## New results for convergence of Adomian’s method applied to integral equations.(English)Zbl 0756.65083

Consider the general functional equation (1) $$y-N(y)=f$$, where $$N$$ is a nonlinear operator from a Hilbert space $$H$$ into itself and $$f$$ is a given element in $$H$$. The paper presents a new convergence proof of Adomian’s method which consists in solving (1) by means of representing $$y$$ as an infinite series. Some applications of this technique to nonlinear integral equations are given.

### MSC:

 65J15 Numerical solutions to equations with nonlinear operators 65R20 Numerical methods for integral equations 45G10 Other nonlinear integral equations 47J25 Iterative procedures involving nonlinear operators
Full Text:

### References:

 [1] Adomian, G., Nonlinear Stochastic Systems Theory and Applications to Physics (1989), Kluwer · Zbl 0659.93003 [3] Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Math. Comput. Modelling, 13, 7, 17-43 (1990) · Zbl 0713.65051 [4] Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 2, 31-38 (1989) · Zbl 0697.65051 [5] Goursat, E., Cours d’Analyse Mathématique, Tome I (1902), Gauthier-Villars: Gauthier-Villars Paris · JFM 46.0375.13 [6] Jaggi, S.; Caron, A., The solution of an integral equation with its application to the human joint, Int. J. Biomed. Comput., 14, 3, 249-262 (1983) [7] Cherruault, Y., New deterministic methods for global optimization and applications to biomedicine, Int. J. Biomed. Comput., 27, 3/4, 215-229 (1991) [8] Cherruault, Y., Mathematical modelling in biomedicine, Optimal Control of Biomedical Systems (1986), Reidel Kluwer [9] Kumar, S., The numerical solution of Hammerstein equations by a method based on polynomial collocation, J. Austral. Soc. Ser. B, 31, 319-329 (1990) · Zbl 0707.65101 [10] Caron, A., Méthode de résolution numérique d’équations intégrales non linéaires à noyaux réguliers par un schéma itératif sous contrainte de régularité, C.R.A.S., 649-652 (30.3.1981), Serie I
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.