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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Bibliographic Details
Published in:Applied mathematics and computation Vol. 183; no. 1; pp. 260 - 271
Main Authors: Malek, A., Shekari Beidokhti, R.
Format: Journal Article
Language:English
Published: New York, NY Elsevier Inc 01.12.2006
Elsevier
Subjects:
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
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
ISSN:0096-3003, 1873-5649
Online Access:Get full text
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Summary: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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2006.05.068