Approximate solution of nonlinear differential equations with convolution product nonlinearities.

*(English)*Zbl 0932.65143Summary: A new iteration method is proposed to solve nonlinear problems. Special attention is paid to nonlinear differential equations with convolution product nonlinearities. The results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems. Furthermore, the first-order approximations are more accurate than perturbation solutions at high-order approximation.

##### MSC:

65R20 | Numerical methods for integral equations |

45J05 | Integro-ordinary differential equations |

45G10 | Other nonlinear integral equations |

##### Keywords:

nonlinear integro-ordinary differential equations; variational iteration method; Lagrange multiplier method; nonlinear differential equations with convolution product nonlinearities
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\textit{J. He}, Comput. Methods Appl. Mech. Eng. 167, No. 1--2, 69--73 (1998; Zbl 0932.65143)

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##### References:

[1] | He, J.H., A new approach to nonlinear partial differential equations, Comm. nonlinear sci. numer. simul., 2, 4, 230-235, (1997) |

[2] | He, J.H., A variational iteration approach to nonlinear problems and its application, Mech. applic., 20, 1, 30-31, (1998) |

[3] | J.H. He, Variational iteration method: A new approach to nonlinear analytical technique, J. Shanghai Mech. to appear. |

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[6] | Adomian, G.; Roch, R., On the solution of nonlinear differential equations with convolution product nonlinearities, J. math. anal. appl., 114, 171-175, (1986) · Zbl 0588.34004 |

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