Research Papers

Galerkin Approximations for Neutral Delay Differential Equations

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

Department of Mechanical Engineering,
Indian Institute of Technology Hyderabad,
Ordnance Factory Estate, Andhra Pradesh 502205, India
e-mail: vcprakash@iith.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received January 1, 2012; final manuscript received August 10, 2012; published online October 1, 2012. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 8(2), 021014 (Oct 01, 2012) (5 pages) Paper No: CND-12-1001; doi: 10.1115/1.4007446 History: Received January 01, 2012; Revised August 10, 2012

In this work, Galerkin approximations are developed for a system of first order nonlinear neutral delay differential equations (NDDEs). The NDDEs are converted into an equivalent system of hyperbolic partial differential equations (PDEs) along with the nonlinear boundary constraints. Lagrange multipliers are introduced to enforce the boundary constraints. The explicit expressions for the Lagrange multipliers are derived by exploiting the equivalence of partial derivatives in space and time at a given point on the domain. To illustrate the method, comparisons are made between numerical solution of NDDEs and its Galerkin approximations for different NDDEs.

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Grahic Jump Location
Fig. 1

Solution from analytical and the Galerkin approximation. (a) Comparison. (b) % Error.

Grahic Jump Location
Fig. 2

Comparison between solutions obtained from numerical and Galerkin approximation with N=100: (a) with initial function z1(t)=2,t≤0; (b) with initial function z1(t)=3+t,t≤0

Grahic Jump Location
Fig. 3

Computation time for solving Eq. (26) for 100(s) with increasing number of terms in the Galerkin approximation with initial function z1(t)=2,t≤0

Grahic Jump Location
Fig. 4

(a) Actual z1(t) and observed z∧1(t) responses. (b) Tracking error z1(t)-z∧1(t).

Grahic Jump Location
Fig. 5

Comparison between solutions obtained from numerical and Galerkin approximation with N=100: (a) for z1(t); (b) for z2(t)




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