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Zbl 1110.45005
Cui, Minggen; Du, Hong
Representation of exact solution for the nonlinear Volterra-Fredholm integral equations. (English)
[J] Appl. Math. Comput. 182, No. 2, 1795-1802 (2006). ISSN 0096-3003

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 $\Vert \cdot\Vert _{W^{1}_{2}[a,b]}$. The intrinsic merit of the method given in this paper lies in its speedy convergence.
[Mouffak Benchohra (Sidi Bel Abbes)]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
65R20 Integral equations (numerical methods)

Keywords: series solution; nonlinear Volterra-Fredholm integral equation; reproducing kernel space; exact solution; numerical approximate; convergence

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