Skip to main content
SpringerLink
Log in

An Iterative Scheme for Solving Arbitrary-Order Nonlinear Volterra Integro-Differential Equations Involving Delay

  • Research Paper
  • Published:
Iranian Journal of Science Aims and scope Submit manuscript

Abstract

This paper introduces an iterative-based numerical scheme for solving nonlinear fractional-order Volterra integro-differential equations involving delay. Additionally, we provide sufficient conditions for the existence and uniqueness of the solution. The composite trapezoidal rule is applied to approximate the integral involved in the equation, followed by discretizing the Caputo fractional derivative operator of arbitrary order \(\alpha \in (0,1)\) by using the classical L1 scheme. Further, the Daftardar-Gejji and Jafari method is employed to solve the implicit algebraic equation. The convergence analysis and error bounds of the proposed scheme are presented. It is shown that the approximate solution converges to the exact solution with order \(( 2-\alpha ).\) We illustrate the efficacy and applicability of the proposed method through a couple of examples.

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

References

  • Agarwal RP, de Andrade B, Siracusa G (2011) On fractional integro-differential equations with state-dependent delay. Comput Math Appl 62(3):1143–1149

    Article  MathSciNet  MATH  Google Scholar 

  • Bellen A, Zennaro M (2003) Numerical methods for delay differential equations. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  • Bohner M, Tunç O, Tunç C (2021) Qualitative analysis of Caputo fractional integro-differential equations with constant delays. Comput Appl Math 40(6):1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Daftardar-Gejji V, Jafari H (2006) An iterative method for solving nonlinear functional equations. J Math Anal Appl 316(2):753–763

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh B, Mohapatra J (2022) Analysis of finite difference schemes for volterra integro-differential equations involving arbitrary order derivatives. J Appl Math Comput. https://doi.org/10.1007/s12190-022-01817-9

    Article  Google Scholar 

  • Diethelm K (2010) The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  • Dos Santos JPC, Arjunan MM, Cuevas C (2011) Existence results for fractional neutral integro-differential equations with state-dependent delay. Comput Math Appl 62(3):1275–1283

    Article  MathSciNet  MATH  Google Scholar 

  • Erfanian M, Gachpazan M, Beiglo H (2017) A new sequential approach for solving the integro-differential equation via Haar wavelet bases. Comput Math Math Phys 57:297–305. https://doi.org/10.1134/S096554251702004X

    Article  MathSciNet  MATH  Google Scholar 

  • Jothimani K, Valliammal N, Ravichandran C (2018) Existence result for a neutral fractional integrodifferential equation with state dependent delay. J Appl Nonlinear Dyn 7:371–381

    Article  MathSciNet  MATH  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, San Diego. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Lin Y, Xu C (2007) Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225(2):1533–1552

    Article  MathSciNet  MATH  Google Scholar 

  • Linz P (1985) Analytical and numerical methods for volterra equations. SIAM, Bangkok

    Book  MATH  Google Scholar 

  • Ma X, Huang C (2013) Numerical solution of fractional integro-differential equations by a hybrid collocation method. Appl Math Comput 219(12):6750–6760

    MathSciNet  MATH  Google Scholar 

  • Mansouri L, Azimzadeh Z (2022) Numerical solution of fractional delay Volterra integro-differential equations by Bernstein polynomials, Math Sci 1–12

  • Mohamed EM, Raslan KR, Ali KK, Abd El Salam MA (2020) On general form of fractional delay integro-differential equations. Arab J Basic Appl Sci 27(1):313–323

    Article  Google Scholar 

  • Nemati S, Lima PM, Sedaghat S (2020) Legendre wavelet collocation method combined with the Gauss-Jacobi quadrature for solving fractional delay-type integro-differential equations. Appl Numer Math 149:99–112

    Article  MathSciNet  MATH  Google Scholar 

  • Panda A, Santra S, Mohapatra J (2022) Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations. J Appl Math Comput 68(3):2065–2082

    Article  MathSciNet  MATH  Google Scholar 

  • Peykrayegan N, Ghovatmand M, Skandari MHN (2021) An efficient method for linear fractional delay integro-differential equations. Comput Appl Math 40(7):1–33

    Article  MathSciNet  MATH  Google Scholar 

  • Peykrayegan N, Ghovatmand M, Skandari MHN, Baleanu D (2022) An approximate approach for fractional singular delay integro-differential equations. AIMS Math 7(5):9156–9171

    Article  MathSciNet  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academie Press, New York

    MATH  Google Scholar 

  • Ren Y, Qin Y, Sakthivel R (2010) Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integr Equ Oper Theory 67:33–49

    Article  MathSciNet  MATH  Google Scholar 

  • Rezabeyk S, Abbasbandy S, Shivanian E (2020) Solving fractional-order delay integro-differential equations using operational matrix based on fractional-order Euler polynomials. Math Sci 14(2):97–107

    Article  MathSciNet  MATH  Google Scholar 

  • Rihan FA, Doha EH, Hassan MI, Kamel N (2009) Numerical treatments for Volterra delay integro-differential equations. Comput Appl Math 9(3):292–318

    MathSciNet  MATH  Google Scholar 

  • Saeedi H, Moghadam MM, Mollahasani N, Chuev G (2011) A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Commun Nonlinear Sci Numer Simulat 16(3):1154–1163

    Article  MathSciNet  MATH  Google Scholar 

  • Santra S, Mohapatra J (2021) Numerical analysis of Volterra integro-differential equations with Caputo fractional derivative. Iran J Sci Technol Trans Sci 45:1815–1824

    Article  MathSciNet  Google Scholar 

  • Shahmorad S, Ostadzad MH, Baleanu D (2020) A Tau-like numerical method for solving fractional delay integro-differential equations. Appl Numer Math 151:322–336

    Article  MathSciNet  MATH  Google Scholar 

  • Smith HL (2011) An introduction to delay differential equations with applications to the life sciences. Springer, New York

    Book  MATH  Google Scholar 

  • Tarasov VE (2010) Fractional dynamics: application of fractional calculus to dynamics of particles. Fields and Media, Springer, Science and Business Media, Heidelberg

    Book  MATH  Google Scholar 

  • Tunç O, Tunç C (2023) Solution estimates to Caputo proportional fractional derivative delay integro-differential equations. Rev Real Acad Cienc Exactas Fis Nat Ser A Mat 117:12. https://doi.org/10.1007/s13398-022-01345-y

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao J, Cao Y, Xu Y (2017) Sinc numerical solution for pantograph Volterra delay-integro-differential equation. Int J Comput Math 94(5):853–865

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author would like to thank the Council of Scientific & Industrial Research (CSIR), Government of India (File No.: 09/983(0046)/2020-EMR-I), for financial support to carry out his research work at NIT Rourkela.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

    Bappa Ghosh & Jugal Mohapatra

Authors
  1. Bappa Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jugal Mohapatra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jugal Mohapatra.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Ethical approval

N/A.

Informed consent

On behalf of the authors, Dr. Jugal Mohapatra shall be communicating the manuscript.

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

Ghosh, B., Mohapatra, J. An Iterative Scheme for Solving Arbitrary-Order Nonlinear Volterra Integro-Differential Equations Involving Delay. Iran J Sci 47, 851–861 (2023). https://doi.org/10.1007/s40995-023-01446-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-023-01446-2

Keywords

MSC Classification

Search

Navigation