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.


45G10 Other nonlinear integral equations
65R20 Numerical methods for integral equations
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