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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

$u\left(x\right)=f\left(x\right)+Gu\left(x\right),$

where

$Gu\left(x\right)={\lambda }_{1}{\int }_{a}^{x}{K}_{1}\left(x,\xi \right){N}_{1}\left(u\left(\xi \right)\right)\phantom{\rule{0.166667em}{0ex}}d\xi +{\lambda }_{2}{\int }_{a}^{b}{K}_{2}\left(x,\xi \right){N}_{2}\left(u\left(\xi \right)\right)\phantom{\rule{0.166667em}{0ex}}d\xi ,$

$u\left(x\right)$ is the unknown function, $u\left(x\right),\phantom{\rule{4pt}{0ex}}f\left(x\right)\in {W}_{2}^{1}\left[a,b\right],\phantom{\rule{4pt}{0ex}}{N}_{1}\left(·\right),{N}_{2}\left(·\right)$ are the continuous nonlinear terms in a reproducing kernel space ${W}_{2}^{1}\left[a,b\right]$. Here ${W}_{2}^{1}\left[a,b\right]$ is the space of absolutely continuous functions whose first derivative belongs of ${L}^{2}\left[a,b\right]$. 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 ${\parallel ·\parallel }_{{W}_{2}^{1}\left[a,b\right]}$. The intrinsic merit of the method given in this paper lies in its speedy convergence.

##### MSC:
 45G10 Nonsingular nonlinear integral equations 65R20 Integral equations (numerical methods)