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A survey of numerical methods for solving nonlinear integral equations. (English) Zbl 0760.65118
The author gives a survey of numerical methods for solving nonlinear integral equations of the second kind such as the following: $x(t)=y(t)+\int_ D K(t,s)f(s,x(s))ds,\quad t\in D.$ Projection methods (such as Nyström technique and iterated projection method) are described and convergence results are given. Also Galerkin’s method is detailed and the concrete example $x(t)=y(t)+\int^ 1_ 0(t+s+x(s))^{-1}ds,\quad 0\leq t\leq 1$ is solved numerically.
We also find a discrete collocation method, a discrete Galerkin method and grid methods, but I did not find a description of a new and powerful method for solving nonlinear functional equations: Adomian’s method. This technique is a decomposition method giving the exact solution as a series of functions. Practically one uses a truncated series and thus an approximated solution [see Ph.D. thesis of Lionel Gabet, Ecole Centrale de Paris (1992)].

##### MSC:
 65R20 Numerical methods for integral equations 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 45G10 Other nonlinear integral equations
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##### References:
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