Analytical Approach to Differential Equations with Piecewise Continuous Arguments via Modified Piecewise Variational Iteration Method

Qi Wang

School of Applied Mathematics, Guangdong University of Technology, Guangzhou, China.

DOI: 10.4236/jamp.2014.21005                

Abstract

In the present article, we apply the modified piecewise variational iteration method to obtain the approximate analytical solutions of the differential equations with piecewise continuous arguments. This technique provides a sequence of functions which converges to the exact solution of the problem. Moreover, this method reduces the volume of calculations because it does not need discretization of the variables, linearization or small perturbations. The results seem to show that the method is very reliable and convenient for solving such equations.

Keywords

Delay Differential Equations; Piecewise Continuous Arguments; Variational Iteration Method; Approximation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Busenberg and K. L. Cooke, “Models of Vertically Transmitted Diseases with Sequential-Continuous Dynamics, Nonlinear Phenomena in Mathematical Sciences,” In: V. Lakshmikantham, Ed., Academic Press, New York, 1982, pp. 179-187.
[2]	K. L. Cooke and J. Wiener, “Retarded Differential Equations with Piecewise Constant Delays,” Journal of Mathematical Analysis and Applications, Vol. 99, 1984, pp. 265-297. http://dx.doi.org/10.1016/0022-247X(84)90248-8
[3]	Y. K. Huang, “A Nonlinear Equation with Piecewise Constant Argument,” Applied Analysis, Vol. 33, 1989, pp. 183-190. http://dx.doi.org/10.1080/00036818908839871
[4]	J. L. Hong, R. Obayab and A. Sanzb, “Almost Periodic Type Solutions of Some Differential Equations with Piecewise Constant Argument,” Nonlinear Analysis, Vol. 45, 2001, pp. 661-688. http://dx.doi.org/10.1016/S0362-546X(98)00296-X
[5]	L. Dai and M. C. Singh, “A New Approach with Piecewise Constant Arguments to Approximate and Numerical Solutions of Oscillatory Problems,” Journal of Sound and Vibration, Vol. 263, 2003, pp. 535-548. http://dx.doi.org/10.1016/S0022-460X(02)01065-9
[6]	G. Papaschinopoulos, G. Stefanidou and P. Efraimidis, “Existence, Uniqueness and Asymptotic Behavior of the Solutions of a Fuzzy Differential Equation with Piecewise Constant Argument,” Information Science, Vol. 177, 2007, pp. 3855-3870. http://dx.doi.org/10.1016/j.ins.2007.03.006
[7]	M. Pinto, “Asymptotic Equivalence of Nonlinear and Quasi Linear Differential Equations with Piecewise Constant Arguments,” Mathematical and Computer Modelling, Vol. 49, 2009, pp. 1750-1758. http://dx.doi.org/10.1016/j.mcm.2008.10.001
[8]	L. Dai and M. C. Singh, “On Oscillatory Motion of Spring-mass Systems Subjected to Piecewise Constant Forces,” Journal of Sound and Vibration, Vol. 173, 1994, pp. 217-232. http://dx.doi.org/10.1006/jsvi.1994.1227
[9]	N. M. Murad and A. Celeste, “Linear and Nonlinear Characterization of Loading Systems Under Piecewise Discontinuous Disturbances Voltage: Analytical and Numerical Approaches,” Proceedings of International Conference on Power Electronics Systems and Applications, November 2004, pp. 291-297.
[10]	J. Wiener and V. Lakshmikantham, “A Damped Oscillator with Piecewise Constant Time Delay,” Nonlinear Studies, Vol. 1, 2000, pp. 78-84.
[11]	R. Yuan, “On Favard’s Theorems,” Journal of Differential Equations, Vol. 249, 2010, pp. 1884-1916. http://dx.doi.org/10.1016/j.jde.2010.07.014
[12]	E. Ait Dads and L. Lhachimi, “Pseudo Almost Periodic Solutions for Equation with Piecewise Constant Argument,” Journal of Computational and Applied Mathematics, Vol. 371, 2010, pp. 842-854. http://dx.doi.org/10.1016/j.jmaa.2010.06.032
[13]	M. U. Akhmet, D. Arugaslanc and E. Y1lmaz, “Method of Lyapunov Functions for Differential Equations with Piecewise Constant Delay,” Journal of Computational and Applied Mathematics, Vol. 235, 2011, pp. 4554-4560. http://dx.doi.org/10.1016/j.cam.2010.02.043
[14]	M. Z. Liu, M. H. Song and Z. W. Yang, “Stability of Runge-Kutta Methods in the Numerical Solution of Equation u'(t)=au(t)+a0u([t]),” Journal of Computational and Applied Mathematics, Vol. 166, 2004, pp. 361-370. http://dx.doi.org/10.1016/j.cam.2003.04.002
[15]	W. J. Lv, Z. W. Yang and M. Z. Liu, “Stability of Runge-Kutta Methods for the Alternately Advanced and Retarded Differential Equations with Piecewise Continuous Arguments,” Computers & Mathematics with Applications, Vol. 54, 2007, pp. 326- 335. http://dx.doi.org/10.1016/j.camwa.2006.07.018
[16]	M. Z. Liu, J. F. Gao and Z. W. Yang, “Preservation of Oscillations of the Runge-Kutta Method for Equation x'(t)+ax(t)+a1x([t-1])=0,” Computers & Mathematics with Applications, Vol. 58, 2009, pp. 1113-1125. http://dx.doi.org/10.1016/j.camwa.2009.07.030
[17]	J. H. He, “Variational Iteration Method for Delay Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 2, 1997, pp. 235-236. http://dx.doi.org/10.1016/S1007-5704(97)90008-3
[18]	J. H. He, “Approximate Solution of Nonlinear Differential Equations with Convolution Product Non-Linearities,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, 1998, pp. 69-73. http://dx.doi.org/10.1016/S0045-7825(98)00109-1
[19]	J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Applied Mathematics and Computation, Vol. 114, 2000, pp. 115-123. http://dx.doi.org/10.1016/S0096-3003(99)00104-6
[20]	M. Inokuti, H. Sekine and T. Mura, “General Use of the Lagrange Multiplier in Non-linear Mathematical Physics,” In: Variational Methods in the Mechanics of Solids, Pergamon Press, Oxford, 1978, pp. 156-162.
[21]	J. H. He, “Variational Iteration Method-Some Recent Results and New Interpretation,” Journal of Computational and Applied Mathematics, Vol. 207, 2007, pp. 3-17. http://dx.doi.org/10.1016/j.cam.2006.07.009
[22]	J. H. He and X. H. Wu, “Variational Iteration Method: New Development and Applications,” Computers & Mathematics with Applications, Vol. 54, 2007, pp. 881-894. http://dx.doi.org/10.1016/j.camwa.2006.12.083
[23]	J. H. He, G. C. Wu and F. Austin, “The Variational Iteration Method Which Should Be Followed,” Nonlinear Science Letters A, Vol. 1, 2010, pp. 1-30.
[24]	G. C. Wu, “A Fractional Variational Iteration Method for Solving Fractional Nonlinear Differential Equations,” Computers & Mathematics with Applications, Vol. 61, 2011, pp. 2186-2190. http://dx.doi.org/10.1016/j.camwa.2010.09.010
[25]	J. F. Lu, “An Analytical Approach to the Fornberg-Whitham Type Equations by Using the Variational Iteration Method,” Computers & Mathematics with Applications, Vol. 61, 2011, pp. 2010-2013. http://dx.doi.org/10.1016/j.camwa.2010.08.052
[26]	M. M. Khader, “Numerical and Theoretical Treatment for Solving Linear and Nonlinear Delay Differential Equations Using Variational Iteration Method,” Arab Journal of Mathematical Sciences, Vol. 19, No. 2, 2013, pp. 243-256. http://dx.doi.org/10.1016/j.ajmsc.2012.09.004
[27]	G. Yang and R. Y. Chen, “Choice of an Optimal Initial Solution for a Wave Equation in the Variational Iteration Method,” Computers & Mathematics with Applications, Vol. 61, 2011, pp. 2053-2057. http://dx.doi.org/10.1016/j.camwa.2010.08.068
[28]	A. F. Elsayed, “Comparison between Variational Iteration Method and Homotopy Perturbation Method for Thermal Diffusion and Diffusion Thermo Effects of Thixotropic Fluid through Biological Tissues with Laser Radiation Existence,” Applied Mathematical Modelling, Vol. 37, no.6, 2013, pp. 3660-3673. http://dx.doi.org/10.1016/j.apm.2012.07.016
[29]	E. S. Fahmy, H. A. Abdusalam and K. R. Raslan, “On the Solutions of the Time-delayed Burgers Equation,” Nonlinear Analysis, Vol. 69, 2008, pp. 4775-4786. http://dx.doi.org/10.1016/j.na.2007.11.027
[30]	M. Dehghan and R. Salehi, “Solution of a Nonlinear Time-delay Model in Biology via Semi-analytical Approaches,” Computer Physics Communications, Vol. 181, 2010, pp. 1255-1265. http://dx.doi.org/10.1016/j.cpc.2010.03.014
[31]	Z. H. Yu, “Variational Iteration Method for Solving the Multi-Pantograph Delay Equation,” Physical Letter A, Vol. 372, 2008, pp. 6475-6479. http://dx.doi.org/10.1016/j.physleta.2008.09.013
[32]	A. Saadatmandi and M. Dehghan, “Variational Iteration Method for Solving a Generalized Pantograph Equation,” Computers & Mathematics with Applications, Vol. 58, 2009, pp. 2190-2196. http://dx.doi.org/10.1016/j.camwa.2009.03.017
[33]	X. M. Chen and L. J. Wang, “The Variational Iteration Method for Solving a Neutral Functional-Differential Equation with Proportional Delays,” Computers & Mathematics with Applications, Vol. 59, 2010, pp. 2696-2702. http://dx.doi.org/10.1016/j.camwa.2010.01.037
[34]	F. Z. Geng, Y. Z. Lin and M. G. Cui, “A Piecewise Variational Iteration Method for Riccati Differential Equations,” Computers & Mathematics with Applications, Vol. 58, 2009, pp. 2518-2522. http://dx.doi.org/10.1016/j.camwa.2009.03.063
[35]	F. Z. Geng, “A Piecewise Variational Iteration Method for Treating a Nonlinear Oscillator of a Mass Attached to a Stretched Elastic Wire,” Computers & Mathematics with Applications, Vol. 62, 2011, pp. 1641-1644. http://dx.doi.org/10.1016/j.camwa.2011.05.004