Collocation Method for Nonlinear Volterra-Fredholm Integral Equations

Jafar Ahmadi Shali, Parviz Darania, Ali Asgar Jodayree Akbarfam

Department of Mathematics and Computer Science, University of Tabriz, Tabriz, Iran.
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.

DOI: 10.4236/ojapps.2012.22016                

Abstract

A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of Volterra- Fredholm integral equations.

Keywords

Collocation Method; Nonlinear Volterra-Fredholm Integral Equations; Convergence Analysis; Chelyshkov Polynomials

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	V. S. Chelyshkov, “Alternative Orthogonal Polynomials and Quadratures,” Electronic Transactions on Numerical Analysis, Vol. 25, No. 7, 2006, pp. 17-26.
[2]	H. Brunner, “Collocation Methods for Volterra Integral and Related Functional Equations (Cambridge Monographs on Applied and Computational Mathematics),” Vol. 15, Cambridge University Press, Cambridge, 2004.
[3]	H. Brunner, “Implicitly Linear Collocation Methods for Nonlinear Volterra Equations,” Applied Numerical Mathematics, Vol. 9, No. 3-5, 1992, pp. 235-247. doi:10.1016/0168-9274(92)90018-9
[4]	H. Brunner, “High-Order Collocation Methods for Singular Volterra Functional Equations of Neutral Type,” Applied Numerical Mathematics, Vol. 57, No. 5-7, 2007, pp. 533-548. doi:10.1016/j.apnum.2006.07.006
[5]	H. Brunner, “The Numerical Solution of Weakly Singular Volterra Functional Integro-Differential Equations with Variable Delays,” Communications on Pure and Applied Analysis, Vol. 5, No. 2, 2006, pp. 261-276. doi:10.3934/cpaa.2006.5.261
[6]	A. T. Diogo, S. McKee and T. Tang, “A Hermite-Type Collocation Method for the Solution of an Integral Equation with a Certain Weakly Singular Kernel,” IMA Journal of Numerical Analysis, Vol. 11, No. 4, 1991, pp. 595605. doi:10.1093/imanum/11.4.595
[7]	K. E. Atkinson and W. Han, “Theoretical Numerical Analysis,” Springer, Berlin, 2009.