Numerical solution for high order differential equations using a hybrid neural network-optimization method.

*(English)*Zbl 1105.65340Summary: This paper reports a novel hybrid method based on optimization techniques and neural networks methods for the solution of high order ordinary differential equations. Here neural networks is considered as a part of large field called neural computing or soft computing. This means that we propose a new solution method for the approximated solution of high order ordinary differential equations using innovative mathematical tools and neural-like systems of computation. This hybrid method can result in improved numerical methods for solving initial/boundary value problems, without using preassigned discretisation points. The mixture of feed forward neural networks and optimization techniques, based on Nelder-Mead method is used to introduce the close analytic form of the solution for the differential equation. Excellent test results are obtained for the solution of lower and higher order differential equations. The model finds approximation solution for the differential equation inside and outside the domain of consideration for the close enough neighborhood of initial/boundary points. Numerical examples are described to demonstrate the method.

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

##### Keywords:

feed forward artificial neural networks; multidimensional optimization; Nelder-Mead method; numerical examples##### Software:

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\textit{A. Malek} and \textit{R. Shekari Beidokhti}, Appl. Math. Comput. 183, No. 1, 260--271 (2006; Zbl 1105.65340)

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