Variational iteration method for singular perturbation initial value problems

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Abstract

In this paper, the variational iteration method (VIM) is applied to solve singular perturbation initial value problems (SPIVPs). The obtained sequence of iterates is based on the use of Lagrange multipliers. Some convergence results of VIM for solving SPIVPs are given. Moreover, the illustrative examples show the efficiency of the method.

Introduction

Singular perturbation initial value problems often arise in many scientific and engineering fields such as automatic control, biology, medical science, economics, etc. (cf. [4], [19]). These problems are characterized by a small parameter ϵ multiplying the derivatives, and there exist initial boundary layers where the solutions change rapidly.

The variational iteration method was first proposed by He [9], [10], [11], and has been extensively worked out over a number of years by numerous authors. This method solves the problems without any need to discrete the variables. Therefore, there is no need to compute the round off errors and one is not faced with necessity of large computer memory and time. Applications of the method have been enlarged due to its flexibility, convenience and efficiency. The VIM was first applied to autonomous ordinary differential systems, delay differential equations, and fractional differential equations by He et al. [7], [8], [9]. Salkuyeh [3] studied the convergence of VIM for solving linear system of ODEs with constant coefficients. Wazwaz [1] used the VIM for analytic treatment of linear and nonlinear ODEs. Saadatmandi, Dehghan [2] and Yu [21] applied the VIM to solve pantograph equations. Xu [14], Saadati, Dehghan, Vaezpour and Saravi [20] considered the convergence of VIM for solving integral equations. Though Darvishi, Khani and Soliman [16] applied the VIM to some stiff ODEs, but these stiff problems do not contain the singular perturbation problems. Tatari and Dehghan [17], Mamode [18] considered the VIM for solving second order initial value problems. Lu [13] applied VIM to solve two point boundary value problems. Rafei, Ganji, Daniali and Pashaei [15], Marinca, Herisanu and Bota [22] applied the VIM to Oscillations. For more comprehensive survey on this method and its applications, the reader is refer to the review articles [5], [6] and the references therein.

In this paper, we apply the VIM to SPIVPs to obtain the analytical or approximate analytical solutions. The convergence results of VIM for solving SPIVPs are obtained. Some illustrative examples confirm the theoretical results.

In the rest parts of the text, we denote x(t)=x(t,ϵ), y(t)=y(t,ϵ) for simplicity, here, ϵ is the singular perturbation parameter. The vectors (a1,a2,,an)T=ab=(b1,b2,,bn)T means each component aibi (i=1,2,,n). denotes the standard Euclidean norm of a vector.

Section snippets

Case 1

Consider the following singular perturbation initial value problem{x(t)=f(t,x(t),y(t)),0tT,ϵy(t)=g(t,x(t),y(t)),ϵ0ϵ1,x(0)=x0,y(0)=y0, where t[0,T] is the ‘time’, xRn1 and yRn2 are the state variables, ϵ is the singular perturbation parameter, ϵ0>0 is a sufficiently small constant. f:[0,T]×Rn1×Rn2Rn1, g:[0,T]×Rn1×Rn2Rn2 are given continuous mappings which satisfy the following Lipschitz conditionsf(t,x1,y1)f(t,x2,y2)l1(t)x1x2+l2(t)y1y2,g(t,x1,y1)g(t,x2,y2)k1(t)x1x2+k2(

Illustrative examples

In this section, some illustrative examples are given to show the efficiency of the VIM for solving SPIVPs.

Example 1

Consider the linear SPIVP (cf. [19]){ϵy(t)=(t1)y(t),0<ϵ1,0t0.9,y(0)=0.2.

Using the VIM in the previous section, we construct the following correction functional as Case 2yn+1(t)=yn(t)+0tλ(s,t)(yn(s)s1ϵyn(s))ds. To find the optimal value λ, we haveδyn+1(t)=δyn(t)+δ0tλ(s,t)(yn(s)s1ϵyn(s))ds, which leads toδyn+1(t)=δyn(t)+λ(s,t)δyn(s)|s=t0t(λ(s,t)s+s1ϵλ(s,t))δyn(s)ds.

Conclusion

The VIM used in this paper is the variational iteration algorithm-I, there are also variational iteration algorithm-II and variational iteration algorithm-III [12]. In this paper, we apply the VIM to obtain the analytical or approximate analytical solutions of SPIVPs. The convergence results of VIM for solving SPIVPs are given. The illustrative examples show the efficiency of the method. When considering the system (2.10), the choice of correction functionals of Case 1 or Case 2 rely on the

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This work is supported by projects NSF of China (10971175), Specialized Research Fund for the Doctoral Program of Higher Education of China (20094301110001), NSF of Hunan Province (09JJ3002) and Hunan Provincial Innovation Foundation for Postgraduate (S2008yjscx02).

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