﻿ Analytical Approach to Differential Equations with Piecewise Continuous Arguments via Modified Piecewise Variational Iteration Method Journal of Applied Mathematics and Physics, 2014, 2, 26-31 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 [3-7]. 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 [1,8-10]. Several important properties of the analytic solution of EPCA as well as nu- merical methods have been studied in [11-16]. In this paper, we consider the following two EPCA: 01000u'(t)au(t)au([t]),t,u()u ,=+≥= (1) and the coupled system 2345x'(t )ax(t )ay([t]),y'(t)ay(t )ax([t]),= += + (2) with initial value 000TX()(x ,y)=, where ia 01 5(i, ,,)= are real constants and [.] denotes the greatest integer function and TX(t)(x(t),y(t))=. 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 [17-19] as a mod-ification of a general Lagrange multiplier method . VIM is one of the non-perturbation methods that does not require any small or large parameter. An elementary introduction of VIM is given in . 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 [24-28] 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 , delay logistic equation  and pantograph equation [31-33] 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 LuNug(x ),+= (3) where L and N are linear and nonlinear operators, respectively, and g( x) is the inhomogeneous term. In VIM, a correction functional for (3) can be written as 10xnnn nu(x)u (x)(s)[Lu(s)Nu(s)g(s)]ds,λ+=+ +−∫ (4) where λ is a general Lagrange’s multiplier, which can be identified optimally via integration by parts and the variational theory, and nu denotes the restricted variation, i.e. 0nuδ=. It is to be noted that the Lagrange multiplier λ can be a constant or a function. After determining the Lagrange multiplier λ, an iteration for-mula, without restricted variation, should be used for the determination of the successive approximations 1nu (x)+ of the solution u(x ). The zeroth approximation 0u can be selected freely. Consequently, the solu-tion is given by nnu( x )lim u( x )→∞= (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 ( )110nt'nnnnu(t)u(t)(s)u (s)au (s)au([s])ds.λ+=+ −−∫ (6) To find the optimal value of λ we have 10nt'nnu(t)u (t)(s)u(s)ds,δδ δλ+= +∫ (7) that results ( )101tn nnstu(t)u (t)'(s)u(s)ds.δλ δδλ+==+−∫ (8) Thus we have the following stationary conditions 1| 00stst,'( s ).λλ==+== (9) This in turn gives 1λ= −. So we obtain the following iteration formula ( )1 010nt'nnn nu(t)u (t)u(s)au(s)au([s])ds,+=− −−∫ (10) Q. WANG OPEN ACCESS JAMP 28 and the approximation solution is given by 1nnu(t )limu(t )+→∞=. (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 0[,X] is divided into some equal subintervals, then the in-order ap-proximation ii,nu (x) are obtained on these subintervals. Following this way, we introduce the modified piece-wise variational iteration method (MPVIM). In our method, the interval 0[, )∞ is divided into lots of subin-tervals 1[k,k)+ with unit length, where kN∈. On the interval 01[ ,], let ( )11101011 10t',n ,,n,n,nu(t)u(t)u (s)au (s)au([s])ds,+=− −−∫ 100,u(t)u(),= (12) where 01t[ ,]∈. Then we can obtain the 1n-order approximation 11,nu (t) on 01[,]. On the interval 12[,], let ( )2120202120t',n ,,n,n,nu(t)u(t)u (s)au (s)au ([s])ds,+=− −−∫ 120 11, ,nu(t)u().= (13) The integration in (13) can be computed in 01[ ,] and 1[ ,t], respectively. Then the 2n-order approxima-tion 22,nu (t) on 12[,] can be obtained. In a similar way, on the interval 1[k ,k]−, 34k ,,= let ( )100 10t'k,nk,k,nk,nk,nu(t)u(t)u (s)au (s)au ([s])ds,+=−−−∫ 1011kk ,k,nu(t)u( k).−−= − (14) The integration in (14) can be computed in a series of subintervals: 01[,],12[ ,],,1[k ,t]−. Then we can obtain the kn-order approximation kk ,nu (t) on1[k ,k]−. Therefore, according to (12)-(14), the approximation of (1) on the entire interval 0[, )∞ can be obtained. 3.2. System (2) According to VIM, the iteration formula for (2) can be constructed as follows ( )1 230nt'nnn nx(t)x (t)x(s)ax (s)ay ([s])ds,+=−−−∫ ( )1 450nt'nnn ny(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 01[ ,], let ( )111012 1310t',n ,,n,n,nx(t)x (t)x(s)ax (s)ay ([s])ds,+=−−−∫ ( )11101415 10t',n ,,n,n,ny(t)y(t)y (s)ay (s)ax ([s])ds,+=− −−∫ 100,x(t)x(),= 100,y(t )y().= Then we can obtain the 1n-order approximation 11,nX (t) on 01[,], where TX(t) (x(t),y(t))=. On the interval 12[,], let ( )212022 2320t',n ,,n,n,nx(t)x (t)x (s) ax (s) ay([s])ds,+=− −−∫ ( )21202425 20t',n ,,n,n,ny(t)y (t)y (s) ay(s) ax ([s])ds,+=− −−∫ Q. WANG OPEN ACCESS JAMP 29 120 11, ,nx(t )x(),= 120 11, ,ny(t )y().= Then we can obtain the 2n-order approximation 22,nX (t) on 12[,]. Similarly, on the interval 1[k ,k]−, 34k,,= let ( )102 30t'k,nk,k,nk,nk,nx(t)x (t)x(s) ax(s) ay([s])ds,+=−−−∫ ( )104 50t'k,nk,k,nk,nk,ny(t)y(t)y(s) ay(s) ax ([s])ds,+=− −−∫ 1011kk ,k,nx(t)x( k),−−= − 1011kk ,k,ny(t)y(k).−−= − Then we can obtain the kn-order approximation kk ,nX (t) on 1[k ,k]−. Therefore, according to (16)-(18), the approximation of coupled system (2) on the entire interval 0[, )∞ can be obtained. 4. Results and Discussion In this section, we apply the MPVIM presented in Section 3 and the classical θ-methods to two concrete EPCA. Numerical results show that the MPVIM is very effective. For (1), we choose 02a=, 11a= − and 01u=. According to (12)-(14), taking 3k= and 5in=, 1i, ,k.= We can obtain the approximations of (1) on 03[,]. 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 21a=, 32a= −,42a=, 51a= − and 001xy= =. In Fig ure 2 we compare the 5th-order approximation of MPVIM with the numerical solution. Figure 1. A comparison of the results of the exact solution (upper), the 5th-order 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 5th-order 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 105time tu(t ) 00.1 0.20.3 0.40.5 0.6 0.70.8 0.9105time tun(t) 00.1 0.20.3 0.40.5 0.6 0.70.8 0.910246time tun-0.8 -0.6 -0.4 -0.200.2 0.4 0.6 0.8112345xn(t)yn(t)-0.6 -0.4 -0.2 00.2 0.4 0.6 0.8 11234xnyn Q. WANG OPEN ACCESS JAMP 30 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. 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