024-00265-5

Keywords

Mathematics Subject Classification

Search

Navigation