In this work, approximations for state dependent delay differential equations (DDEs) are developed using Galerkin's approach. The DDE is converted into an equivalent partial differential equation (PDE) with a moving boundary, where the length of the domain dependents on the solution of the PDE. The PDE is further reduced into a finite number of ordinary differential equations (ODEs) using Galerkin's approach with time dependent basis functions. The nonlinear boundary condition is represented by a Lagrange multiplier, whose expression is derived in closed form. We also demonstrate the validity of the developed method by comparing the numerical solution of the ODEs to that of the original DDE.
Issue Section:
Research Papers
References
1.
Driver
, R.
, 1977
, Ordinary and Delay Differential Equations
, Springer-Verlag
, New York
.2.
Tsimring
, L.
, and Pikovsky
, A.
, 2001
, “Noise-Induced Dynamics in Bistable Systems With Delay
,” Phys. Rev. Lett.
, 87
, p. 250602
.10.1103/PhysRevLett.87.2506023.
Yeung
, M.
, and Strogatz
, S.
, 1999
, “Time Delay in the Kuramoto Model of Coupled Oscillators
,” Phys. Rev. Lett.
, 82
(3
), pp. 648
–651
.10.1103/PhysRevLett.82.6484.
Guillouzic
, S.
, L'Heureux
, I.
, and Longtin
, A.
, 1999
, “Small Delay Approximation of Stochastic Delay Differential Equations
,” Phys. Rev. E
, 59
(4
), pp. 3970
–3982
.10.1103/PhysRevE.59.39705.
Kuang
, Y.
, 1993
, Delay Differential Equations: With Applications in Population Dynamics
, Academic
, Boston
.6.
Li
, C.
, and Chen
, G.
, 2004
, “Synchronization in General Complex Dynamical Networks With Coupling Delays
,” Physica A
, 343
, pp. 263
–278
.10.1016/j.physa.2004.05.0587.
Bocharov
, G.
, and Rihan
, F.
, 2000
, “Numerical Modeling in Biosciences Using Delay Differential Equations
,” J. Comput. Appl. Math.
, 125
(1–2
), pp. 183
–199
.10.1016/S0377-0427(00)00468-48.
Richard
, J.
, 2003
, “Time-Delay Systems: An Overview of Some Recent Advances and Open Problems
,” Automatica
, 39
(10
), pp. 1667
–1694
.10.1016/S0005-1098(03)00167-59.
Safonov
, L.
, Tomer
, E.
, Strygin
, V.
, Ashkenazy
, Y.
, and Havlin
, S.
, 2002
, “Multifractal Chaotic Attractors in a System of Delay-Differential Equations Modeling Road Traffic
,” Chaos
, 12
(4
), pp. 1006
–1014
.10.1063/1.150790310.
Hartung
, F.
, Krisztin
, T.
, Walther
, H.
, and Wu
, J.
, 2006
, “Functional Differential Equations With State-Dependent Delays: Theory and Applications
,” Handbook of Differential Equations: Ordinary Differential Equations, 3, pp. 435
–545
.11.
Nayfeh
, A.
, and Mook
, D.
, 1995
, Nonlinear Oscillations
, Wiley-VCH
, New York
.12.
Koto
, T.
, 2004
, “Method of Lines Approximations of Delay Differential Equations
,” Comput. Math. Appl.
, 48
(1–2
), pp. 45
–59
.10.1016/j.camwa.2004.01.00313.
Maset
, S.
, 2003
, “Numerical Solution of Retarded Functional Differential Equations as Abstract Cauchy Problems
,” J. Comput. Appl. Math.
, 161
(2
), pp. 259
–282
.10.1016/j.cam.2003.03.00114.
Khalil
, H.
, and Grizzle
, J.
, 2002
, Nonlinear Systems
, Prentice Hall
, New Jersey
.15.
Sanders
, J.
, Verhulst
, F.
, and Murdock
, J.
, 2007
, Averaging Methods in Nonlinear Dynamical Systems
, Springer-Verlag
, New York
.16.
Nayfeh
, A.
, 1973
, Perturbation Methods
, Vol. 6
, Wiley
, New York
.17.
Morton
, K.
and Mayers
, D.
, 2005
, Numerical Solution of Partial Differential Equations: An Introduction
, Cambridge University
Press, Cambridge, UK
.18.
Carr
, J.
, 1981
, Applications of Centre Manifold Theory
, Springer
, New York.
19.
Wahi
, P.
, and Chatterjee
, A.
, 2005
, “Galerkin Projections for Delay Differential Equations
,” ASME J. Dyn. Syst., Meas., Control
, 127
(1
), pp. 80
–87
.10.1115/1.187004220.
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
.10.1016/j.cnsns.2005.08.00621.
de Jesus Kozakevicius
, A.
, and Kalmár-Nagy
, T.
, 2010
, “Weak Formulation for Delay Equations
,” 9th Brazilian Conference on Dynamics, Control and Their Applications
.22.
Vyasarayani
, C. P.
, 2012
, “Galerkin Approximation for Higher Order DelayDifferential Equations
,” ASME J. Comput. Nonlinear Dyn.
, 7
(3
), p. 031004
.10.1115/1.400593123.
Vyasarayani
, C. P.
, 2012
, “Galerkin Approximations for Neutral Delay Differential Equations
,” ASME J. Comput. Nonlinear Dyn.
, 8
, p. 021014
.10.1115/1.400744624.
Wang
, P.
, and Wei
, J.
, 1987
, “Vibrations in a Moving Flexible Robot Arm
,” J. Sound Vib.
, 116
(1
), pp. 149
–160
.10.1016/S0022-460X(87)81326-325.
Shampine
, L.
, 2005
, “Solving ODEs and DDEs With Residual Control
,” Appl. Numer. Math.
, 52
, pp. 113
–127
.10.1016/j.apnum.2004.07.00326.
Kuang
, Y.
, and Smith
, H.
, 1992
, “Slowly Oscillating Periodic Solutions of Autonomous State-Dependent Delay Equations
,” Topol. Methods Nonlinear Anal.
, 19
(9
), pp. 855
–872
.10.1016/0362-546X(92)90055-J27.
Copyright © 2013 by ASME
You do not currently have access to this content.