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The theoretical foundation of the Adomian method. (English) Zbl 0805.65056

This paper deals with decomposition methods first introduced by G. Adomian. The author tries to find a theoretical foundation of the method but he uses some operations in Banach spaces which are not valid and have to be precisely defined and studied. Nevertheless the consequences of this study are interesting and allow to understand better why Adomian’s method converges.
More useful and practical theorems of convergence have been proved since the publication of this paper [see e.g. K. Abbaoui and Y. Cherruault, Comput. Math. Appl. 28, No. 5, 103-109 (1994)].

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
65R20 Numerical methods for integral equations
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References:

[1] Adomian, G., Stochastic systems, (1983), Academic Press · Zbl 0504.60067
[2] Adomian, G., Nonlinear stochastic operator equations, (1986), Academic Press · Zbl 0614.35013
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[12] Cherruault, Y.; Saccomandi, G.; Somé, B., New results for convergence of Adomian’s method applied to integral equations, Mathl. comput. modelling, 16, 2, 83-93, (1992) · Zbl 0756.65083
[13] Cartan, H., Théorie elémentaire des fonctions analytiques, (1985), Hermann
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