A reliable modification of Adomian decomposition method.

*(English)*Zbl 0928.65083The purpose of this paper is to show that, although the modified technique needs only a slight variation from the standard Adomian method, the results are improved and the convergence of the series solution is accelerated. Some illustrative examples are treated proving the performance of the modified algorithms.

This interesting paper has some relationships with results previously published by K. Abbaoui and Y. Cherruault [cf. Comput. Math. Appl. 28, No. 5, 103-109 (1995; Zbl 0809.65073)]. These works proved that when the Adomian method did not converge a change in the first term of the series solution could involve the convergence of the technique. Furthermore, it has also be proved that the convergence was accelerated by modifying the choice of the first term.

The author also asserts that his method minimizes the size of calculation needed. This is possible but not proved in this paper. May be that a good choice of the decomposition \(f= f_1+ f_2\) could minimize the calculations size. It would be an interesting following of the present paper.

This interesting paper has some relationships with results previously published by K. Abbaoui and Y. Cherruault [cf. Comput. Math. Appl. 28, No. 5, 103-109 (1995; Zbl 0809.65073)]. These works proved that when the Adomian method did not converge a change in the first term of the series solution could involve the convergence of the technique. Furthermore, it has also be proved that the convergence was accelerated by modifying the choice of the first term.

The author also asserts that his method minimizes the size of calculation needed. This is possible but not proved in this paper. May be that a good choice of the decomposition \(f= f_1+ f_2\) could minimize the calculations size. It would be an interesting following of the present paper.

Reviewer: Yves Cherruault (Paris)

##### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

##### Keywords:

Adomian decomposition method; accelerated convergence; noise terms; nonlinear differential and integral equations; numerical examples
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\textit{A.-M. Wazwaz}, Appl. Math. Comput. 102, No. 1, 77--86 (1999; Zbl 0928.65083)

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##### References:

[1] | G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994 · Zbl 0802.65122 |

[2] | Adomian, G., A review of the decomposition method and some recent results for nonlinear equation, Math. comput. model., 13, 7, 17-43, (1992) · Zbl 0713.65051 |

[3] | Adomian, G.; Rach, R., Noise terms in decomposition series solution, Comput. math. appl., 24, 11, 61-64, (1992) · Zbl 0777.35018 |

[4] | Cherruault, Y.; Saccomandi, G.; Some, B., New results for convergence of Adomian’s method applied to integral equations, Math. comput. model., 16, 2, 85-93, (1992) · Zbl 0756.65083 |

[5] | Wazwaz, A.M., Necessary conditions for the appearance of noise terms in decomposition series, Appl. math. comput., 81, 265-274, (1997) · Zbl 0882.65132 |

[6] | A.M. Wazwaz, A First Course In Integral Equations, World Scientific, Singapore, 1997 · Zbl 0924.45001 |

[7] | Wazwaz, A.M., A reliable technique for solving the weakly singular second-kind Volterra-type integral equations, Appl. math. comput., 80, 287-299, (1996) · Zbl 0880.65122 |

[8] | Wazwaz, A.M., On the solution of the fourth order parabolic equation by the decomposition method, Int. J. computer math., 57, 213-217, (1995) · Zbl 1017.65518 |

[9] | E. Yee, Application of the decomposition method to the solution of the reaction-convection-diffusion equation, Appl. Math. Comput. 56 (1993) 1-27 · Zbl 0773.76055 |

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