Variational Iteration Method for Solving Differential Equations with Piecewise Constant Arguments

Full Text (PDF, 197KB), PP.36-43

Author(s)

Qi Wang 1,*

1. Faculty of Applied Mathematics,Guangdong University of Technology, Guangzhou, Guangdong 510006, China

2. Faculty of Environmental Science and Engineering, Guangdong University of Technology, Guangzhou, Guangdong 510006, China

* Corresponding author.

Received: 18 Nov. 2011 / Revised: 26 Jan. 2012 / Accepted: 15 Mar. 2012 / Published: 6 Apr. 2012

Index Terms

Variational iteration method, Piecewise constant arguments, Approximate analytical solution

Abstract

In this paper, variational iteration method is applied for ﬁnding the solution of differential equations with piecewise constant arguments. A correction functional is constructed by a general Lagrange multiplier, which can be identiﬁed by variational theory. This technique provides a sequence of functions which converges to the exact solution of the problem without discretization of the variables. The ﬂexibility and adaptation provided by the method have been verified by an example.

Cite This Paper

Qi Wang, Fenglian Fu,"Variational Iteration Method for Solving Differential Equations with Piecewise Constant Arguments", IJEM, vol.2, no.2, pp.36-43, 2012. DOI: 10.5815/ijem.2012.02.06

Reference

[1] G.Q. Wang, “Periodic solutions of a neutral differential equation with piecewise constant arguments,” J. Math. Anal. Appl., vol. 326, pp. 736-747, 2007.

[2] M.U. Akhmet, “On the reduction principle for differential equations with piecewise constant argument of generalized type,” J. Math. Anal. Appl., vol. 336, pp. 646-663, 2007.

[3] M.U. Akhmet, “Asymptotic behavior of solutions of differential equations with piecewise constant arguments,” Appl. Math. Lett.., vol. 21, pp. 951-956, 2008.

[4] M.U. Akhmet, “Stability of differential equations with piecewise constant arguments of generalized type,” Nonlinear Anal., vol.68, pp. 794-803, 2008.

[5] D. Altintan, “Extension of the logistic equation with piecewise constant arguments and population dynamics,” M.Sc.Thesis, Middle East Technical University, 2006.

[6] K.L. Cooke and J. Weiner, “A survey of differential equation with piecewise continuous argument,” Lecture Notes in Mathematics, Springer, Berlin, 1991.

[7] L. Dai and L. Fan, “Analytical and numerical approaches to characteristics of linear and nonlinear vibratory systems under piecewise discontinuous disturbances,” Commun. Nonlinear Sci. Numer. Simul., vol. 9, pp. 417–429, 2004.

[8]  F. Gurcan and F. Bozkurt, “Global stability in a population model with piecewise constant arguments,”J. Math. Anal. Appl., vol. 360, pp. 334–342, 2009.

[9] K.L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays,” J. Math. Anal. Appl., vol. 99, pp. 265–297, 1984.

[10] S.M. Shah and J. Wiener, “Advanced differential equations with piecewise constant argument deviations,” Int. J. Math. Math. Sci., vol. 6, pp. 671–703, 1983.

[11] A.F. Ivanov, “Global dynamics of a differential equation with piecewise constant argument,” Nonlinear Anal., vol. 71, pp. e2384–e2389, 2009.

[12] Y. Muroya, “New contractivity condition in a population model with piecewise constant arguments,” J. Math. Anal. Appl., vol. 346, pp. 65–81, 2008.

[13] Y.H. Xia, Z.K. Huang, and M.A. Han, “Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument,” J. Math. Anal. Appl., vol. 333, pp. 798–816, 2007.

[14] J. Wiener, Generalized Solutions of Functional Differential Equations,  Singapore: World Scientiﬁc, 1993.

[15] M.H. Song, Z.W. Yang, and M.Z. Liu, “Stability of -methods for advanced differential equations with piecewise continuous arguments,”Comput.Math. Appl., vol. 49, pp. 1295-1301, 2005.

[16] M.Z. Liu, S.F. Ma, and Z.W. Yang, “Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments,” Appl. Math. Comput.., vol. 191, pp. 57-66, 2007.

[17] Q. Wang, Q.Y. Zhu, and M.Z. Liu, “Stability and oscillations of numerical solutions of differential equations with piecewise continuous arguments of alternately advanced and retarded type,” J. Comput. Appl. Math.., vol. 235, pp. 1542-1552, 2011.

[18] J.H. He, “Variational iteration method-A kind of non-linear analytical technique: Some examples,” Internat. J. Non-Linear Mech., vol. 34, no. 4, pp. 699-708, 1999.

[19] J.H. He, “Variational iteration method-Some recent results and new interpretations,” J. Comput. Appl. Math.., vol. 207, no. 1, pp. 3-17, 2007.

[20] J.H. He and X.H. Wu, “Variational iteration method: New development and applications,"Comput. Math. Appl., vol. 54, no.7-8, pp. 881-894, 2007.

[21] J.H. He, G.C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Sci. Lett. A, vol. 1, no.1, pp. 1-30, 2010.

[22] M.M. Al-Sawalha, M.S.M. Noorani, and I. Hashim, “On accuracy of Adomian decompositionmethod for hyperchaotic Rössler system,” Chaos Solitons Fractals., vol. 40, no.4, pp. 1801-1807, 2009.

[23] M.M. Al-Sawalha and M.S.M. Noorani, “Application of the differential transformation method for the solution of the hyperchaotic Rössler system,” Comm. Non. Sci. Num. Simu., vol. 14, pp. 1509-1514, 2009.

[24] F.M. Allan, “Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method,” Chaos Solitons Fractals., vol. 39, no.4, pp. 1744-1752, 2009.

[25] J.H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” Int. J. Nonlinear Sci. Numer. Simu., vol. 6, pp. 207-208, 2005.

[26] J.H. He, “Some asymptotic methods for strongly nonlinear equations,” Int. J. Modern Phys.B, vol. 20, no.10, pp. 1141–1199, 2006.

[27] J.H. He, “Approximate analytical solution for seepage ﬂow with fractional derivatives in porous media,” Comput. Meth. Appl. Mech. Eng., vol. 167, pp. 57–68, 1998.

[28] J.H. He, “Variational iteration method for autonomous ordinary differential systems,” Appl. Math. Comput.., vol. 114, pp. 115–123, 2000.

[29] J.H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” Int. J. Modern Phys.B, vol. 22, no.21, pp. 3487–3578, 2008.