Skip to main content
SpringerLink
Log in

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This article discussed and analyzed a numerical technique based on fractional-order Lagrange polynomials to solve a class of fractional-order non-linear Volterra-Fredholm integro-differential equations. The fractional derivative has been considered of Caputo type. The existence and uniqueness of the continuous solution have been discussed for the given problem. In this approach, first using the Laplace transform, fractional-order Lagrange polynomials operational matrices of fractional integration have been derived. Then using these operational matrices, the continuous problem has been reduced into a system of algebraic equations. The error analysis also has been carried out and an upper error bound estimate for the approximate solution has been given in \(L^2\)-norm. It is also shown that as the number of fractional-order Lagrange polynomials increases, the approximation error approaches to zero rapidly. Further, some numerical examples are discussed to verify the accuracy and efficiency of the proposed numerical technique and to validate 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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Benson, D.A., Meerschaert, M.M., Revielle, J.: Fractional calculus in hydrologic modeling: a numerical perspective. Adv. Water Resour. 51, 479–497 (2013)

    Article  Google Scholar 

  2. Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50(1), 15–67 (1997)

    Article  Google Scholar 

  3. Kumar, S., Gupta, V.: An application of variational iteration method for solving fuzzy time-fractional diffusion equations. Neural Comput. Appl. 33, 17659–17668 (2021)

    Article  Google Scholar 

  4. Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73(1), 5–59 (1996)

    Article  MATH  Google Scholar 

  5. Magin, R.L.: Fractional calculus in bioengineering, part 2. Crit. Rev. Biomed. Eng. 32(2), 105–193 (2004)

    Article  Google Scholar 

  6. Hall, M.G., Barrick, T.R.: From diffusion-weighted MRI to anomalous diffusion imaging. Magn. Reson. Med. 59(3), 447–455 (2008)

    Article  Google Scholar 

  7. Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14(9–10), 1487–1498 (2008)

    Article  Google Scholar 

  8. Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics, arXiv preprint arXiv:1201.0863

  9. Panda, R., Dash, M.: Fractional generalized splines and signal processing. Signal Process. 86(9), 2340–2350 (2006)

    Article  MATH  Google Scholar 

  10. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, 1st Edition, Vol. 204 of North-Holland Mathematics Studies, Elsevier (North-Holland) Science, Amsterdam, (2006)

  11. Podlubny, I.: Fractional Differential Equations, 1st Edition, Vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego (1999)

  12. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York, NY, USA (1993)

    MATH  Google Scholar 

  13. Abel, N.H.: Solution de quelques problemesa laide d’integrales définies, ed. Oeuvres Completes 1, 11–27 (1823)

    Google Scholar 

  14. Singh, S., Kumar, S., Metwali, M.M.A., Aldosary, S.F., Nisar, K.S.: An existence theorem for nonlinear functional Volterra integral equations via Petryshyn’s fixed point theorem. AIMS Math. 7(4), 5594–5604 (2022)

    Article  Google Scholar 

  15. Williams, W.K., Vijaykumar, V., Udhayakumar, R., Panda, S.K., Nisar, K.S.: Existence and controllability of nonlocal mixed Volterra-Fredholm type fractional delay integro-differential equations of order \(1 < r < 2\). Numer. Methods Partial Differ. Equ. (2020). https://doi.org/10.1002/num.22697

    Article  Google Scholar 

  16. Vijaykumar, V., Ravichandran, C., Nisar, K.S., Kucche, K.D.: New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order \(1 < r < 2\). Numer. Partial Differ. Equ. (2021). https://doi.org/10.1002/num.22772

    Article  Google Scholar 

  17. Momani, S., Noor, M.A.: Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 182(1), 754–760 (2006)

    MATH  Google Scholar 

  18. Ray, S.S.: Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1295–1306 (2009)

    Article  MATH  Google Scholar 

  19. Mustafa, M.M., Ghanim, I.N.: Numerical solution of linear Volterra-Fredholm integral equations using Lagrange polynomials. Math. Theory Model. 4(5), 137–146 (2014)

    Google Scholar 

  20. Shahsavaran, A.: Lagrange functions method for solving nonlinear Hammerstein Fredholm-volterra integral equations. Appl. Math. Sci. 5(49), 2443–2450 (2011)

    MATH  Google Scholar 

  21. Saadatmandi, A., Dehghan, M.: A Legendre collocation method for fractional integro-differential equations. J. Vib. Control 17(13), 2050–2058 (2011)

    Article  MATH  Google Scholar 

  22. Das, P., Rana, S., Ramos, H.: Homotopy perturbation method for solving Caputo-type fractional-order Volterra-Fredholm integro-differential equations. Comput. Math. Methods 1(5), e1047 (2019)

    Article  Google Scholar 

  23. Ahsan, S., Nawaz, R., Akbar, M., Nisar, K.S., Abualnaja, K.M., Mahmoud, E.E., Abdel-Aty, A.H.: Numerical solution of two-dimensional fractional order Volterra integro-differential equations. AIP Adv. 11(3), 035232 (2021)

    Article  Google Scholar 

  24. Akbar, M., Nawaz, R., Ahsan, S., Baleanu, D., Nisar, K.S.: Analytical solution of system of Volterra integral equations using OHAM. J. Math. 2020, 8845491 (2020)

    Article  MATH  Google Scholar 

  25. Akbar, M., Nawaz, R., Ahsan, S., Nisar, K.S., Abdel-Aty, A.H., Eleuch, H.: New approach to approximate the solution for the system of fractional order Volterra integro-differential equations. Result Phys. 19, 103453 (2020)

    Article  Google Scholar 

  26. Lepik, Ü.: Solving fractional integral equations by the Haar wavelet method. Appl. Math. Comput. 214(2), 468–478 (2009)

    MATH  Google Scholar 

  27. Saeedi, H., Mollahasani, N., Moghadam, M., Chuev, G.: An operational Haar wavelet method for solving fractional Volterra integral equations. Int. J. Appl. Math. Comput. Sci. 21(3), 535–547 (2011)

    Article  MATH  Google Scholar 

  28. Zhu, L., Fan, Q.: Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2333–2341 (2012)

    Article  MATH  Google Scholar 

  29. Kadalbajoo, M.K., Gupta, V.: Hybrid finite difference methods for solving modified Burgers and Burgers-Huxley equations. Neural Parallel Sci. Comput. 18(3–4), 409–422 (2010)

    MATH  Google Scholar 

  30. Gupta, V., Kadalbajoo, M.K.: A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh. Commun. Nonlinear Sci. Numer. Simul. 16(4), 1825–1844 (2011)

    Article  MATH  Google Scholar 

  31. Gupta, V., Kadalbajoo, M.K.: Qualitative analysis and numerical solution of Burgers’ equation via B-spline collocation with implicit Euler method on piecewise uniform mesh. J. Numer. Math. 24(2), 73–94 (2016)

    Article  MATH  Google Scholar 

  32. Javidi, M., Ahmad, B.: Numerical solution of fractional partial differential equations by numerical Laplace inversion technique. Adv. Differ. Equ. 2013(1), 375 (2013)

    Article  MATH  Google Scholar 

  33. Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)

    MATH  Google Scholar 

  34. Hamoud, A.A., Ghadle, K.P., Issa, G. M Sh. B.: Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations. Int. J. Appl. Math. 31(3), 333–348 (2018)

    Article  MATH  Google Scholar 

  35. Sabermahani, S., Ordokhani, Y., Yousefi, S.: Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations. Comput. Appl. Math. 37(3), 3846–3868 (2018)

    Article  MATH  Google Scholar 

  36. Kreyszig, E.: Introductory functional analysis with applications, vol. 1. Wiley, New York, USA (1978)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the anonymous referees for the valuable suggestions. The authors also acknowledge the DST-FIST program (Govt. of India) for providing the financial support for setting up the computing lab facility under the scheme “Fund for Improvement of Science and Technology” (FIST - No. SR/FST/MS-I/2018/24).

Author information

Authors and Affiliations

  1. Centre for Mathematical and Financial Computing, Department of Mathematics, The LNM Institute of Information Technology, Jaipur, 302031, India

    Saurabh Kumar & Vikas Gupta

Authors
  1. Saurabh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vikas Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vikas Gupta.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Gupta, V. An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations. J. Appl. Math. Comput. 69, 251–272 (2023). https://doi.org/10.1007/s12190-022-01743-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-022-01743-w

Keywords

Search

Navigation