## 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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