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Research Papers

Galerkin Approximations for Higher Order Delay Differential Equations

[+] Author and Article Information
C. P. Vyasarayani1

Systems Design Engineering,  University of Waterloo, Waterloo, Ontario N2L 3G1, Canadacpvyasar@engmail.uwaterloo.ca

1

Author to whom correspondence should be addressed.

J. Comput. Nonlinear Dynam 7(3), 031004 (Mar 19, 2012) (5 pages) doi:10.1115/1.4005931 History: Received June 15, 2011; Revised January 09, 2012; Published March 13, 2012; Online March 19, 2012

Abstract

In this work, Galerkin approximations are developed for a system of n first order nonlinear delay differential equations (DDEs) and also for an nth order nonlinear scalar DDE. The DDEs are converted into an equivalent system of partial differential equations of the same order along with the nonlinear boundary constraints. Lagrange multipliers are then introduced and explicit expressions for the Lagrange multipliers are derived to enforce the nonlinear boundary constraints. To illustrate the method, comparisons are made between the numerical solution of nonlinear DDEs and its Galerkin approximations for different parameter values.

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Figures

Figure 5

Time response of z(t) of Eq. 29 obtained from direct numerical integration as compared to a0 (t) obtained from the Galerkin approximation with N = 7. The system parameters are b1  = 0, b2  = 1.5, b3  = 0.1, b4  = 0, F = 1, α=1, β=0.5, and ω=6π.

Figure 6

Time response of z(t) of Eq. 29 obtained from direct numerical integration as compared to a0 (t) obtained from the Galerkin approximation with N = 7. The system parameters are b1  = 0.05, b2  = 0.75, b3  = 1, b4  = 0.1, F = 1, α=1, β=0.5, and ω=4π.

Figure 7

The least square error Σk=1k=10000(a0(tk)-z(tk))2 as a function of the number of terms (N + 1) in the Galerkin approximation. The parameters are the same as those used to generate Fig. 6.

Figure 4

Time response of z(t) of Eq. 29 obtained from direct numerical integration as compared to a0 (t) obtained from the Galerkin approximation with N = 7. The system parameters are b1  = 0.01, b2  = 1, b3  = 1, b4  = 0, F = 0.5, α=1, β=0.5, and ω=2π.

Figure 3

Maximum absolute error between a10 (t) and z1 (t) over a 100 s simulation for different values of ω. All other system system parameters are the same as those used to generate Fig. 2.

Figure 2

Time response of (a) z1 (t), and (b) z2 (t) of Eq. 28 obtained from direct numerical integration as compared to a10 (t) and a20 (t) obtained from the Galerkin approximation with N = 7. The system parameters are b1  = 1, b2  = 1, b3  = 1, b4  = 1, b5  = 1, b6  = 1, b7  = 1, b8  = 1, α1  = 1, α2  = 1, α3  = 1, α4  = 1, F = 1, and ω=10π.

Figure 1

Time response of (a) z1 (t), and (b) z2 (t) of Eq. 28 obtained from direct numerical integration as compared to a10 (t) and a20 (t) obtained from the Galerkin approximation with N = 7. The system parameters are b1  = 2, b2  = 1, b3  = 0.1, b4  = 1, b5  = 0.75, b6  = 2, b7  = 0.1, b8  = 0.1, α1  = 0.1, α2  = 0.3, α3  = 0.5, α4  = 1, F = 1.8, and ω=2π.

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