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Representation of exact solution for the nonlinear Volterra-Fredholm integral equations. (English) Zbl 1110.45005

This paper is concerned with the existence of the exact solution of the following nonlinear Volterra-Fredholm integral equation

u(x)=f(x)+Gu(x),

where

Gu(x)=λ 1 a x K 1 (x,ξ)N 1 (u(ξ))dξ+λ 2 a b K 2 (x,ξ)N 2 (u(ξ))dξ,

u(x) is the unknown function, u(x),f(x)W 2 1 [a,b],N 1 (·),N 2 (·) are the continuous nonlinear terms in a reproducing kernel space W 2 1 [a,b]. Here W 2 1 [a,b] is the space of absolutely continuous functions whose first derivative belongs of L 2 [a,b]. The exact solution is given by the form of series. Its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained. The error of the approximate solution is monotonously decreasing in the sense of · W 2 1 [a,b] . The intrinsic merit of the method given in this paper lies in its speedy convergence.

MSC:
45G10Nonsingular nonlinear integral equations
65R20Integral equations (numerical methods)