Skip to main content
SpringerLink
Log in

Analytical and numerical solution techniques for a class of time-fractional integro-partial differential equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

This article investigates the analytical and numerical solutions of a class of non-autonomous time-fractional integro-partial differential initial-boundary-value problems (IBVPs) with fractional derivative of Caputo-type. The existence and uniqueness of the analytical solution of the IBVP are established by using the Sumudu decomposition method and the maximum-minimum principle, respectively. To obtain the numerical solution, first, we semi-discretize the IBVP by discretizing the time fractional derivative by using the L1-scheme and the integral term by using the trapezoidal rule on a graded mesh, and then we approximate the spatial derivatives by using the cubic spline method over a uniform mesh. The stability and convergence analysis of the numerical method are established. The performance of the proposed technique is validated through numerical experiments, and the results are compared with the method presented in Santra and Mohapatra (J. Comput. Appl Math. 400, 113746, 13, 2022).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Abdelkawy, M.A., Amin, A.Z.M., Lopes, A.M.: Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations. Comput. Appl. Math. 41(1), 2,21 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adomian, G.: Solving Frontier Problems of Physics: the Decomposition Method, vol. 60 of Fundamental Theories of Physics. Kluwer Academic Publishers Group, Dordrecht (1994)

    MATH  Google Scholar 

  3. Caballero, J., Mingarelli, A.B., Sadarangani, K.: Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer. Elect. J. Differ. Eq. 2006(57), 1–11 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Cherruault, Y.: Convergence of Adomian’s method. Kybernetes 18(2), 31–38 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dabas, J., Chauhan, A.: Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay. Math. Comput. Modelling 57(3-4), 754–763 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diagana, T.: Semilinear Evolution Equations and Their Applications. Springer, Cham (2018)

    Book  MATH  Google Scholar 

  7. Graef, J. R., Tunç, C., Şevli, H.: Razumikhin qualitative analyses of Volterra integro-fractional delay differential equation with Caputo derivatives. Commun. Nonlinear Sci. Numer Simul 103, 106037, 12 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hu, Y., Luo, Y., Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. Comput. Appl Math. 215(1), 220–229 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Katatbeh, Q.D., Belgacem, F.B.M.: Applications of the Sumudu transform to fractional differential equations. Nonlinear Stud. 18(1), 99–112 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351(1), 218–223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ma, X., Huang, C.: An accurate Legendre collocation method for third-kind Volterra integro-differential equations with non-smooth solutions. Numer Algorithms 88(4), 1571–1593 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Qiu, W., Xu, D., Guo, J.: The Crank-Nicolson-type sinc-Galerkin method for the fourth-order partial integro-differential equation with a weakly singular kernel. Appl. Numer. Math. 159, 239–258 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rahman, M.: Integral Equations and Their Applications. WIT Press, Southampton (2007)

    MATH  Google Scholar 

  14. Santra, S., Mohapatra, J.: A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type. J. Comput. Appl Math. 400, 113746, 13 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. Saqib, M., Khan, I., Shafie, S.: Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms. Adv. Difference Equ. 18, 52 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sloan, I. H.: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer Anal. 23(5), 1052–1061 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Vyawahare, V.A., Nataraj, P.S.V.: Fractional-order modeling of neutron transport in a nuclear reactor. Appl. Math Model. 37(23), 9747–9767 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, T., Qin, M., Zhang, Z.: The Puiseux expansion and numerical integration to nonlinear weakly singular Volterra integral equations of the second kind. J. Sci. Comput. 82(3), 64,28 (2020)

    Article  MathSciNet  Google Scholar 

  20. Watugala, G.K.: Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Internat. J. Math. Ed. Sci. Tech 24(1), 35–43 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wazwaz, A.-M.: Linear and Nonlinear Integral Equations. Higher Education Press. Springer, Heidelberg (2011)

    Book  Google Scholar 

  22. Zhang, P., Hao, X.: Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces. Adv. Difference Equ. 7, 247 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to acknowledge the referee for his/her valuable comments and suggestions, which helped to improve the article. Furthermore, the first author would like to acknowledge the fellowship and amenities support provided by IIT Guwahati in his research.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Sandip Maji & Srinivasan Natesan

Authors
  1. Sandip Maji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Srinivasan Natesan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

S.N. provided the methodology along with the model problem and re-editing and correcting the manuscript. S.M. implemented the scheme and obtained the error analysis and the numerical experiments and writing the manuscript.

Corresponding author

Correspondence to Srinivasan Natesan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Ethical approval

Not applicable.

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

Maji, S., Natesan, S. Analytical and numerical solution techniques for a class of time-fractional integro-partial differential equations. Numer Algor 94, 229–256 (2023). https://doi.org/10.1007/s11075-023-01498-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-023-01498-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation