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An efficient methodology for the physical sciences. (English) Zbl 0744.65039
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.

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
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References:
[1] DOI: 10.1016/0895-7177(90)90125-7 · Zbl 0713.65051
[2] Adomian G., Non-linear Stochastic Operator Equations (1986) · Zbl 0609.60072
[3] Cherrault Y., Kybemetes 18
[4] DOI: 10.1007/BFb0041214
[5] DOI: 10.1007/BF00733348
[6] DOI: 10.2514/3.9904
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