Collocation Method for Nonlinear Volterra-Fredholm Integral Equations


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.

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J. Shali, P. Darania and A. Akbarfam, "Collocation Method for Nonlinear Volterra-Fredholm Integral Equations," Open Journal of Applied Sciences, Vol. 2 No. 2, 2012, pp. 115-121. doi: 10.4236/ojapps.2012.22016.

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The authors declare no conflicts of interest.


[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.

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