Convergence of Adomian’s method.

*(English)*Zbl 0697.65051G. Adomian and his collaborators have developed in some papers a new method for solving nonlinear functional equations. The solution of this method is given by a series in which each term is a polynomial adapted to the nonlinearity and determined in a recurrent manner.

In this paper the author proposes a new definition of the technique, in order to use results, as fixed point theorems, for proving convergence of the series and the fact that the sum of the series is the solution of the given equations.

It is easy to see that the technique proposed is powerful and the use of this one is not difficult. The series solution converges with remarkable rapidity under some reasonable assumptions. When these assumptions cannot be true then a modified technique is proposed and so these approaches are better than most of the numerical methods, suggested in the literature, for solving nonlinear problems.

In this paper the author proposes a new definition of the technique, in order to use results, as fixed point theorems, for proving convergence of the series and the fact that the sum of the series is the solution of the given equations.

It is easy to see that the technique proposed is powerful and the use of this one is not difficult. The series solution converges with remarkable rapidity under some reasonable assumptions. When these assumptions cannot be true then a modified technique is proposed and so these approaches are better than most of the numerical methods, suggested in the literature, for solving nonlinear problems.

Reviewer: A.Donescu

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

47J25 | Iterative procedures involving nonlinear operators |

##### Keywords:

Adomian’s method; Hilbert space; nonlinear functional equations; fixed point theorems; convergence; series solution
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##### References:

[1] | DOI: 10.1016/0270-0255(84)90004-6 · Zbl 0556.93005 |

[2] | DOI: 10.1016/0022-247X(85)90102-7 · Zbl 0552.60060 |

[3] | DOI: 10.1016/0022-247X(85)90226-4 · Zbl 0579.60060 |

[4] | Adomian G., Journal of Mathematical Analaysis and Applications 91 (1) (1983) |

[5] | DOI: 10.1016/0096-3003(81)90002-3 · Zbl 0466.65046 |

[6] | Adomian G., Journal of Mathematical Analysis and Applications 91 (1) (1983) |

[7] | DOI: 10.1007/978-94-009-5209-6 |

[8] | DOI: 10.1016/0377-0427(84)90022-0 · Zbl 0549.65034 |

[9] | DOI: 10.1016/0377-0427(84)90013-X · Zbl 0547.65053 |

[10] | Cherruault Y., Optimal Control of Biomedical Systems (1986) |

[11] | DOI: 10.1073/pnas.81.9.2938 · Zbl 0534.92003 |

[12] | Siboney M., Rapport INRIA (1968) |

[13] | Adomian G., Journal of Mathematical Analysis and Applications 111 (1) (1985) |

[14] | DOI: 10.1016/0022-247X(85)90178-7 · Zbl 0606.34009 |

[15] | Saaty T.L., Non-linear Mathematics (1964) · Zbl 0198.00102 |

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