Skip to main content
SpringerLink
Log in

A computational method for solving a problem with parameter for linear systems of integro-differential equations

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

Abstract

This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical 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

Similar content being viewed by others

References

  • Akhmetov MU, Zafer A, Sejilova RD (2002) The control of boundary value problems for quasi lin-ear impulsive integro-differential equations. Nonl Anal 48(2):271–286

    MATH  Google Scholar 

  • Alimhan K, Otsuka N, Adamov AA, Kalimoldayev MN (2015) Global practical output tracking of inherently nonlinear systems using continuously differentiable controllers. Math Probl Eng 2015(932097):1–10

    MathSciNet  MATH  Google Scholar 

  • Atkinson K, Han W, Stewart D (2009) Numerical solution of ordinary differential equations. John Wiley and Sons, Hoboken

    MATH  Google Scholar 

  • Burgos C, Cortes J-C, Debbouche A, Villafuerte L, Villanueva R-J (2019) Random fractional generalized Airy differential equations: a probabilistic analysis using mean square calculus. Appl Math Comput 352:15–29

    MathSciNet  MATH  Google Scholar 

  • Butcher JC (2000) Numerical methods for ordinary differential equations in the 20th century. J Comp App Math 125:1–29

    MathSciNet  MATH  Google Scholar 

  • Dauylbaev MK, Mirzakulova AE (2017) Boundary-value problems with initial jumps for singularly perturbed integrodifferential equations. J Math Sci 222(3):214–225

    MathSciNet  MATH  Google Scholar 

  • Dauylbayev MK, Atakhan N (2015) The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations. Miskolc Math Notes 16(2):747–761

    MathSciNet  MATH  Google Scholar 

  • Dzhumabaev DS (2010) A method for solving the linear boundary value problem for an integro-differential equation. Comput Math Math Phys 50:1150–1161

    MathSciNet  Google Scholar 

  • Dzhumabaev DS (2013) An algorithm for solving the linear boundary value problem for an integro-differential equation. Comput Math Math Phys 53:736–758

    MathSciNet  Google Scholar 

  • Dzhumabaev DS (2015) Necessary and sufficient conditions for the solvability of linear boundary-value problems for the Fredholm integro-differential equation. Ukr Math J 66:1200–1219

    MATH  Google Scholar 

  • Dzhumabaev DS (2016) On one approach to solve the linear boundary value problems for Fredholm integro-differential equations. J Comput Appl Math 294:342–357

    MathSciNet  MATH  Google Scholar 

  • Dzhumabaev DS, Bakirova EA (2013) Criteria for the unique solvability of a linear two-point boundary value problem for systems of integro-differential equations. Diff Equ 49(9):1087–11021

    MathSciNet  MATH  Google Scholar 

  • Dzhumabayev DS (1989) Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. USSR Comp Math Math Phys 29:34–46

    Google Scholar 

  • Harrat A, Nieto JJ, Debbouche A (2018) Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential. J Comput Appl Math 344:725–737

    MathSciNet  MATH  Google Scholar 

  • Hartman P (1964) Ordinary differential equations. Wiley, New York, London, Sydney

    MATH  Google Scholar 

  • Kiguradze, I.T.: Boundary value problems for systems of ordinary differential equations, Sovremennye problem matematiki. Noveishie dostizhenia. (Itogi nauki i techniki. VINITI AN SSSR). Moscow, 30, 3-103. (1987) [in Russ.]

  • Kim H, Sakthivel R, Debbouche A, Torres DFM (2020) Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equations. Chaos, Solitons & Fractals 131:109542

    MathSciNet  Google Scholar 

  • Liu S, Debbouche A, Wang JR (2018) ILC method for solving approximate controllability of fractional differential equations with noninstantaneous impulses. J Comput Appl Math 339:343–355

    MathSciNet  MATH  Google Scholar 

  • Luchka AYu, Nesterenko OB (2008) Projection method for the solution of integro-differential equations with restrictions and control. Nonlinear Oscillations 11(2):219–228

    MathSciNet  MATH  Google Scholar 

  • Manimaran J, Shangerganesh L, Debbouche A, Antonov V (2019) Numerical solutions for time-fractional cancer invasion system with nonlocal diffusion. Front Phys 7:93

    Google Scholar 

  • Minglibayeva BB (2003) Coefficient criteria of unique solvability to linear two-point boundary value problems with parameter. Math J 3(2):55–62 [in Russ.]

    MathSciNet  Google Scholar 

  • Minglibayeva BB, Dzhumabaev DS (2004) Well-posedness of linear two-point boundary value problems with parameter. Math J 4(1):41–51 [in Russ.]

    Google Scholar 

  • Nesterenko OB (2014) Modified projection-iterative method for weakly nonlinear integro differential equations with parameters. J Math Sci 198(3):328–335

    MathSciNet  MATH  Google Scholar 

  • Ronto M, Samoilenko AM (2000) Numerical-analytic methods in the theory of boundary-value problems, - NJ. World Scientific, River Edge

    MATH  Google Scholar 

  • Stoer J, Bulirsch R (2002) Introduction to numerical analysis. Springer Science and Business Media, Berlin

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions which have helped to improve the manuscript.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, 125, Pushkin Str., 050010, Almaty, Kazakhstan

    Anar T. Assanova, Elmira A. Bakirova, Zhazira M. Kadirbayeva & Roza E. Uteshova

  2. Institute of Information and Computational Technologies, 125, Pushkin Str., 050010, Almaty, Kazakhstan

    Anar T. Assanova & Elmira A. Bakirova

  3. International Information Technology University, 34A, Jandossov Str., 050008, Almaty, Kazakhstan

    Zhazira M. Kadirbayeva & Roza E. Uteshova

Authors
  1. Anar T. Assanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elmira A. Bakirova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhazira M. Kadirbayeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Roza E. Uteshova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anar T. Assanova.

Additional information

Communicated by Antonio José Silva Neto.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research is supported by Ministry of Education and Science of Republic Kazakhstan Grant No. AP 05132455. The project leading to this application has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 873071.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Assanova, A.T., Bakirova, E.A., Kadirbayeva, Z.M. et al. A computational method for solving a problem with parameter for linear systems of integro-differential equations. Comp. Appl. Math. 39, 248 (2020). https://doi.org/10.1007/s40314-020-01298-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01298-1

Keywords

Mathematics Subject Classification

Search

Navigation