Numerical solution for high order differential equations using a hybrid neural network—Optimization method
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 ne...
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| Vydáno v: | Applied mathematics and computation Ročník 183; číslo 1; s. 260 - 271 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York, NY
Elsevier Inc
01.12.2006
Elsevier |
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| ISSN: | 0096-3003, 1873-5649 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Malek, A. Shekari Beidokhti, R. |
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| Keywords | Multidimensional optimization Nelder–Mead method Ordinary differential equations Feed forward artificial neural networks Mixed distribution Neural computation Differential equation Initial value problem Optimization method Numerical method Neural network Nelder-Mead method Computing Many-dimensional calculations Partial differential equation Discretization method Boundary value problem Applied mathematics Numerical solution Mathematical programming |
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| SubjectTerms | Applied sciences Computer science; control theory; systems Exact sciences and technology Feed forward artificial neural networks Global analysis, analysis on manifolds Language theory and syntactical analysis Mathematical analysis Mathematics Multidimensional optimization Nelder–Mead method Numerical analysis Numerical analysis. Scientific computation Ordinary differential equations Sciences and techniques of general use Theoretical computing Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| Title | Numerical solution for high order differential equations using a hybrid neural network—Optimization method |
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