0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Driver, R., 1977, Ordinary and Delay Differential Equations, Springer-Verlag, Berlin.
Bocharov, G., and Rihan, F., 2000, “Numerical Modelling in Biosciences Using Delay Differential Equations,” J. Comput. Appl. Math., 125(1–2), pp. 183–199. [CrossRef]
Balachandran, B., Kalmar-Nagy, T., and Gilsinn, D., 2009, Delay Differential Equations: Recent Advances and New Directions, Springer, New York.
Richard, J., 2003, “Time-Delay Systems: An Overview of Some Recent Advances and Open Problems,” Automatica, 39(10), pp. 1667–1694. [CrossRef]
Pfeiffer, F., and Glocker, C., 1996, Multibody Dynamics With Unilateral Contacts, Wiley, New York.
Wen, L., and Liu, X., 2011, “Numerical Stability of One-Leg Methods for Neutral Delay Differential Equations,” BIT Numer. Math., 52(1), pp. 1–19.
Wang, W., Wen, L., and Li, S., 2010, “Stability of Linear Multistep Methods for Nonlinear Neutral Delay Differential Equations in Banach Space,” J. Comput. Appl. Math., 233(10), pp. 2423–2437. [CrossRef]
Shampine, L., 2008, “Dissipative Approximations to Neutral DDEs,” Appl. Math. Comput., 203(2), pp. 641–648. [CrossRef]
Koto, T., 2004, “Method of Lines Approximations of Delay Differential Equations,” Comput. Math. Appl., 48(1–2), pp. 45–59. [CrossRef]
Maset, S., 2003, “Numerical Solution of Retarded Functional Differential Equations as Abstract Cauchy Problems,” J. Comput. Appl. Math., 161(2), pp. 259–282. [CrossRef]
Nayfeh, A., and Mook, D., 1995, Nonlinear Oscillations, Wiley-VCH, Berlin.
Nayfeh, A., 1981, Introduction to Perturbation Techniques, Wiley, New York.
Govaerts, W., 2000, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia.
Dhooge, A., Govaerts, W., and Kuznetsov, Y., 2003, “MATCONT: A MATLAB Package for Numerical Bifurcation Analysis of ODEs,” ACM Trans. Math. Softw., 29(2), pp. 141–164. [CrossRef]
Wahi, P., and Chatterjee, A., 2005, “Galerkin Projections for Delay Differential Equations,” J. Dyn. Syst., Meas., Control, 127(1), pp. 80–87. [CrossRef]
Ghosh, D., Saha, P., and Roy Chowdhury, A., 2007, “On Synchronization of a Forced Delay Dynamical System via the Galerkin Approximation,” Commun. Nonlinear Sci. Numer. Simul., 12(6), pp. 928–941. [CrossRef]
de Jesus Kozakevicius, A., and Kalmár-Nagy, T., 2010, “Weak Formulation for Delay Equations,” 9th Brazilian Conference on Dynamics, Control and Their Applications.
Vyasarayani, C. P., 2012, “Galerkin Approximation for Higher Order Delay Differential Equations,” ASME J. Comput. Nonlinear Dyn., 7(3), p. 031004. [CrossRef]
Khalil, H., and Grizzle, J., 2002, Nonlinear Systems, Prentice Hall, New Jersey.
Paul, C., 1994, A Test Set of Functional Differential Equations, Mathematics Department, University of Manchester, Numerical Analysis Report No. 243.

Figures

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)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In