Solving differential-algebraic equations through variational iteration method with an auxiliary parameter

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Solving differential-algebraic equations through variational iteration method with an auxiliary parameter H. Ghaneai, M.M. Hosseini PII: DOI: Reference:

S0307-904X(15)00635-6 10.1016/j.apm.2015.10.002 APM 10800

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Applied Mathematical Modelling

Received date: Revised date: Accepted date:

2 December 2013 9 May 2015 5 October 2015

Please cite this article as: H. Ghaneai, M.M. Hosseini, Solving differential-algebraic equations through variational iteration method with an auxiliary parameter, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.10.002

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Highlights • The VIM with an auxiliary parameter is applied to differential-algebraic equations.

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• Convergence of the VIM with an auxiliary parameter is studied.

• Present method provides a simple way to adjust and control the convergence region. • Numerical results explicitly reveal the complete reliability of proposed method.

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• Proposed method is particularly suitable for inverse problems.

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Solving differential-algebraic equations through variational iteration method with an auxiliary parameter H. Ghaneaia , M. M. Hosseinia,b a

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Faculty of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran Department of Applied Mathematics, Faculty of mathematics and Computer, Shahid Bahonar University of Kerman, P. O Box 76169-14111, Kerman, Iran

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[email protected], hosse− [email protected]

Abstract

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This paper reports on the importance of auxiliary parameter which is introduced into the well-known variational iteration method to obtain solutions of differential-algebraic equations. The convergence of the proposed method, namely variational iteration method with an auxiliary parameter, is studied. In addition, the proposed method is applied on two differential-algebraic equations to elucidate the solution procedure and to choose the auxiliary parameter optimally. Comparison with results by the standard variational iteration method shows that the auxiliary parameter is very effective in controlling the convergence region of approximate solution.

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Keywords: Differential-algebraic equations, Variational iteration method, Auxiliary parameter. 1. Introduction

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Differential-Algebraic Equations (DAEs) are systems of differential equations (sometimes also referred to descriptors, singular or semi-state systems), where the unknown functions satisfy additional algebraic equations [1]. In other words, they consist of a set of differential equations with additional algebraic constraints. Many physical problems are governed by a system of differential-algebraic equations. For electrical networks, constrained mechanical systems of rigid bodies, singular instance perturbation and discretization Preprint submitted to Applied Mathematical Modelling

November 10, 2015

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of partial differential equations, control theory, etc. [2, 3]. In recent years, finding the solution of these equations has been the subject of many investigators. In this paper, we consider linear variable coefficients system DAEs:  A(t)Y˙ + B(t)Y = f(t), t ∈ [0, T ] . (1) Y (0) = y0 ,

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where A(t) is a singular matrix, B(t) is a nonsingular matrix and f(t) : [0, T ] −→ Cm , is a supposed continuous function. Implicit RungeKutta and the Backward Differentiation Formula (BDF) methods are the most important numerical methods for certain classes of DAEs [4]. But these methods cannot be applied to approximate the solution to all DAEs [5]. Sometimes a DAE can be converted into a system of ODEs. However, numerical stability of the system is often undermined in the process so that, even if all DAEs can be converted into ODEs, it is not usually desirable to do so [6, 7, 8]. A considerable research has been done on numerical methods for DAEs. Ascher, Petzold, Campbell and Gear have done extensive work for numerical solution of this class of equations [9, 10, 11]. The related Predicted Sequential Regularization Method (PSRM) and Sequential Regularization Method (SRM) are numerical methods designed to deal with certain classes of DAEs [12, 13]. Waveform relaxation (WR) methods to solve the initial value problems of DAEs have been proposed and investigated by many authors[14, 15, 16]. The approximate analytical solutions to differential-algebraic equations were presented by Hosseini using the Adomian decomposition method [17]. Moreover, Homotopy perturbation method is used for approximate solutions of DAEs [18]. The variational iteration method (VIM) has been proposed by Ji-Huan He, which was further developed by the originator himself [19, 20, 21, 22]. This method is applied to solve various kinds of functional equations. In fact, through conducting this method a large class of nonlinear problems converge rapidly to approximate solutions. Moreover, various nonlinear equations can be solved by this method[23, 24, 25]. Yilmaz and Inc proposed a variational iteration algorithm in which an auxiliary parameter was presented to control the convergence rate, but they did not give a general rule for the best selection of the auxiliary parameter [25]. This improved method was further developed by Hosseini et al. through introducing some profitable rules for optimal determination of the auxiliary parameter [23, 27, 28, 29]. 3

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Here, we are going to implement variational iteration method with an auxiliary parameter to obtain approximate solution of differential-algebraic equations. In the proposed method the residual function and the error of residual function are defined to choose the auxiliary parameter optimally. Also, the convergence of the variational iteration method with an auxiliary parameter is studied according to the alternative approach of this method. 2. Vriational iteration method with an auxiliary parameter

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In this section, the variational iteration method with an auxiliary parameter is described. Consider the following general nonlinear equation: Hu = Lu + N u + Ru + g(x, t) = 0,

(2)

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where L shows the highest order derivative that is assumed to be easily invertible, R indicates a linear differential operator of order less than L, N u represents the nonlinear terms, and g illustrates the source inhomogeneous term. Ji-Huan He proposed the variational iteration method in which a correction functional for (2), can be written as: Z t λ (τ ) Hu n (x, τ ) dτ , (3) un+1 (x, t) = un (x, t) +

u(x, t) = lim un (x, t), n→∞

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In the above equation λ is a Lagrange multiplier which can be identified optimally via variational theory [20], un is the nth approximate solution, and u˜n denotes a restricted variation, i.e., ∂ u˜n = 0. The approximations un+1 (x, t), n ≥ 0, of the solution u(x, t), will be readily obtained upon using the determined Lagrangian multiplier and any selected function u0 (x, t), providing that Lu0 (x, t) = 0. In fact, the correction functional (3) will give several approximations such as follows, (4)

In summary, the following variational iteration formula for (2) is given:  u0 (x, t) is an arbitrary f unction, Rt (5) un+1 (x, t) = un (x, t) + 0 λ (τ ) Hu n (x, τ ) dτ , n ≥ 0. 4

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An unknown auxiliary parameter can be inserted into the variational iteration algorithm, equation (3):   u0 (x, t) is an arbitrary f unction, Rt u1 (x, t, h) = u0 (x, t) + h 0 λ (τ ) Hu n (x, τ ) dτ , (6) Rt  un+1 (x, t, h) = un (x, t, h) + h 0 λ (τ ) Hu n (x, τ, h) dτ , n ≥ 1.

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3. Convergence analysis

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The successive approximate solutions un+1 (x, t, h) , n ≥ 1 include the auxiliary parameter h. The accuracy of the method depends on an assumption in which the approximation un+1 (x, t, h) , n ≥ 1 converges to the exact solution. here, the auxiliary parameter ensures that the assumption can be satisfied. Totally, it is straightforward to choose a proper value of h which ensures that the approximate solutions are convergent and this is done through the error of norm two of the residual function [26, 27, 28, 29]. In fact, the suggested technology, that is, variational iteration method with an auxiliary parameter [30] is very simple, easier to apply and is capable to approximate the solution more precisely in a large solution domain.

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In this section, the convergence of the variational iteration method with an auxiliary parameter is studied according to the alternative approach of this method which present in the following. This approach can be implemented, in a reliable and efficient way and also can handle the nonlinear differential equation (2). The linear operator L is defined as L = dtd +α, when the variational iteration method with an auxiliary parameter is applied to solve the differential-algebraic equation (1). Now, the operator A is defined as follows: Z t Au(t, h) = h λ (τ ) Hu(τ, h)dτ , (7) 0

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and the components vn , sn , n ≥ 0, are defined below,  v0 (t) = u0 (t), s0 (t) = v0 (t),  v1 (t, h) = As0 (t), s1 (t, h) = s0 (t) + v1 (t, h), 5

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and in general for n ≥ 1,  vn+1 (t, h) = Asn (t, h), sn+1 (t, h) = sn (t, h) + vn+1 (t, h),

u(t, h) = lim sn (t, h) = v0 (t) + n→∞

∞ X n=1

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then, consequently,

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vn (t, h).

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The zeroth approximation u0 (t) can be freely chosen, if it satisfies the initial conditions of the problem and P∞Lu0 (t) = 0. For the approximation purpose, the solution u(t, h) = v0 (t)+ n=1 vn (t, h), is approximated by the N th-order P truncated series uN (t, h) = v0 (t) + N n=1 vn (t, h). The approximate solutions uN (t, h) , contains the auxiliary parameter h. It is the auxiliary parameter that ensures that the assumption can be satisfied, in general, by means of the error of norm two of the residual function. The sufficient conditions for convergence of the method and the error estimate will be introduced in this section. The main results are proposed in the following theorems [29, 31].

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Theorem 3.1 Let A, defined in (7), be an operator from a Hilbert space H ˜ 6= 0, 0 < γ < 1, such that, to H. If ∃h

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  k As0 (t) k ≤ γ k s0 (t) k, ˜ k≤ γ k As0 (t) k, k As1 (t, h)  ˜ k≤ γ k Asn−1 (t, h) ˜ k, k Asn (t, h)

n = 2, 3, 4, · · · ,

˜ = v0 (t) + u(t) = lim sn (t, h) n→∞

∞ X

˜ vn (t, h),

n=1

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Then the series solution defined in (9),

converges. Proof. We show that {sn }∞ n=0 is a Cauchy sequence in the Hilbert space H. For this purpose, consider, k sn+1 − sn k = k vn+1 k = k Asn k≤ γ k Asn−1 k ≤ γ 2 k Asn−2 k≤ · · · ≤ γ n+1 k s0 k= γ n+1 k v0 k . 6

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For every n ≥ j, we have,

and since 0 < γ < 1, we get, lim k sn − sj k= 0.

n,j→∞

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k sn − sj k = k (sn − sn−1 ) + (sn−1 − sn−2 ) + · · · + (sj+1 − sj ) k ≤ k sn − sn−1 k + k sn−1 − sn−2 k + · · · + k sj+1 − sj k ≤ γ n k v0 k +γ n−1 k v0 k + · · · + γ j+1 k v0 k n−j = 1−γ γ j+1 k v0 k, 1−γ

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Therefore, {sn }∞ n=0 is a Cauchy sequence in the Hilbert space H and it implies that the series solution ˜ = v0 (t) + u(t) = lim sn (t, h) n→∞

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˜ vn (t, h),

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converges. This completes the proof of Theorem 3.1. 

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Lemma 3.1 Let L, defined in (2), be as, L = dtd + α, and λ identified optimally via variational theory. If k, be a function from a Hilbert space H to H, then,  Z t λ (τ ) k(s)dτ = −k(t), L 0

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Proof. Suppose that L, defined in (2), be as, L = dtd + α. It is easy to verify that λ(τ ) = −eα(τ −t) . Thus, nR o R  R  t t t L 0 λ (τ ) k(τ )dτ = dtd 0 −eα(τ −t) k(τ )dτ + α 0 −eα(τ −t) k(τ )dτ =



R t ∂ (−eα(τ −t) )

+ α

∂t

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k(τ )dτ

R

t −eα(τ −t) k(τ )dτ 0

 


+ −eα(τ −t) k(τ ) |τ =t

= −k(t).



Theorem 3.2 Let L, defined in (2), be as, L = dtd + α, if we have u(t) = P ˜ v0 (t) + ∞ n=1 vn (t, h), then u(t), is an exact solution of the nonlinear problem 7

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˜ = 0, lim v(t, h)

n→∞

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proof. Suppose that the series solution (9), converges, then we have,

n h h i X i ˜ ˜ − vj+1 (t, h) ˜ = v0 (t) − vn+1 (t, h), ˜ v0 (t) − v1 (t, h) + vj (t, h) j=1

and so,

∞ h i i X ˜ − vj+1 (t, h) ˜ = v0 (t)− lim vn+1 (t, h) ˜ = v0 (t), ˜ + vj (t, h) v0 (t) − v1 (t, h)

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Applying the operator L to both sides of (11) we obtain,

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∞ h i X h i ˜ ˜ − vj+1 (t, h) ˜ = L [v0 (t)] = 0, L v0 (t) − v1 (t, h) + L vj (t, h) j=1

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from definitions (7), (8) and Lemma 3.1 we have, h i ˜ L v0 (t) − v1 (t, h) = L [v0 (t)] − L [As0 (t)] = n R o ˜ t λ (τ ) Hs0 (τ )dτ = hHs ˜ 0 (t), = −L h 0

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h i h i h i ˜ − v2 (t, h) ˜ = L v1 (t, h) ˜ − L v2 (t, h) ˜ L v1 (t, h) ˜ = L[As n 0R(t)] − L[As1 (t, h)] o ˜ t λ (τ ) Hs0 (τ )dτ =L h n R0 o ˜ t λ (τ ) Hs1 (τ, h)dτ ˜ −L h n 0 o ˜ ˜ = h Hs1 (t, h) − Hs0 (t) ,

similarly, for j ≥ 2, we get, h i n o ˜ ˜ ˜ ˜ ˜ L vj (t, h) − vj+1 (t, h) = h Hsj (t, h) − Hsj−1 (t, h) . 8

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Consequently, we obtain, h i h i h i ˜ + L v1 (t, h) ˜ − v2 (t, h) ˜ + Pn L vj (t, h) ˜ − vj+1 (t, h) ˜ L v0 (t) − v1 (t, h) j=2 n o ˜ 0 (t) + h ˜ Hs1 (t, h) ˜ − Hs0 (t) = hHs n o ˜ ˜ ˜ +h Hsn (t, h) − Hs1 (t, h) h i P ˜ n (t, h) ˜ = hH ˜ ˜ , = hHs v0 (t) + nj=1 vj (t, h) therefore,

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h i h i ˜ − vj+1 (t, h) ˜ ˜ + P∞ L vj (t, h) L v0 (t) − v1 (t, h) j=1 h i P∞ ˜ ˜ = hH v0 (t) + j=1 vj (t, h) ,

according to (12) we have, " # ∞ X ˜ ˜ = 0. hH v0 (t) + vj (t, h) j=1

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˜ Since the P∞auxiliary˜ parameter h is nonzero, we can observe that u(t) = v0 (t) + n=1 vn (t, h), is an exact solution of problem (2). This completes the proof of Theorem 3.2. 

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P ˜ Theorem 3.3 Suppose that the series solution u(t) = v0 (t) + ∞ n=1 vn (t, h), defined in (9), is convergent to exactP solution of the nonlinear problem (2). If ˜ the truncated series uN (t) = v0 (t) + N n=1 vn (t, h), is used as an approximate solution, then the maximum error is estimated as, k u(t) − uN (t) k ≤

1 γ N +1 k v0 k, 1−γ

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Proof. From Theorem 3.1 inequality (10), for every n ≥ N, we have, k sn − sN k ≤

1 − γ n−N N +1 γ k v0 k, 1−γ

since, limn→∞ sn = u(t), and 0 < γ < 1, we get, k u(t) − uN (t) k ≤

1 γ N +1 k v0 k . 1−γ 9

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i = 0, 1, 2, · · · .

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This completes the proof of Theorem 3.3.  In summary, we can define, ( kvi+1 k , k vi k6= 0, kvi k βi = 0, k vi k= 0,

Now, if 0 < βi < 1 for i = 0, 1, 2, · · · , then the series solution v0 (t) + P ∞ ˜ n=1 vn (t, h), of problem (2) converges to an exact solution, u(t). Moreover, as stated in Theorem 3.3, the maximum absolute truncation error is estimated to be, 1 β N +1 k v0 k, 1−β

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k u(t) − uN (t) k ≤

4. Numerical Examples

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where β = max {βi , i = 0, 1, 2, · · ·} . Notice that, the first finite terms do not affect the convergence of series solution. In other words, if the first finite βi ’s, i = 0, 1, 2, · · · , l, are not less than P∞one and˜ βi < 1, for i > l, then, of course the series solution v0 (t) + n=1 vn (t, h), of problem (2), converges to an exact solution [29, 31].

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To elucidate the solution procedure, two examples are given. In each case, the standard variational iteration (VIM) method is applied on the differential-algebraic equations. The obtained results show that the solutions of differential-algebraic equations are under investigation, and the standard VIM is not applicable. Therefore, the proposed method is tested on the aforementioned differential-algebraic equations. Comparison with results by exact solutions indicates that the large domains will not decrease the effectiveness of the proposed method.

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Example 4.1 Consider the following differential-algebraic equations [23]:       −t  0 0 1 1 e + sin(t)  0  Y (t) + Y (t) = , t ∈ [0, 10] ,   0 1 −1 − sin(t)  1     1   ,  Y (0) = 0

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 −t  e which admits the solution Y(t)= . Take t ∈ [0, 10]. According to sin(t) the standard VIM we have the following variational iteration formula:  y2,n+1 (t) = e−t + sin(t) − y1,n (t) ,  0 Rt y1,n+1 (t) = y1,n (t) − 0 eτ −t y1,n (τ ) + y1,n (τ ) − y2,n (τ ) + sin(τ ) dτ .

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Beginning with y1,0 (t) = y1 (0) = 1, y2,0 (t) = y2 (0) = 0, we stop the solution procedure at y1,10 (t) and y2,10 (t), then we obtain the following approximations,   ( 3 2 2 t5 + t2 e−t , y1,10 (t) = −1 + 2 + t + t2 + t6 + 120 1 y2,10 (t) = 78 sin(t) + 81 cos(t) + 48 (−6 + 6t2 + 4t3 + t4 ) e−t .

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Figures 1, 2, indicate that the solutions y1,10 (t) and y2,10 (t) are not valid for large values of t, of course, the accuracy can be improved if the iteration procedure continues and the exact solution can be obtained when n tends to infinite. Now, using the recursive scheme (6), we successively have,  y1,0 (t) = y1 (0) = 1, y2,0 (t) = y2 (0) = 0, and,

y1,1 (t, h) = 1 + h2 (e−t + cos(t) − sin(t) − 2) , y2,1 (t, h) = e−t − sin(t) − 1,

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and in general for n > 1,   y2,n+1 (t, h) = e−t + sin(t) −Ry1,n (t,h) , t ∂ y (t, h) = y1,n (t, h) − h 0 eτ −t ∂τ y1,n (τ, h) + y1,n (τ, h)  1,n+1 −y2,n (τ, h) + sin(τ )} dτ .

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In order to find a proper value of h for the approximate solutions (13), we define the following residual functions, r1,10 (t, h) = y2,10 (t, h) + y1,10 (t, h) − e−t − sin(t), ∂ r2,10 (t, h) = ∂t y1,10 (t, h) + y1,10 (t, h) − y2,10 (t, h) + sin(t),

and the following error of residual function, Z 10
e10 (h) = |r1,10 (t, h)|2 + |r2,10 (t, h)|2 dt. 0

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1 Exact Solution VIM with an auxiliary parameter

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−0.2 −0.4 −0.6 −0.8 −1

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VIM 0.6 We apply a numerical integration to calculate theStandard approximate e10 (h). For 0.4 obtaining an optimal value of h, we choose the minimum point of the error residual function0.2(14). The minimum point of e10 (h) , as h = 0.30849, is obtained by using0 Maple software. By Substitution h = 0.30849, we obtain

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Figure 1: The comparison between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter for y1 (t) in example 4.1.

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Exact Solution VIM with an auxiliary parameter Standard VIM

0.8 0.6 0.4

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y2(t)

0.2 0 −0.2

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−0.4 −0.6 −0.8

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Figure 2: The comparison between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter for y2 (t) in example 4.1.

the following successive approximations,

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y1,10 (t) = −0.00159 + 0.00502 sin (t) − 0.00004 cos (t) + (1.00163 + 0.02160 t − 0.06988 t2 + 0.03298 t3 − 0.00279 t4 + 0.00002 t5 ) e−t , y2,10 (t) = −0.00271 + 0.99336 sin (t) + 0.00353 cos (t) + (−0.00081 − 0.03033 t + 0.09127 t2 − 0.03051 t3 + 0.00136 t4 ) e−t .

The comparisons between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter, are shown in Figures

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4.2 Consider the following differential-algebraic equations [23]:      0 −1 t 0 t6 − t4 , 0  Y 0 (t) +  1 0 − 1  Y (t) =  0 2 3 4 5 6 7 t −t − 1 − t 4t + 12t + 8t − 2t − t

    0      Y (0) = 0  ,    0

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1 and 2. Example   0 0      0 0     t 1 

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 t4 + t5 It is easy to verify that Y (t) =  t4 + t5 . t4 + t5 We take the solution domain as t ∈ [0, 1]. According to the standard VIM, we obtain the following approximations, 

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y1,10 (t) = −19315994756040 t − 19315994756040 t2 − 9678610509720 t3 −3240523693095 t4 − 815291548353 t5 − 164268696762 t6 − 27570618546 t7 −3956223097 t8 − 493868599 t9 − 54220829 t10 − 5260884 t11 − 450122 t12 −33532 t13 − 2102 t14 − 98 t15 − 2 t16 + (19315994756040 t + 20613131700 t3 t7 − 48836 t8 +578102056 t4 − 426234387 t5
+ 3735204 t6 − 18004735 6 7 238727 9 11501 10 2387 11 t − 240 t − 15 t + 8 t e ,

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y2,10 (t) = −220624667065680 − 220624667065680 t − 110172179616168 t2 −36643787913352 t3 − 9125261416791 t4 − 1811245271079 t5 − 297515352054 t6 −41397319583 t7 − 4945855560 t8 − 509926533 t9 − 45149663 t10 − 3361064 t11 −198132 t12 − 7462 t13 + 93 t14 + 45 t15 + 3 t16 + (220624667065680 −140153916672 t2 + 13163985744 t3 − 10520071686 t4 + 3443792 t5 − 16367080 t6 − 31043879 t7 + 41894981 t8 − 941639 t9 3 35 270
120 64289 10 217 11 217 12 + 30 t + 4 t − 16 t et ,

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y3,10 (t) = −319799790576 t − 319799790576 t2 − 160767367872 t3 − 54167650679 t4 −13762657106 t5 − 2813455007 t6 − 482222476 t7 − 71302149 t8 − 9284150 t9 −1080184 t10 − 113270 t11 − 10719 t12 − 911 t13 − 67 t14 − 3 t15 + 319799790576t + 867472584 t3 + 213000 t4 + 3716541 t5 − 274736 t6 5
− 74617 t7 + 12039 t8 − 7031 t9 et . 2 5 4

Similarly, the solutions y1,10 (t) and y2,10 (t) and y3,10 (t) are not valid when time tends to 1, as illustrated in Figures 3,4 and 5. Using the iteration 13

formulation (6), we successively have,   y1,0 (t) = y1 (0) = 0, y2,0 (t) = y2 (0) = 0,  y3,0 (t) = y3 (0) = 0,

and,

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 y1,1 (t, h) = t4 − t6 ,    y2,1 (t, h) = h (−5208 et + 5208 + 5208 t + 218 t4 +46 t5 + 9 t6 + 868 t3 + t7 + 2604 t2 ) ,    y3,1 (t, h) = 0,

and in general for n > 1,

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 y1,n+1 (t, h) = ty2,n (t, h) + t4 − t6 ,      y3,n+1 (t, h) = y1,n (t, h) , R t t−τ  ∂y2,n (τ,h) − y2,n (τ, h) + τ ∂y1,n∂τ(τ,h) y (t, h) = y (t, h) − h e 2,n+1 2,n  ∂τ 0      −τ y1,n (τ, h) + τ ∂y3,n∂τ(τ,h) − τ 2 y3,n (τ, h) − g(τ ) dτ ,

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where g(τ ) = 4τ 3 + 12τ 4 + 8τ 5 − 2τ 6 − τ 7 . We define the residual functions of y1,10 (t, h) and y2,10 (t, h) and y3,10 (t, h) as,

(15)

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 r1,10 (t, h) = y1,10 (t, h) − ty2,10 (t, h) − t4 + t6 ,    r (t, h) = y (t, h) − y (t, h) , 2,10 3,10 1,10 ∂y2,10 (t,h) (t,h) − y2,10 (t, h) + t ∂y1,10 − ty1,10 (t, h) r3,10 (t, h) =  ∂t ∂t   ∂y3,10 (t,h) 2 3 − t y3,10 (t, h) − 4t − 12t4 − 8t5 + 2t6 + t7 . +t ∂t

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For obtaining an optimal value of h, we choose the global minimum point of the error of residual function (15): Z 1
e10 (h) = |r1,10 (t, h)|2 + |r2,10 (t, h)|2 + |r3,10 (t, h)|2 dt. 0

The minimum point of e10 (h) , as h = 0.13754, is obtained by using Maple software. Thus, we select h = 0.13754, and obtain the following successive

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approximations,

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y1,10 (t) = −5023041369.1 t − 5023041369.1 t2 − 2519453486.9 t3 − 845157962.9 t4 −213281395.5 t5 − 43177779 t6 − 7300829.4 t7 − 1059575 t8 − 134522.7 t9 −15134.1 t10 − 1519.4 t11 − 136.207 t12 − 10.878 t13 − 0.76171 t14 − 0.04054 t15 −0.00102 t16 + (5023041369.1 t + 7932802.4 t3 + 51599.9 t4 − 29995.2 t5 +1162.29 t6 − 914.219 t7 + 15.939 t8 − 7.8854 t9 − .27438 t10 + 0.10677 t11 ) et ,

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y2,10 (t) = −131847188413.53 − 131847188413.53 t − 65963371140.96 t2 −22019545271.45 t3 − 5514841771.07 t4 − 1104065393.96 t5 − 183694633.65 t6 −26054740.80 t7 − 3203894.46 t8 − 345168.92 t9 − 32741.18 t10 − 2731.509 t11 −198.692 t12 − 12.3261791 t13 − 0.608030 t14 − 0.018869 t15 − 0.00016098 t16 + (131847188413.53 + 39776934.20 t2 + 5236934.99 t3 − 3916480.81 t4 +7348.999 t5 − 5763.965 t6 − 258.9664, t7 + 90.01825 t8 −1.613238 t9 + 0.6480060 t10 + 0.01941403 t11 − 0.0048535 t12 ) et ,

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y3,10 (t) = 365661789.58 t + 365661789.58 t2 + 182803965.80 t3 + 60950350.90 t4 +15229737.53 t5 + 3033231.78 t6 + 499185.75 t7 + 69310.87 t8 + 8193.82 t9 +822.24 t10 + 68.82 t11 + 4.548 t12 + 0.1626 t13 − 0.009503 t14 − 0.001073 t15 + (−365661789.58 t + 26928.98 t3 − 33647.29 t4 + 26353.85 t5 −68.7635 t6 + 56.3138 t7 + 4.88742 t8 − 2.13869 t9 ) et .

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The comparison between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter, are shown in Figures 3,4 and 5.

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5. Conclusion

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The attention has here been restricted to a class of differential-algebraic equations. The VIM with an auxiliary parameter investigated and sufficient conditions which guarantee the convergence of method are presented. This technique provides a simple way to adjust and control the convergence region of approximate solution of differential-algebraic equations for any values of t. An optimal auxiliary parameter can be determined by minimizing of the error of the residual function. The applications of this method for solving two examples are described. Numerical results and graphical representations explicitly reveal the complete reliability of proposed method. 15

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4 Exact solution VIM with an auxiliary parameter Standard VIM

2 0

−4 −6 −8 −10

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

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y1(t)

−2

0.8

0.9

1

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Figure 3: The comparison between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter for y1 (t) in example 4.2.

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Exact Solution VIM with an auxiliary parameter Standard VIM

1.5

y2(t)

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1

0.5

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0

−0.5

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−1

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

Figure 4: The comparison between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter for y2 (t) in example 4.2.

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4 Exact solution VIM with an auxiliary parameter Standard VIM

2 0

3

y (t)

−2

−6 −8 −10

0

0.1

0.2

0.3

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−4

0.4

0.5 t

0.6

0.7

0.8

0.9

1

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Figure 5: The comparison between exact solution and 10th-order approximation solutions by VIM and VIM with an auxiliary parameter for y3 (t) in example 4.2.

References

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[1] Gear, C. W., Petzold, L. R. (1984). ODE methods for the solution of differential-algebraic systems. SIAM Journal on Numerical Analysis, 21(4), 716-728.

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[2] Brenan, K. E., Campbell, S. L. V., Petzold, L. R. (1989). Numerical solution of initial-value problems in differential-algebraic equations (Vol. 14). Siam. [3] Campbell, S. L. V., Campbell, S. L. (1980). Singular systems of differential equations (Vol. 1). San Francisco: Pitman.

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[4] Hairer, E., Lubich, C., Roche, M. (1989). The Numerical Solution of DifferentialAlgebraic Systems by RungeKutta Methods. SLNM 1409. SpringerVerlag, Berlin. [5] Gear, C. W. (1971). Numerical initial value problems in ordinary differential equations. Prentice Hall PTR. 17

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[6] Brenan, K. E., Campbell, S. L. V., Petzold, L. R. (1989). Numerical solution of initial-value problems in differential-algebraic equations (Vol. 14). Siam.

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