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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 $Gu(x)=\lambda_{1}\int_{a}^{x}K_{1}(x,\xi)N_{1}(u(\xi))\,d\xi +\lambda_{2}\int_{a}^{b}K_{2}(x,\xi)N_{2}(u(\xi))\,d\xi,$ $$u(x)$$ is the unknown function, $$u(x), \;f(x)\in W^{1}_{2}[a,b], \;N_{1}(\cdot), N_{2}(\cdot)$$ are the continuous nonlinear terms in a reproducing kernel space $$W^{1}_{2}[a,b]$$. Here $$W^{1}_{2}[a,b]$$ 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 $$\| \cdot\| _{W^{1}_{2}[a,b]}$$. The intrinsic merit of the method given in this paper lies in its speedy convergence.

##### MSC:
 45G10 Other nonlinear integral equations 65R20 Numerical methods for integral equations
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##### References:
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