Variational Iteration Method for Solving Systems of Linear Delay Differential Equations

In this paper, using a model transformation approach a system of linear delay differential equations (DDEs) with multiple delays is converted to a non-delayed initial value problem. The variational iteration method (VIM) is then applied to obtain the approximate analytical solutions. Numerical results are given for several examples involving scalar and second order systems. Comparisons with the classical fourth-order Runge-Kutta method (RK4) verify that this method is very effective and convenient.

• Sara Barati
• Karim Ivaz

• Variational iteration method
• delay differential equations
• multiple delays
• Runge-Kutta method.

[1] H. Gorecki, S. Fuksa, P. Grabowski and A. Korytowski. Analysis and
Synthesis of Time Delay Systems. New York: John Wiley and Sons, pp.
369, 1989.
[2] J. K. Hale and S. M. V. Lunel. Introduction to Functional Differential
Equations. New York: Springer-Verlag, 1993.
[3] J. P. Richard. Time delay systems: an overview of some recent advances
and open problems. Automatica, 39 (2003) 1667-1964.
[4] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins
Univ. Press, Baltimore, 1989.
[5] J. Lam. Model reduction of delay systems using Pade approximants. Int.
J. Control, 57(2) (1993) 377-391.
[6] F. M. Asl and A. G. Ulsoy. Analysis of a system of linear delay differential
equations. J. Dyn. Syst. Meas. Control, 125 (2003) 215-223.
[7] S. Yi and A. G. Ulsoy. Solution of a system of linear delay differential
equations using the matrix Lambert function. Proc. 25th American
Control Conference, Minnearpolis, MN, Jun. (2006) 2433-2438.
[8] J. H. He. A new approach to nonlinear partial differential equations.
Commun. Nonlinear Sci. Numer. Simul. , 2 (1997) 230-235.
[9] J. H. He. Variational iteration method for delay differential equations.
Commun. Nonlinear Sci. Numer. Simul. , 2 (1997) 235-236.
[10] J. H. He. Approximate analytical solution for seepage flow with fractional
derivatives in porous media. Comput. Methods Appl. Mech. Engrg.
, 167 (1998) 27-68.
[11] J. H. He. Approximate solution of nonlinear differential equations with
convolution product nonlinearities. Comput. Methods Appl. Mech. Engrg.
, 167 (1998) 69-73.
[12] J. H. He. Variational iteration method-a kind of nonlinear analytical
technique: some examples. Int. J. Nonlinear Mech. , 34 (1999) 699-708.
[13] J. H. He. Variational iteration method for autonomous ordinary differential
systems. Appl. Math. Comput. , 114 (2000) 115-123.
[14] X. Chen, L. Wang. The variational iteration method for solving a neutral
functional-differential equation with proportional delays. Computers and
Mathematics with Applications, 59(8) (2010) 2696-2702.
[15] S. T. Mohyud-Din, A. Yildirim. Variational iteration method for delay
differential equations using He-s polynomials. Z. Naturforsch. , 65(12)
(2010) 1045-1048.
[16] S. P. Yang, A. G. Xiao. Convergence of the variational iteration method
for solving multi-delay differential equations. Computers and Mathematics
with Applications, 61(8) (2011) 2148-2151.
[17] Z. H. Yu. Variational iteration method for solving the multi-pantograph
delay equation. Phys. Lett. A. , 372 (2008) 6475-6479.
[18] E. Jarlebring. The Spectrum of Delay Differential Equations: Numerical
Methods, Stability and Perturbation. Ph. D. Thesis, TU Braunschweig,
2008.
[19] J. M. Heffernan, R. M. Corless. Solving some delay differential equations
with computer algebra. Mathematical Scientist, 31(1) (2006) 21-34.
[20] S. Yi, A. G. Ulsoy and P. W. Nelson. Solution of systems of linear delay
differential equations via Laplace transformation. Proc. 45th IEEE Conf.
on Decision and Control, San Diego, CA, Dec. (2006), pp. 2535-2540.