Journal of Applied Mathematics and Physics, 2014, 2, 2631 Published Online January 2014 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2014.21005 OPEN ACCESS JAMP 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 Email: bmwzwq@126.com Received October 2013 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 perturba tions. 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 1. Introduction Differential equations with piecewise continuous arguments (EPCA) are special type of delay differential equa tions (DDEs). The theory of EPCA was initiated in [1,2] and developed by many authors [37]. These systems have been under intensive investigation for the last twenty years. EPCA describe hybrid dynamical systems and combine properties of both differential and difference equations. They are appeared in modeling of various problems in real life such as biology, mechanics, and electronics. For some applications of this equation we refer the interested reader to [1,810]. Several important properties of the analytic solution of EPCA as well as nu merical methods have been studied in [1116]. In this paper, we consider the following two EPCA: 01 0 0 0 u'(t)au(t)au([t]),t, u()u , =+≥ = (1) and the coupled system 23 45 x'(t )ax(t )ay([t]), y'(t)ay(t )ax([t]), = + = + (2) with initial value , where are real constants and [.] denotes the greatest integer function and . In this work, we apply the modified piecewise variational iteration method (MPVIM) to systems (1) and (2) to obtain approximate analytical solutions. The VIM gives several successive approximations by using the iteration of the correction functional. This method was proposed by the Chinese researcher Jihuan He [1719] as a mod ification of a general Lagrange multiplier method [20]. VIM is one of the nonperturbation methods that does not require any small or large parameter. An elementary introduction of VIM is given in [21]. The main con cepts in VIM, such as general Lagrange multiplier, restricted variation, correction functional are explained sys temically. For more comprehensive survey on this method and its applications, the reader is referred to the re view articles [22,23] and the references therein.
Q. WANG OPEN ACCESS JAMP 27 The VIM has been favorably applied to various kinds of linear and nonlinear problems. The main property of the method is in flexibility and ability to solve linear and nonlinear equations accurately and conveniently. The flexibility and adaptation provided by this method have made the method a strong candidate for approximate analytical solutions. The VIM plays an important role in recent researches for solving various kinds of problems (see for example [2428] and the references therein). However, the researches on the application of VIM on DDE are relatively fewer. As far as we know, only delay Burgers equation [29], delay logistic equation [30] and pantograph equation [3133] are considered. As for the analytical study of EPCA with VIM, up to now, there are almost no results published. Therefore, we will conduct this study. The organization of this paper is as follows. In Section 2, we simply provide the mathematical framework of the VIM. In Section 3, we apply the modified piecewise variational iteration method on the systems (1) and (2) after analyzing the conventional VIM and piecewise variational iteration method. Some numerical results are given in Section 4. Finally, in Section 5, a brief conclusion is provided. 2. He’s Variational Iteration Method In this section, we introduce the basic idea underlying the VIM for solving nonlinear equations. Consider the general differential equation (3) where and are linear and nonlinear operators, respectively, and is the inhomogeneous term. In VIM, a correction functional for (3) can be written as 10 xn nn n u(x)u (x)(s)[Lu(s)Nu(s)g(s)]ds, λ + =+ +− ∫ (4) where is a general Lagrange’s multiplier, which can be identified optimally via integration by parts and the variational theory, and denotes the restricted variation, i.e. . It is to be noted that the Lagrange multiplier can be a constant or a function. After determining the Lagrange multiplier , an iteration for mula, without restricted variation, should be used for the determination of the successive approximations of the solution . The zeroth approximation can be selected freely. Consequently, the solu tion is given by (5) 3. The Application of VIM In this section the application of VIM is discussed for solving systems (1) and (2). 3.1. System (1) We consider system (1), according to the VIM, the correction function is given by ( ) 11 0 n t'nn nn u(t)u(t)(s)u (s)au (s)au([s])ds. λ + =+ −− ∫ (6) To find the optimal value of we have 10 n t' nn u(t)u (t)(s)u(s)ds, δδ δλ + = + ∫ (7) that results ( ) 10 1 t n nn st u(t)u (t)'(s)u(s)ds. δλ δδλ += =+− ∫ (8) Thus we have the following stationary conditions 1 0 0 st st , '( s ). λ λ = = += = (9) This in turn gives . So we obtain the following iteration formula ( ) 1 01 0 n t' nnn n u(t)u (t)u(s)au(s)au([s])ds, + =− −− ∫ (10)
Q. WANG OPEN ACCESS JAMP and the approximation solution is given by . (11) During the process of computation, the greatest integer function [.] causes us many problems. To overcome them, we recall a modified VIM: the piecewise variational iteration method (PVIM), which was introduced by Geng [34,35]. In PVIM, the interval is divided into some equal subintervals, then the order ap proximation are obtained on these subintervals. Following this way, we introduce the modified piece wise variational iteration method (MPVIM). In our method, the interval is divided into lots of subin tervals with unit length, where . On the interval , let ( ) 11101011 1 0 t' ,n ,,n,n,n u(t)u(t)u (s)au (s)au([s])ds, + =− −− ∫ (12) where . Then we can obtain the order approximation on . On the interval , let ( ) 212020212 0 t' ,n ,,n,n,n u(t)u(t)u (s)au (s)au ([s])ds, + =− −− ∫ (13) The integration in (13) can be computed in and , respectively. Then the order approxima tion on can be obtained. In a similar way, on the interval , let ( ) 100 1 0 t' k,nk,k,nk,nk,n u(t)u(t)u (s)au (s)au ([s])ds, + =−−− ∫ 1 01 1 k k ,k,n u(t)u( k). − − = − (14) The integration in (14) can be computed in a series of subintervals: , . Then we can obtain the order approximation on . Therefore, according to (12)(14), the approximation of (1) on the entire interval can be obtained. 3.2. System (2) According to VIM, the iteration formula for (2) can be constructed as follows ( ) 1 23 0 n t' nnn n x(t)x (t)x(s)ax (s)ay ([s])ds, + =−−− ∫ ( ) 1 45 0 n t' nnn n y(t)y(t)y(s)ay (s)ax ([s])ds. + =− −− ∫ (15) Similar to Subsection 3.1, in view of MPVIM we have the following formulas. On the interval , let ( ) 111012 131 0 t' ,n ,,n,n,n x(t)x (t)x(s)ax (s)ay ([s])ds, +=−−− ∫ ( ) 11101415 1 0 t' ,n ,,n,n,n y(t)y(t)y (s)ay (s)ax ([s])ds, + =− −− ∫ Then we can obtain the order approximation on , where . On the interval , let ( ) 212022 232 0 t' ,n ,,n,n,n x(t)x (t)x (s) ax (s) ay([s])ds, + =− −− ∫ ( ) 21202425 2 0 t' ,n ,,n,n,n y(t)y (t)y (s) ay(s) ax ([s])ds, + =− −− ∫
Q. WANG OPEN ACCESS JAMP 29 Then we can obtain the order approximation on . Similarly, on the interval , let ( ) 102 3 0 t' k,nk,k,nk,nk,n x(t)x (t)x(s) ax(s) ay([s])ds, + =−−− ∫ ( ) 104 5 0 t' k,nk,k,nk,nk,n y(t)y(t)y(s) ay(s) ax ([s])ds, + =− −− ∫ 1 01 1 k k ,k,n x(t)x( k), − − = − 1 01 1 k k ,k,n y(t)y(k). − − = − Then we can obtain the order approximation on . Therefore, according to (16)(18), the approximation of coupled system (2) on the entire interval can be obtained. 4. Results and Discussion In this section, we apply the MPVIM presented in Section 3 and the classical methods to two concrete EPCA. Numerical results show that the MPVIM is very effective. For (1), we choose , and . According to (12)(14), taking and , We can obtain the approximations of (1) on . The numerical results are depicted in Figure 1. This figure shows the comparison of approximation obtained by using the present method with the exact solu tion and the numerical solution. Moreover, for (2), we choose , , , and . In Fig ure 2 we compare the 5thorder approximation of MPVIM with the numerical solution. Figure 1. A comparison of the results of the exact solution (upper), the 5thorder MPVIM solution (middle) and the numerical solution (lower) with θ = 0.6 and m = 20 to (1). Figure 2. A comparison of the results of the 5thorder MPVIM solution (upper) and the numerical solution (lower) with θ = 0.3 and m = 20 to (2). 00.1 0.20.3 0.40.5 0.6 0.70.8 0.9 1 0 5 time t u(t ) 00.1 0.20.3 0.40.5 0.6 0.70.8 0.91 0 5 time t u n (t) 00.1 0.20.3 0.40.5 0.6 0.70.8 0.91 0 2 4 6 time t u n 0.8 0.6 0.4 0.200.2 0.4 0.6 0.81 1 2 3 4 5 x n (t) y n (t) 0.6 0.4 0.2 00.2 0.4 0.6 0.8 1 1 2 3 4 x n y n
Q. WANG OPEN ACCESS JAMP The above numerical examples demonstrate that the present method is quite effective and simple. 5. Conclusions An efficient algorithm based on the VIM has been successfully applied to the EPCA. As can be seen from the numerical results, implementing only a few steps in the MPVIM, the approximate analytical solutions with high accuracy can be obtained. It can be concluded that the MPVIM is a powerful and promising tool for solving such kinds of delay diffe rential equations. This method can also be extended to the EPCA of the advanced type and mixed type, which are our future research issues. Acknowledgements The author would like to thank the reviewers for his/her hard work. In addition, we thank Professor Fazhan Geng for his helpful assistance. REFERENCES [1] S. Busenberg and K. L. Cooke, “Models of Vertically Transmitted Diseases with SequentialContinuous Dynamics, Nonlinear Phenomena in Mathematical Sciences,” In: V. Lakshmikantham, Ed., Academic Press, New York, 1982, pp. 179187. [2] K. L. Cooke and J. Wiener, “Retarded Differential Equations with Piecewise Constant Delays,” Journal of Mathematical Anal ysis and Applications, Vol. 99, 1984, pp. 265297. http://dx.doi.org/10.1016/0022247X(84)902488 [3] Y. K. Huang, “A Nonlinear Equation with Piecewise Constant Argument,” Applied Analysis, Vol. 33, 1989, pp. 183190. 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 Con stant Argument,” Nonlinear Analysis, Vol. 45, 2001, pp. 661688. http://dx.doi.org/10.1016/S0362546X(98)00296X [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. 535548. http://dx.doi.org/10.1016/S0022460X(02)010659 [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. 38553870. 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. 17501758. http://dx.doi.org/10.1016/j.mcm.2008.10.001 [8] L. Dai and M. C. Singh, “On Oscillatory Motion of Springmass Systems Subjected to Piecewise Constant Forces,” Journal of Sound and Vibration, Vol. 173, 1994, pp. 217232. 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. 291297. [10] J. Wiener and V. Lakshmikantham, “A Damped Oscillator with Piecewise Constant Time Del ay,” Nonlinear Studies, Vol. 1, 2000, pp. 7884. [11] R. Yuan, “On Favard’s Theorems,” Journal of Differential Equations, Vol. 249, 2010, pp. 18841916. 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. 842854. 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 Con stant Delay,” Journal of Computational and Applied Mathematics, Vol. 235, 2011, pp. 45544560. http://dx.doi.org/10.1016/j.cam.2010.02.043 [14] M. Z. Liu, M. H. Song and Z. W. Yang, “Stability of RungeKutta Methods in the Numerical Solution of Equation u'(t)=au(t)+a0u([t]),” Journal of Computational and Applied Mathematics, Vol. 166, 2004, pp. 361370. http://dx.doi.org/10.1016/j.cam.2003.04.002 [15] W. J. Lv, Z. W. Yang and M. Z. Liu, “Stability of RungeKutta Methods for the Alternately Advanced and Retarded Differen tial 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 RungeKutta Method for Equation x'(t)+ax(t)+a1x([t1])=0,” Computers & Mathematics with Applications, Vol. 58, 2009, pp. 11131125.
Q. WANG OPEN ACCESS JAMP 31 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 Numeri cal Simulation, Vol. 2, 1997, pp. 235236. http://dx.doi.org/10.1016/S10075704(97)900083 [18] J. H. He, “Approximate Solution of Nonlinear Differential Equations with Convolution Product NonLinearities,” Computer Methods in Applied Mechanics and Engineering, Vol. 167, 1998, pp. 6973. http://dx.doi.org/10.1016/S00457825(98)001091 [19] J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Applied Mathematics and Computa tion, Vol. 114, 2000, pp. 115123. http://dx.doi.org/10.1016/S00963003(99)001046 [20] M. Inokuti, H. Sekine and T. Mura, “General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics,” In: Varia tional Methods in the Mechanics of Solids, Pergamon Press, Oxford, 1978, pp. 156162. [21] J. H. He, “Variational Iteration MethodSome Recent Results and New Interpretation,” Journal of Computational and Applied Mathematics, Vol. 207, 2007, pp. 317. 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. 881894. 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. 130. [24] G. C. Wu, “A Fractional Variational Iteration Method for Solving Fractional Nonlinear Differential Equations,” Computers & Mathematics with Applications, Vol. 61, 2011, pp. 21862190. http://dx.doi.org/10.1016/j.camwa.2010.09.010 [25] J. F. Lu, “An Analytical Approach to the FornbergWhitham Type Equations by Using the Variational Iteration Method,” Computers & Mathematics with Applications, Vol. 61, 2011, pp. 20102013. 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. 243256. 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. 20532057. 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 Ma thematical Modelling, Vol. 37, no.6, 2013, pp. 36603673. 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 Timedelayed Burgers Equation,” Nonlinear Analy sis, Vol. 69, 2008, pp. 47754786. http://dx.doi.org/10.1016/j.na.2007.11.027 [30] M. Dehghan and R. Salehi, “Solution of a Nonlinear Timedelay Model in Biology via Semianalytical Approaches,” Computer Physics Communications, Vol. 181, 2010, pp. 12551265. http://dx.doi.org/10.1016/j.cpc.2010.03.014 [31] Z. H. Yu, “Variational Iteration Method for Solving the MultiPantograph Delay Equation,” Physical Letter A, Vol. 372, 2008, pp. 64756479. 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. 21902196. 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 FunctionalDifferential Equation with Proportional Delays,” Computers & Mathematics with Applications, Vol. 59, 2010, pp. 26962702. 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. 25182522. 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. 16411644. http://dx.doi.org/10.1016/j.camwa.2011.05.004
