Skip to main content
SpringerLink
Log in

On the approximate solution of nonlinear Volterra-Fredholm integral equations on a complex domain by Dzyadyk’s method

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

In 1980–1984, V. K. Dzyadyk suggested and modified an iterative approximation method (IA-method) for numerical solution of the Cauchy problem y′=f(x,y), y(x 0)=x0. Particular cases of nonlinear mixed Volterra-Fredholm integral equations of the second kind arise in the mathematical simulation of the space-time development of an epidemic. This paper is concerned with the approximate solution of integral equations of this type by the Dzyadyk method on complex domains. Finally, we test this method numerically by four different 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

Similar content being viewed by others

References

  1. V. K. Dzyadyk, “Polynomial approximation to the solution of the Cauchy and Goursat problems with application,” Coll. Math. Soc. J. Bolyai. Fun. Sek. Oper., No. 35, 441–448 (1980).

    MathSciNet  Google Scholar 

  2. V. K. Dzyadyk, Approximation-Iterative Method for the Approximate Solution of the Cauchy Problem for Ordinary Differential Equations [in Russian], Preprint No. 27.84, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984).

    Google Scholar 

  3. V. K. Dzyadyk, Approximation Methods for the Solution of Differential and Integral Equations [in Russian], Naukova Dumka, Kiev (1988).

    Google Scholar 

  4. S. F. Karpenko, Application of the Approximation-Iterative Method to the Approximation of Solutions of Integral Equations of Certain Types [in Russian], Preprint No. 21.85. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985).

    Google Scholar 

  5. V. K. Dzyadyk and Yu. I. Romanenko, Approximation-Iterative Method for Polynomial Approximation of Solutions of a Nonlinear Cauchy Problem for Equations of Hyperbolic Type [in Russian], Preprint No. 63.86, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986).

    Google Scholar 

  6. A. M. Bassov, Application of the Approximation-Iterative Method to the Solution of Boundary-Value Problems for Second Order Ordinary Differential Equations [in Russian], Preprint No. 13.89, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1989).

    Google Scholar 

  7. V. K. Dzyadyk, A. M. Bassov, and M. M. Rizk, Theory and Application of Approximation-Iterative Method and Its Comparison with Methods of Runge-Kutta Type [in Russian], Preprint No. 39.91, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991).

    Google Scholar 

  8. V. K. Dzyadyk and Ya. F. Vassilenko, Application of the Approximation-Iterative Method to the Solution of Stiff Problems for Ordinary Differential Equations [in Russian], Preprint No. 55.91, Institute of Mathematics. Ukrainian Academy of Sciences, Kiev (1991).

    Google Scholar 

  9. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover, New York (1970).

    MATH  Google Scholar 

  10. Ya. F. Vassilenko, Application of the Approximation-Iterative Method to the Solution of Implicit Ordinary Differential Equations [in Russian] Preprint No. 39.91, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1988).

    Google Scholar 

  11. S. N. Bernshtein, Extreme Properties of Polynomials [in Russian], ONTI NKTP SSSR, Moscow-Leningrad (1937).

    Google Scholar 

Download references

Authors
  1. M. M. Rizk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. L. Zaher
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

University of Cairo, Giza, Egypt. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11, pp. 1519–1528, November, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rizk, M.M., Zaher, S.L. On the approximate solution of nonlinear Volterra-Fredholm integral equations on a complex domain by Dzyadyk’s method. Ukr Math J 49, 1705–1717 (1997). https://doi.org/10.1007/BF02487509

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487509

Keywords

Search

Navigation