Adomian, G.; Rach, R. C.; Meyers, R. E. An efficient methodology for the physical sciences. (English) Zbl 0744.65039 Kybernetes 20, No. 7, 24-34 (1991). This very interesting paper gives a new decomposition method for solving nonlinear functional equations of different kinds (integral, differential, partial differential, …).Considering the general equation \(u-N(u)=f\) where \(N\) is a general nonlinear operator acting on a Hilbert or Banach space, the method consists in searching the exact solution under the form \(u=\sum^ \infty_{i=0}u_ i\) and to develop \(N(u)\) such as \(N(u)=\sum^ \infty_{i=0}A_ i\) where the \(A_ i\)’s are special polynomials ( Adomian’s polynomials) which can be generally easily calculated.The \(A_ i\)’s depend only on \(u_ 0,\dots,u_ i\). The series’ terms are calculated by means of the relations \(u_ 0=f\), \(u_ 1=A_ 0,\dots,u_ i=A_{i-1},\dots\). The authors give applications of this technique to ordinary differential equations ( Schrödinger equation with a generalized potential) and to partial differential equations. Reviewer: Y.Cherruault (Paris) Cited in 1 ReviewCited in 28 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65Z05 Applications to the sciences 65L05 Numerical methods for initial value problems involving ordinary differential equations 35G05 Linear higher-order PDEs 34A34 Nonlinear ordinary differential equations and systems 47J25 Iterative procedures involving nonlinear operators Keywords:Hilbert space; decomposition method; nonlinear functional equations; nonlinear operator; Banach space; Adomian’s polynomials; Schrödinger equation PDFBibTeX XMLCite \textit{G. Adomian} et al., Kybernetes 20, No. 7, 24--34 (1991; Zbl 0744.65039) Full Text: DOI References: [1] DOI: 10.1016/0895-7177(90)90125-7 · Zbl 0713.65051 · doi:10.1016/0895-7177(90)90125-7 [2] Adomian G., Non-linear Stochastic Operator Equations (1986) · Zbl 0609.60072 [3] Cherrault Y., Kybemetes 18 [4] DOI: 10.1007/BFb0041214 · doi:10.1007/BFb0041214 [5] DOI: 10.1007/BF00733348 · doi:10.1007/BF00733348 [6] DOI: 10.2514/3.9904 · doi:10.2514/3.9904 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.