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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.

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)

### 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
Full Text:
DOI

### 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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