A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions

Authors:
J. A. Rad;S. Kazem;K. Parand
Affiliations:
Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin, Tehran 19839, Iran;Department of Mathematics, Imam Khomeini International University, Ghazvin 34149-16818, Iran;Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin, Tehran 19839, Iran
Venue:
Computers & Mathematics with Applications
Year:
2012

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Abstract

In this research, a new numerical method is applied to investigate the nonlinear controlled Duffing oscillator. This method is based on the radial basis functions (RBFs) to approximate the solution of the optimal control problem by using the collocation method. We apply Legendre-Gauss-Lobatto points for RBFs center nodes in order to use the numerical integration method more easily; then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the dynamic systems, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.