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.
Similar content being viewed by others
References
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).
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).
V. K. Dzyadyk, Approximation Methods for the Solution of Differential and Integral Equations [in Russian], Naukova Dumka, Kiev (1988).
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).
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).
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).
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).
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).
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover, New York (1970).
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).
S. N. Bernshtein, Extreme Properties of Polynomials [in Russian], ONTI NKTP SSSR, Moscow-Leningrad (1937).
Additional information
University of Cairo, Giza, Egypt. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11, pp. 1519–1528, November, 1997.
Rights 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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02487509