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)].


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


[1] Adomian, G., Stochastic systems, (1983), Academic Press · Zbl 0504.60067
[2] Adomian, G., Nonlinear stochastic operator equations, (1986), Academic Press · Zbl 0614.35013
[3] Bellman, R.E.; Adomian, G., Partial differential equations—new methods for their treatment and application, (1986), Reidel
[4] Adomian, G., Nonlinear stochastic systems theory and applications to physics, (1989), Kluwer · Zbl 0659.93003
[5] Adomian, G.; Rach, R., Nonlinear stochastic differential-delay equations, J. math. anal. and applic., 91, 1, (1983) · Zbl 0504.60067
[6] Adomian, G.; Rach, R., Polynomial nonlinearities in differential equations, J. math. anal. and applic., 109, 1, (1985) · Zbl 0606.34009
[7] Adomian, G.; Rach, R., Algebraic computation and the decomposition method, Kybernetes, 15, 1, (1986) · Zbl 0604.60064
[8] Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Mathl. comput. modelling, 13, 7, 17-43, (1990) · Zbl 0713.65051
[9] Adomian, G., An analytical solution of the stochastic Navier-Stokes system, Foundations of physics, 21, 7, 831-843, (1991)
[10] Adomian, G., An efficient methodology for the physical sciences, Kybernetes, 20, 7, 24-34, (1991) · Zbl 0744.65039
[11] Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 2, 31-38, (1989) · Zbl 0697.65051
[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
[14] Ramis, E.; Deschamps, C.; Odoux, J., Cours de mathématiques spéciales, (1977), Masson, tome 4 · Zbl 0471.00003
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.