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Variational iteration technique for solving higher order boundary value problems
Applied Mathematics and Computation 189 (2007) 1929–1942 www.elsevier.com/locate/amc
Variational iteration technique for solving higher order boundary value problems Muhammad Aslam Noor *, Syed Tauseef Mohyud-Din Department of Mathematics, COMSATS Institute of Information Technology, Sector H-8/1, Islamabad 44000, Pakistan
Abstract In this paper, we have shown that higher order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration technique. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. Comparisons are made to confirm the reliability of the technique. Variational iteration technique may be considered as alternative and efficient for finding the approximate solutions of the boundary values problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Variational iteration; Nonlinear problems; Boundary value problems; System of integral equations; Approximate solution
1. Introduction In this paper, we consider the higher order boundary value problems. First, we consider the general fourth-order boundary value problem of the type: uð4Þ ðxÞ ¼ f ðx; u; u0 ; u00 ; u000 Þ
ð1Þ
with the boundary conditions uðaÞ ¼ a1 ;
u0 ðaÞ ¼ a2 ;
uðbÞ ¼ b1 ;
u0 ðbÞ ¼ b2 ;
where f is continuous function on [a, b] and the parameters ai and bi ; i ¼ 1; 2 are real constants. Such types of system arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory, see [4] and references therein. Various numerical methods including finite difference and Bspline were developed for solving fourth-order boundary value problems; see [3,5,16]. Homotopy perturbation method has been used in [13,14] for the solution of fourth-order boundary value problems.
*
Corresponding author. E-mail addresses: [email protected] (M. Aslam Noor), [email protected] (S.T. Mohyud-Din).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.12.071
1930
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
We now consider the general fifth-order boundary value problem: y ðvÞ ðxÞ ¼ gðxÞy þ qðxÞ
ð2Þ
with boundary conditions yðaÞ ¼ A1 ;
y 0 ðaÞ ¼ A2 ;
y 00 ðaÞ ¼ A3 ;
yðbÞ ¼ B1 ;
y 0 ðbÞ ¼ B2 :
This type of boundary value problems arises in the mathematical modeling of the viscoelastic flows and other branches of mathematical, physical and engineering sciences see [1,2,4,12] and the references therein. Several numerical methods including spectral Galerkin and collocation [4,12], decomposition [17] and sixth order B-spline have been developed for solving fifth-order boundary value problems. The use of spline function in the context of fifth-order boundary value problems was studied by Fyfe [6], who used the quintic polynomial spline functions to develop consistency relation connecting the values of solution with fifth-order derivative at the respective nodal points. Recently, fifth-order boundary value problems are solved in [15] by using homotopy perturbation method. He [7–10] developed the variational iteration method for solving nonlinear initial and boundary value problems. It is worth mentioning that the method was first considered by Inokuti et al. [11]. The basic motivation of this paper is to apply the variational iteration method to solve a system of integral equations. It is shown that the method provides the solution in a rapid convergent series. The variational iteration method has been shown [7–11] to solve effectively, easily and accurately a large class of linear and nonlinear, ordinary, partial, deterministic or stochastic differential equations with approximate solutions, which converge rapidly to accurate solutions. It is shown that fourth- and fifth-order boundary value problems are equivalent to the system of integral equations by using a suitable transformation. This alternative equivalent transformation plays an important and fundamental part in solving the boundary value problems. We show that the equivalent system of integral equations can be solved efficiently using the variational iteration method. This clearly indicates that the variational iteration technique may be considered as an alternative method for solving linear and nonlinear problems. 2. Variational iteration method To illustrate the basic concept of the technique, we consider the following general differential equation: Lu þ Nu ¼ gðxÞ;
ð3Þ
where L is a linear operator, N a nonlinear operator and g(x) is a forcing term. According to variational iteration method [7–11], we can construct a correct functional as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kðLun ðsÞ þ N ~ un ðsÞ gðsÞÞ ds; ð4Þ 0
where k is a Lagrange multiplier, which can be identified optimally via variational iteration method. The subscripts n denote the nth approximation, ~ un is considered as a restricted variation. i.e. d~un ¼ 0; (4) is called as a correct functional. The solution of the linear problems can be solved in a single iteration step due to the exact identification of the Lagrange multiplier. The principles of variational iteration method and its applicability for various kinds of differential equations are given [7–11,13]. For the sake of simplicity and to convey the idea of the technique, we consider the following system of differential equations: x0i ðtÞ ¼ fi ðt; xi Þ;
i ¼ 1; 2; 3; . . . ; n
ð5Þ
subject to the boundary conditions. xi ð0Þ ¼ ci ; i ¼ 1; 2; 3; . . . ; n. To solve the system by means of the variational iteration method, we rewrite system (5) in the following form: x0i ðtÞ ¼ fi ðxi Þ þ gi ðtÞ;
i ¼ 1; 2; 3; . . . ; n
subject to the boundary conditions. xi ð0Þ ¼ ci ; i ¼ 1; 2; 3; . . . ; n and gi is defined in (3).
ð6Þ
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
The correct functional for the nonlinear system (6) can be approximated as Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ xi ðtÞ ¼ x1 ðtÞ þ k1 ðx1 ðT Þ; f1 ð~x1 ðT Þ; ~x2 ðT Þ; . . . ; ~xðkÞ n ðT ÞÞ g1 ðT ÞÞ dT ; 0 Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ x2 ðtÞ ¼ x2 ðtÞ þ k2 ðx2 ðT Þ; f2 ð~x1 ðT Þ; ~x2 ðT Þ; . . . ; ~xðkÞ n ðT ÞÞ g2 ðT ÞÞ dT ; 0
.. . xnðkþ1Þ ðtÞ
¼
xðkÞ n ðtÞ
þ
Z
t
ðkÞ
1931
ð7Þ
ðkÞ
kn ðx0ðkÞ x1 ðT Þ; ~x2 ðT Þ; . . . ; ~xðkÞ n ðT Þ; fn ð~ n ðT ÞÞ gn ðT ÞÞ dT ; 0
where ki ¼ 1; i ¼ 1; 2; 3; . . . ; n are Lagrange multipliers, ~x1 ; ~x2 ; . . . ; ~xn denote the restricted variations. For ki ¼ 1; i ¼ 1; 2; 3; . . . ; n; we have the following iterative schemes: Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ x1 ðtÞ ¼ x1 ðtÞ ðx1 ðT Þ; f1 ðx1 ðT Þ; x2 ðT Þ; . . . ; xðkÞ n ðT ÞÞ g1 ðT ÞÞ dT ; 0 Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ x2 ðtÞ ¼ x2 ðtÞ ðx2 ðT Þ; f2 ðx1 ðT Þ; x2 ðT Þ; . . . ; xðkÞ n ðT ÞÞ g2 ðT ÞÞ dT ; 0
.. . xnðkþ1Þ ðtÞ ¼ xðkÞ n ðtÞ
Z
t
ðkÞ
ð8Þ
ðkÞ
ðkÞ ðx0ðkÞ n ðT Þ; fn ðx1 ðT Þ; x2 ðT Þ; . . . ; xn ðT ÞÞ gn ðT ÞÞ dT : 0
If we start with the initial approximations xi ð0Þ ¼ ci ; i ¼ 1; 2; 3; . . . ; n then the approximations can be completely determined; finally we approximate the solution. ðkÞ ðnÞ xi ðtÞ ¼ Limk!1 xi ðtÞ by the nth term xi ðtÞ for i ¼ 1; 2; 3; . . . ; n. 3. Applications We first show that fourth- or fifth-order boundary value problem may be reformulated as a system of integral equations. We use the variational iteration method developed in Section 2 to solve the resultant system of integral equations. Example 3.1 [17]. Consider the following linear boundary value problem of fifth order: y ðvÞ ðxÞ ¼ y 15ex 10xex with boundary conditions yð0Þ ¼ 0; y 0 ð0Þ ¼ 1; y 00 ð0Þ ¼ 0;
ð9Þ yð1Þ ¼ 0;
y 0 ð1Þ ¼ e:
The exact solution of the problem is yðxÞ ¼ xð1 xÞex : Using the transformation dy ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ sðxÞ; dx
ds ¼ zðxÞ; dx
we rewrite the fifth-order boundary value problem (9) and (10) as a system of differential equations 8 dy > ¼ qðxÞ; > > dx > > dq > > < dx ¼ f ðxÞ; df ¼ sðxÞ; dx > > > ds > ¼ zðxÞ; > dx > > : dz ¼ y 15ex 10xex dx
with yð0Þ ¼ 0; qð0Þ ¼ 1; f ð0Þ ¼ 0; zð0Þ ¼ A; sð0Þ ¼ B.
ð10Þ
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The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ þ1; i ¼ 1; 2; 3; 4; 5. Rx 8 ðkþ1Þ y ðxÞ ¼ 0 þ 0 qðkÞ ðtÞ dt; > > > > Rx > ðkþ1Þ > ðxÞ ¼ 1 þ 0 f ðkÞ ðtÞ dt; > >q < Rx f ðkþ1Þ ðxÞ ¼ 0 þ 0 sðkÞ ðtÞ dt; > > Rx > > sðkþ1Þ ðxÞ ¼ A þ 0 zðkÞ ðtÞ dt; > > > > Rx : ðkþ1Þ z ðxÞ ¼ B þ 0 ð15et 10tet þ y ðkÞ ðtÞÞ dt; y ð0Þ ðxÞ ¼ 0;
qð0Þ ðxÞ ¼ 1;
f ð0Þ ðxÞ ¼ 0;
sð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: y ð1Þ ðxÞ ¼ þx; qð1Þ ðxÞ ¼ 1; f ð1Þ ðxÞ ¼ þAx; sð1Þ ðxÞ ¼ A þ Bx; zð1Þ ðxÞ ¼ B þ 5 5ex 10xex ; y ð2Þ ðxÞ ¼ þx; A 2 x; 2 B f ð2Þ ðxÞ ¼ Ax þ x2 ; 2 sð2Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex ; 1 zð2Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 ; 2 A 3 ð3Þ y ðxÞ ¼ x þ x ; 6 A B ð3Þ q ðxÞ ¼ 1 x2 þ x3 ; 2 6 B 5 f ð3Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex ; 2 2 x3 sð3Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ ; 6 1 2 ð3Þ x x z ðxÞ ¼ B þ 5 5e 10xe þ x ; 2 A 3 B 4 ð4Þ y ðxÞ ¼ x þ x þ x ; 6 24 A 2 B 3 5 5 ð4Þ q ðxÞ ¼ 1 x þ x 25 15x x2 þ x3 þ 25ex 10xex ; 2 6 2 6 B 5 x4 f ð4Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ ; 2 2 24 x3 ð4Þ x x s ðxÞ ¼ A þ Bx 5 þ 5x þ 5e 10xe þ ; 6 1 2 A 4 ð4Þ x x z ðxÞ ¼ B þ 5 5e 10xe þ x þ x ; 2 24 A B 15 5 5 y ð5Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex ; 6 24 2 6 24 qð2Þ ðxÞ ¼ 1
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
A 2 B 3 5 5 x5 x þ x 25 15x x2 þ x3 þ 25ex 10xex þ ; 2 6 2 6 120 B 5 f ð5Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex ; 2 2 x3 A 5 x; sð5Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 6 120 1 A B 5 x; zð5Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 A B 15 5 5 1 6 x; y ð6Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 A B 5 5 x5 ; qð6Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 B 5 A 6 x; f ð6Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 x3 A 5 B 6 x þ x; sð6Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 1 A B 5 25 x 45 35x x2 zð6Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 5 5 5 x þ 45ex 10xex ; x3 x4 þ 6 24 120 A B 15 5 5 1 6 x; y ð7Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 A B 5 5 x5 A 7 x; þ qð7Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 5040 B 5 A 6 B 7 x þ x; f ð7Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 x3 A 5 B 6 x þ x 55 45x sð7Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 x þ x þ 55ex 10xex ; x2 x3 x4 2 6 24 120 720 1 A B 5 25 x 45 35x x2 zð7Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 x þ 45ex 10xex þ x7 ; x x þ 6 24 120 5040 A B 15 5 5 1 6 A x þ x8 ; y ð8Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 40320 A B 5 5 x5 A 7 B x þ x8 ; þ qð8Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 40320 120 5040 B 5 A 6 B 7 x þ x f ð8Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 45 35 25 15 5 5 6 5 7 x x þ x þ 65ex 10xex ; 65 55x x2 x3 x4 2 6 24 120 720 5040 x3 A 5 B 6 x þ x 55 45x sð8Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 1 x þ x þ 55ex 10xex þ x8 ; x2 x3 x4 2 6 24 120 720 40320
qð5Þ ðxÞ ¼ 1
1933
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1 A B 5 25 x 45 35x x2 zð8Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 x þ 45ex 10xex þ x7 ; x x þ 6 24 120 5040 A B 15 5 5 1 6 B x þ x9 ; y ð9Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 362880 A B 5 5 x5 qð9Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 A 7 B 55 2 45 3 35 4 25 5 15 6 8 x þ x 75 65x x x x x x þ 5040 40320 2 6 24 120 720 5 7 5 x þ x8 þ 75ex 10xex ; 5040 40320 B 5 A 6 B 7 x þ x 65 55x f ð9Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 45 35 25 15 5 5 6 5 7 1 x x þ x þ 65ex 10xex þ x9 ; x2 x3 x4 2 6 24 120 720 5040 362880 x3 A 5 B 6 x þ x 55 45x sð9Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 1 x þ x þ 55ex 10xex þ x8 ; x2 x3 x4 2 6 24 120 720 40320 1 A B 5 25 x 45 35x x2 zð9Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 A x þ 45ex 10xex þ x7 þ x9 ; x x þ 6 24 120 5040 362880 A B 15 5 5 y ð10Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex 6 24 2 6 24 1 6 B 65 55 45 35 5 25 6 x þ x9 85 75x x2 x3 x4 x x þ 720 362880 2 6 24 120 720 15 7 5 5 x x8 þ x9 þ 85ex 10xex ; 5040 40320 362880 A B 5 5 x5 qð10Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 A 7 B 55 45 35 25 5 15 6 x þ x8 75 65x x2 x3 x4 x x þ 5040 40320 2 6 24 120 720 5 7 5 1 x þ x8 þ 75ex 10xex þ x10 ; 5040 40320 3628800 B 5 A 6 B 7 x þ x 65 55x f ð10Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 45 35 25 15 5 5 6 5 7 1 x x þ x þ 65ex 10xex þ x9 ; x2 x3 x4 2 6 24 120 720 5040 362880 x3 A 5 B 6 x þ x 55 45x sð10Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 1 A x þ x þ 55ex 10xex þ x8 þ x10 ; x2 x3 x4 2 6 24 120 720 40320 3628800 1 A B 5 25 x 45 35x x2 zð10Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 7 A B x x x þ 45e 10xe þ x þ x9 þ x10 : x x þ 6 24 120 5040 362880 3628800
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
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Table 1 Error estimates x
Analytical solution yðxÞ ¼ xð1 xÞex
Errorsa (variational iteration)
Errorsa (B-spline)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.000000 0.099465382 0.195424441 0.283470349 0.358037927 0.412180317 0.437308512 0.422840684 0.356086548 0.221364279 0.000000
0.0000000 3E11 2E10 4E10 8E10 1.2E9 2E9 2.2E9 1.9E9 1.4E9 0.000
0.0000000 8.0E3 1.2E3 5.0E3 3.0E3 8.0E3 6.0E3 0.000 9.0E3 9.0E3 0.000000
a
Error = analytical solution numerical solution.
The solution is given as
1 3 1 4 1 5 1 6 1 7 1 1 1 1 8 x þ þ A x þ þ B x9 yðxÞ ¼ x þ Ax þ Bx x x 6 24 8 30 144 896 40320 72576 362880 1 1 1 A 1 x10 þ x11 x12 x12 x13 45360 403200 3991680 622702080 44478720 B 1 x14 þ Oðx15 Þ: 8717029120 544864320
Using the boundary conditions at x = 1, we have A ¼ 2:99999988;
B ¼ 8:00000054:
The series solution is given by yðxÞ ¼ x 0:49999998x3 0:33333355x4 0:125x5 0:03333333x6 0:006944444x7 0:0011904762x8 0:0001736111x9 0:0000220458x10 0:000002488x11 0:000000244x12 0:000000022x13 0:000000001x14 þ Oðx15 Þ; which is exactly the same as obtained in [17] by using the decomposition method and in [15] by using the homotopy perturbation method. Table 1 exhibits a comparison between the errors obtained by using the proposed variational iteration method [7–11] and by using the sixth degree B-spline method [2]. Examining this table closely shows the improvements obtained by using the proposed scheme. Higher accuracy can be obtained by evaluating more components of yðxÞ. Example 3.2 [17]. Consider the following nonlinear boundary value problem of fifth order: y ðvÞ ðxÞ ¼ ex y 2 ðxÞ
ð11Þ
with boundary conditions yð0Þ ¼ y 0 ð0Þ ¼ y 00 ð0Þ ¼ 1;
yð1Þ ¼ y 0 ð1Þ ¼ e:
The exact solution for this problem is yðxÞ ¼ ex : Using the transformation dy ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ sðxÞ; dx
ds ¼ zðxÞ; dx
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we rewrite the above fifth-order boundary value problem as a system of differential equations 8 dy > ¼ qðxÞ; > dx > > > dq > > < dx ¼ f ðxÞ; df ¼ sðxÞ; dx > > > ds > ¼ zðxÞ; > dx > > : dz ¼ ex y 2 ðxÞ dx
with yð0Þ ¼ qð0Þ ¼ f ð0Þ ¼ 1; sð0Þ ¼ A; zð0Þ ¼ B. The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ þ1; i ¼ 1; 2; 3; 4; 5. 8 Rx y ðkþ1Þ ðxÞ ¼ 1 þ 0 qðkÞ ðtÞ dt; > > > > ðkþ1Þ Rx > > ðxÞ ¼ 1 þ 0 f ðkÞ ðtÞ dt; > >q < Rx f ðkþ1Þ ðxÞ ¼ 1 þ 0 sðkÞ ðtÞ dt; > Rx > > > sðkþ1Þ ðxÞ ¼ A þ 0 zðkÞ ðtÞ dt; > > > > Rx : ðkþ1Þ ðxÞ ¼ B þ 0 et ðy ðkÞ ðtÞÞ2 dt; z y ð0Þ ðxÞ ¼ 1;
qð0Þ ðxÞ ¼ 1;
f ð0Þ ðxÞ ¼ 1;
sð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: y ð1Þ ðxÞ ¼ 1 þ x; qð1Þ ðxÞ ¼ 1 þ x; f ð1Þ ðxÞ ¼ 1 þ Ax; sð1Þ ðxÞ ¼ A þ Bx; zð1Þ ðxÞ ¼ B þ 1 ex ; x2 ; 2 A qð2Þ ðxÞ ¼ 1 þ x þ x2 ; 2 B ð2Þ f ðxÞ ¼ 1 þ Ax þ x2 ; 2 sð2Þ ðxÞ ¼ A þ Bx 1 þ x þ ex ; y ð2Þ ðxÞ ¼ 1 þ x þ
zð2Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex ; x2 A 3 þ x; 2 6 A B qð3Þ ðxÞ ¼ 1 þ x þ x2 þ x3 ; 2 6 B 1 f ð3Þ ðxÞ ¼ 1 þ Ax þ x2 þ 1 x þ x2 ex ; 2 2 ð3Þ x s ðxÞ ¼ A þ Bx 1 þ x þ e 6 þ 2x þ 6ex þ 4xex þ x2 ex ; y ð3Þ ðxÞ ¼ 1 þ x þ
1 zð3Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex þ 6 6ex 6xex 3x2 ex x3 ex x4 ex ; 4 x2 A 3 B 4 ð4Þ y ðxÞ ¼ 1 þ x þ þ x þ x ; 24 2 6 A B 1 1 qð4Þ ðxÞ ¼ 1 þ x þ x2 þ x3 1 þ x x2 þ x3 þ ex ; 2 6 2 6
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
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B 2 1 x þ 1 x þ x2 ex þ 12 6x þ x2 12ex 6xex x2 ex ; 2 2 sð4Þ ðxÞ ¼ A þ Bx 1 þ x þ ex 6 þ 2x þ 6ex þ 4xex þ x2 ex 30 þ 6x þ 26ex 1 þ 24xex þ 9x2 ex þ 2x3 ex þ x4 ex ; 4 zð4Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex þ 6 6ex 6xex 3x2 ex x3 ex
f ð4Þ ðxÞ ¼ 1 þ Ax þ
1 A3 720 720ex 720xex 360x2 ex 120x3 ex 30x4 ex 6x5 ex ; x4 ex þ 4 216 x2 A 3 B 4 1 1 1 ð5Þ y ðxÞ ¼ 1 þ x þ þ x þ x þ 1 x þ x2 x3 þ x4 ex ; 24 2 6 24 2 6 A B 1 1 qð5Þ ðxÞ ¼ 1 þ x þ x2 þ x3 1 x x2 þ x3 þ ex 20 þ 12x 3x2 2 6 2 6 1 3 þ x þ 20ex þ 8xex þ x2 ex ; 3 B 1 ð5Þ f ðxÞ ¼ 1 þ Ax þ x2 þ 1 x þ x2 ex þ 12 6x þ x2 12ex 6xex 2 2 1 x2 ex þ 90 30x þ 3x2 90ex 60xex 18x2 ex 3x3 ex x4 ex ; 4 ð5Þ x x x 2 x s ðxÞ ¼ A þ Bx 1 þ x þ e 6 þ 2x þ 6e þ 4xe þ x e 30 þ 6x þ 30ex þ 24xex 1 A3 ð720 þ 720x 720ex þ 720xex þ 9x2 ex þ 2x3 ex þ x4 ex þ 4 216 þ 360x2 ex þ 120x3 ex þ 30x4 ex þ 6x5 ex Þ; zð5Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex þ 6 6ex 6xex 3x2 ex x3 ex 1 A3 ½720 720ex 720xex 360x2 ex 120x3 ex 30x4 ex 6x5 ex x4 ex þ 4 216 B4 þ ð40320 40320ex 40320xex 20160x2 ex 6720x3 ex 1680x4 ex 576 336x5 ex 56x6 ex 8x7 ex x8 ex Þ: The solution is given as
1 2 1 3 1 4 1 5 1 6 1 7 1 1 x þ x þ x þ A yðxÞ ¼ 1 þ x þ x þ Ax þ Bx þ x8 2 6 24 120 720 5040 20160 40320 1 1 1 1 1 1 2 9 10 x þ A Aþ þ x þ x11 18144 362880 3628800 995840 997920 1900800 1 1 1 1 101 A2 þ A Bþ AB þ x12 Oðx13 Þ: 3421440 2280960 6842880 6842880 479001600
Using the boundary condition at x = 1, we have A ¼ 0:9999967742;
B ¼ 1:0000145020:
The series solution is yðxÞ ¼ 1 þ x þ 0:5x2 þ 0:166666236x3 þ 0:04166727092x4 þ 0:008333333333x5 þ 0:00138888888x6 þ 0:000198412x7 þ 0:00002480142729x8 þ 0:00005236x9 þ 0:000000275x10 0:00000898x11 0:000000064x12 þ Oðx13 Þ; which is exactly the same solution as obtained in [17] by using the decomposition method and in [15] by using the homotopy perturbation method.
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Table 2 Error estimates x
Analytical solution
Errorsa (variational iteration)
Errorsa (B-spline)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.000000000 1.105170918 1.221402758 1.349858808 1.491824698 1.648721271 1.822118800 2.013752707 2.225540928 2.459603111 2.718281828
0.0000 1.0E9 2.0E9 1.0E9 2.0E9 3.1E8 3.7E8 4.1E8 3.1E8 1.4E8 0.0000
0.0000 7.0E4 7.2E4 4.1E4 4.6E4 4.7E4 4.8E4 3.9E4 3.1E4 1.6E4 0.0000
a
Error = analytical solution numerical solution.
Table 2 shows the exact values, numerical solutions, and the errors obtained by using the variational iteration method [7–11] and by using the sixth degree B-spline method [2] for x ¼ 0:0; 0:1; 0:2; . . . ; 1:0: The table clearly indicates the improvements as compared with B-spline method. Example 3.3 [14]. We consider the following nonlinear boundary value problem of fourth order: uðivÞ ðxÞ ¼ u2 x10 þ 4x9 4x8 4x7 þ 8x6 4x4 þ 120x 48 with boundary conditions uð0Þ ¼ u0 ð0Þ ¼ 0; uð1Þ ¼ 1; u0 ð1Þ ¼ 1: The exact solution of this problem is
ð12Þ ð13Þ
uðxÞ ¼ x5 2x4 þ 2x2 : Using the transformation du ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ zðxÞ; dx
we rewrite the fourth-order boundary value problem (12) and (13) as a system of differential equations 8 du ¼ qðxÞ; > dx > > > dq < ¼ f ðxÞ; dx df > > dx ¼ zðxÞ; > > : dz ¼ u2 x10 þ 4x9 4x8 4x7 þ 8x6 4x4 þ 120x 48 dx with uð0Þ ¼ qð0Þ ¼ 0; f ð0Þ ¼ A; zð0Þ ¼ B. The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ 1; i ¼ 1; 2; 3; 4. 8 Rx uðkþ1Þ ðxÞ ¼ 0 0 qðkÞ ðtÞ dt; > > > Rx > ðkþ1Þ < q ðxÞ ¼ 0 0 f ðkÞ ðtÞ dt; Rx ðkþ1Þ > ðxÞ ¼ A 0 zðkÞ ðtÞ dt; >f > > Rx : ðkþ1Þ z ðxÞ ¼ B 0 ððuðkÞ ðtÞÞ2 t10 þ 4t9 4t8 4t7 þ 8t6 4t4 þ 120t 48Þ dt; uð0Þ ðxÞ ¼ 0;
qð0Þ ðxÞ ¼ 0;
f ð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: uð1Þ ðxÞ ¼ 0; qð1Þ ðxÞ ¼ Ax;
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
f ð1Þ ðxÞ ¼ A Bx; zð1Þ ðxÞ ¼ B þ uð2Þ ðxÞ ¼
x11 4 4 4 8 4 120 2 x þ 48x; x10 þ x9 þ x8 x7 þ x5 9 8 7 5 2 11 10
A 2 x; 2
B 2 x; 2 x12 2x11 2 1 x8 2x6 þ x10 x9 þ þ 20x3 24x2 ; f ð2Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 x11 4 4 4 8 4 120 2 x þ 48x; x10 þ x9 þ x8 x7 þ x5 zð2Þ ðxÞ ¼ B þ 9 8 7 5 2 11 10 A B uð3Þ ðxÞ ¼ x2 þ x3 ; 2! 3! B x13 x12 2x11 1 10 1 2 7 ð3Þ x x9 þ x 5x4 8x3 ; þ þ q ðxÞ ¼ Ax þ x2 þ 2 63 105 1716 330 495 180 x12 2x11 2 1 x8 2x6 þ x10 x9 þ þ 20x3 24x2 ; f ð3Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 x11 4 4 4 8 4 120 2 A2 x þ 48x x5 ; x10 þ x9 þ x8 x7 þ x5 zð3Þ ðxÞ ¼ B þ 9 8 7 5 2 11 10 20 A 2 B 3 x14 x13 1 12 1 11 1 10 1 8 ð4Þ x x þ x x þ x5 2x4 ; þ u ðxÞ ¼ x þ x 2! 3! 1980 630 420 24024 4290 2970 B x13 x12 2x11 1 10 1 2 7 x x9 þ x 5x4 8x3 ; þ þ qð4Þ ðxÞ ¼ Ax þ x2 þ 2 63 105 1716 330 495 180 x12 2x11 2 1 x8 2x6 A2 6 þ x10 x9 þ þ 20x3 24x2 þ x; f ð4Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 120 x11 4 4 4 8 4 120 2 A2 B2 7 x þ 48x x5 þ x10 þ x9 þ x8 x7 þ x5 x; zð4Þ ðxÞ ¼ B þ 9 8 7 5 2 11 10 20 252 14 13 A B x x 1 12 1 11 1 10 1 8 x x þ x x þ x5 2x4 ; þ uð5Þ ðxÞ ¼ x2 þ x3 2! 3! 1980 630 420 24024 4290 2970 B x13 x12 2x11 1 10 1 2 7 A 7 x x9 þ x 5x4 8x3 x; þ þ qð5Þ ðxÞ ¼ Ax þ x2 þ 2 63 105 840 1716 330 495 180 x12 2x11 2 1 x8 2x6 A2 6 B2 8 þ x10 x9 þ þ 20x3 24x2 þ x þ x; f ð5Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 120 2016 A B x14 x13 1 12 1 11 1 10 1 8 A2 8 x x þ x x þ x5 2x4 þ þ x; uð6Þ ðxÞ ¼ x2 þ x3 2! 3! 1980 630 420 24024 4290 2970 6720 B x13 x12 2x11 1 10 1 2 7 x x9 þ x 5x4 þ þ qð6Þ ðxÞ ¼ Ax þ x2 þ 2 63 105 1716 330 495 180 A2 7 B2 9 x x; 8x3 840 18144 .. . qð2Þ ðxÞ ¼ Ax þ
The solution is uðxÞ ¼ uð0Þ ðxÞ þ uð1Þ ðxÞ þ uð2Þ ðxÞ þ uð3Þ ðxÞ þ ; A B uðxÞ ¼ x2 þ x3 2x4 þ x5 0:002380952380952x8 þ 0:001587301587301x10 2 6 0:000505050501x11 0:0003367x12 þ 0:0002331002331x13 0:000041625043x14 þ
A2 8 B x7 þ x þ 14112 6720
1939
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M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
Table 3 Error estimates x
Exact solution
Series solution
Errora
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0000000000 0.0198100000 0.0771200000 0.1662300000 0.2790400000 0.4062500000 0.5385599999 0.6678700000 0.7884800000 0.8982900000 1.0000000000
0.0000000000 0.0198099999 0.0771199999 0.1662299999 0.2790399999 0.4062499999 0.5385599999 0.6678699999 0.7884799999 0.8982899999 0.9999999999
0.0000E+00 4.579E16 1.5959E15 3.1641E15 4.7739E15 6.0507E15 6.6613E15 6.6613E15 5.2180E15 2.5535E15 3.3306E16
a
Error = Exact solution series solution.
Applying the boundary conditions at x ¼ 1, A ¼ 3:9999999999887983;
B ¼ 3:19339870128 1013 :
The series solution is uðxÞ ¼ 1:99999999993x2 þ 5:322233 1014 x3 2x4 þ x5 0:002380952380952x8 þ 7:04012 1017 x9 þ 4:03323 1017 x10 3:97902 1017 x11 8:94467 1018 x12 þ ; which is exactly the same as obtained in [14] by using the homotopy perturbation method. Table 3 shows the comparison between the exact solution and the series solution obtained by using the proposed algorithm. Higher accuracy can be obtained by evaluating more components of u(x). Example 3.4 [16]. We consider the following boundary value problem of fourth order: 1 uðivÞ ðxÞ ¼ ð1 þ cÞu00 ðxÞ cuðxÞ þ cx2 1 2 with boundary conditions uð0Þ ¼ 1 ¼ u0 ð0Þ;
uð1Þ ¼ 1:5 þ sinhð1Þ;
ð14Þ
u0 ð1Þ ¼ 1 þ coshð1Þ:
ð15Þ
The exact solution of this problem is 1 uðxÞ ¼ 1 þ x2 þ sinhðxÞ: 2 Using the transformation du ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ zðxÞ; dx
we can rewrite the fourth-order boundary value problem as a system of differential equations 8 du ¼ qðxÞ; > dx > > > > < dq ¼ f ðxÞ; dx df > > ¼ zðxÞ; > dx > > : dz ¼ ð1 þ cÞu00 ðxÞ cuðxÞ þ 12 cx2 1 dx
with uð0Þ ¼ qð0Þ ¼ 1; f ð0Þ ¼ A; zð0Þ ¼ B. The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ 1; i ¼ 1; 2; 3; 4.
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
1941
Rx 8 ðkþ1Þ u ðxÞ ¼ 1 0 qðkÞ ðtÞ dt; > > > R > < qðkþ1Þ ðxÞ ¼ 1 x f ðkÞ ðtÞ dt; 0 R x ðkÞ ðkþ1Þ > ðxÞ ¼ A 0 z ðtÞ dt; >f > > Rx : ðkþ1Þ ðxÞ ¼ B 0 ½ð1 þ cÞf ðkÞ ðtÞ cuðkÞ ðtÞ þ 12 ct2 1 dt; z uð0Þ ðxÞ ¼ 1;
qð0Þ ðxÞ ¼ 1;
f ð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: uð1Þ ðxÞ ¼ 1 x; qð1Þ ðxÞ ¼ 1 Ax; f ð1Þ ðxÞ ¼ A Bx;
1 3 z ðxÞ ¼ B þ ð1 AÞx þ c ð1 AÞx x ; 6 ð1Þ
A 2 x; 2 B qð2Þ ðxÞ ¼ 1 Ax þ x2 ; 2
uð2Þ ðxÞ ¼ 1 x þ
x2 x2 1 c ð1 AÞ x4 ; 2 2 24 B 2 1 2 1 ð2Þ z ðxÞ ¼ B ð1 þ cÞ Ax x c x x þ cx3 x; 2 2 6
f ð2Þ ðxÞ ¼ A Bx þ ð1 AÞ
.. . Using essentially the technique of Example 3.3, we can find the other approximate values. Consequently, the approximate solution is given by A 2 B 3 1 A c A B 5 c 5 B x þ x x4 þ x4 x4 þ cx4 þ x x þ cx5 2 6 24 24 24 24 120 120 120 x6 A B c cx6 cx7 þ x8 þ 5040 20160 720 720
uðxÞ ¼ 1 þ x þ
Now using the boundary conditions at x = 1, we have A 1 þ gðcÞ; B 1 þ hðcÞ; where both gðcÞ and hðcÞ grow rapidly with c. In reality, they should go to zero as the number of terms in the series goes to infinity. Scott [16] has solved the problem (14) with boundary conditions (15) for very large values of c with the orthonormalization process. It has been shown in [16] that each time the solutions started to lose their linear independence, one has to perform orthonormalization. In fact, as c got bigger it required more normalization. We would like to mention [16] that Eq. (14) can be rewritten in the following equivalent form: ðivÞ 1 u ðxÞ u00 ðxÞ þ 1 c u00 ðxÞ uðxÞ þ x2 ¼ 0: 2 From this it follows that the solution of the second order system is also a solution of the fourth-order problem. Hence, the solution of (14) is independent of c. In view of these comments and facts, decomposition, homotopy, variational, tanh/cotanh and other like methods could be corrected by a similar process of dividing up the interval into pieces as suggested by Scott.
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Remark. We would like to point out the approximate solution obtained by the variational iteration technique is in good agreement with the exact solution for the small values of the parameter c and continuously depends on the parameter c. We have also solved the Example 3.4 by using the Adomian decomposition, Homotopy perturbation method and Differential transformation method, see [13] for full details. It turns out that all these methods give us the same approximate solution. This is natural since all these techniques are based on finding the solutions of the problems in terms of the series., no matter which technique one uses. Every method has some advantages over the others. Variational technique depends on finding the appropriate and suitable value of the parameter, which is easy to find than finding the derivatives of the Adomian polynomials. In the homotopy method, one usually express the solutions as series with unknown coefficients and substituting the series in the equation and equating the coefficients of the powers on both sides, which may be difficult in some cases. In such cases, variational iteration technique can be considered as an alternative and efficient method for solving linear and nonlinear problems. Acknowledgements We would like to express our deepest gratitude to Prof. M. Scott for his valuable and interesting suggestions and comments as well as providing us some special problems. The research of M. Aslam Noor is supported by the Higher Education Commission, Pakistan, through the research Grant No: I-28/HEC/HRD/2005/90; whereas the research of S. Tauseef Mohyud-Din is also supported by Higher Education Commission, Pakistan, through indigenous PhD scholarship scheme. References [1] R.P. Agarwal, Boundary-value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986. [2] H.N. Caglar, S.H. Caglar, E.E. Twizell, The numerical solution of fifth order boundary-value problems with sixth degree B-spline functions, Appl. Math. Lett. 12 (1999) 25–30. [3] M.M. Chawla, C.P. Katti, Finite difference methods for two-point boundary-value problems involving higher order differential equations, BIT 19 (1979) 27–33. [4] A.R. Davies, A. Karageoghis, T.N. Philips, Spectral Galerkin methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1988) 647–662. [5] E. Doedel, Finite difference methods for nonlinear two-point boundary-value problem, SIAM J. Numer. Anal. 16 (1979) 173–185. [6] D.J. Fyfe, Linear dependence relations connecting equal interval nth degree splines and their derivatives, J. Inst. Math. Appl. 7 (1971) 398–406. [7] J.H. He, Variational iteration method – a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech. 34 (1999) 699–708. [8] J.H. He, Variational method for autonomous ordinary differential equations, Appl. Math. Comput. 114 (2000) 115–123. [9] J.H. He, Variational theory for linear magneto–electro-elasticity, Int. J. Nonlinear Sci. Numer. Simul. 2 (4) (2001) 309–316. [10] J.H. He, Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fract. 19 (4) (2004) 847–851. [11] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: Variational Method in the Mech. of Solids, Pergamon Press, New York, 1978, pp. 156–162. [12] A. Karageoghis, T.N. Philips, A.R. Davies, Spectral collocation methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1998) 805–813. [13] M. Aslam Noor, Some algorithms for solving boundary value problems, Preprint, 2006. [14] S.T. Moyud-Din, M. Aslam Noor, Homotopy perturbation method for solving fourth order boundary value problems, Math. Problems Eng. 2007 (2007) 1–15, Article ID 98602. [15] M. Aslam Noor, S.T. Mohyud-Din, An efficient algorithm for solving fifth order boundary value problems, Math. Comput. Model. 45 (2007) 954–964. [16] M. Scott, Special problems, private communication, 2006. [17] A.M. Wazwaz, The numerical solution of fifth-order boundary-value problems by Adomian decomposition method, J. Comput. Appl. Math. 136 (2001) 259–270.
Further reading [1] S. Momani, M. Aslam Noor, Numerical method of fourth-order fractional integro-differential equations, Appl. Math. Comput. 182 (2006) 754–760.
Variational iteration technique for solving higher order boundary value problems Muhammad Aslam Noor *, Syed Tauseef Mohyud-Din Department of Mathematics, COMSATS Institute of Information Technology, Sector H-8/1, Islamabad 44000, Pakistan
Abstract In this paper, we have shown that higher order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration technique. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. Comparisons are made to confirm the reliability of the technique. Variational iteration technique may be considered as alternative and efficient for finding the approximate solutions of the boundary values problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Variational iteration; Nonlinear problems; Boundary value problems; System of integral equations; Approximate solution
1. Introduction In this paper, we consider the higher order boundary value problems. First, we consider the general fourth-order boundary value problem of the type: uð4Þ ðxÞ ¼ f ðx; u; u0 ; u00 ; u000 Þ
ð1Þ
with the boundary conditions uðaÞ ¼ a1 ;
u0 ðaÞ ¼ a2 ;
uðbÞ ¼ b1 ;
u0 ðbÞ ¼ b2 ;
where f is continuous function on [a, b] and the parameters ai and bi ; i ¼ 1; 2 are real constants. Such types of system arise in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams and plate deflection theory, see [4] and references therein. Various numerical methods including finite difference and Bspline were developed for solving fourth-order boundary value problems; see [3,5,16]. Homotopy perturbation method has been used in [13,14] for the solution of fourth-order boundary value problems.
*
Corresponding author. E-mail addresses: [email protected] (M. Aslam Noor), [email protected] (S.T. Mohyud-Din).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.12.071
1930
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
We now consider the general fifth-order boundary value problem: y ðvÞ ðxÞ ¼ gðxÞy þ qðxÞ
ð2Þ
with boundary conditions yðaÞ ¼ A1 ;
y 0 ðaÞ ¼ A2 ;
y 00 ðaÞ ¼ A3 ;
yðbÞ ¼ B1 ;
y 0 ðbÞ ¼ B2 :
This type of boundary value problems arises in the mathematical modeling of the viscoelastic flows and other branches of mathematical, physical and engineering sciences see [1,2,4,12] and the references therein. Several numerical methods including spectral Galerkin and collocation [4,12], decomposition [17] and sixth order B-spline have been developed for solving fifth-order boundary value problems. The use of spline function in the context of fifth-order boundary value problems was studied by Fyfe [6], who used the quintic polynomial spline functions to develop consistency relation connecting the values of solution with fifth-order derivative at the respective nodal points. Recently, fifth-order boundary value problems are solved in [15] by using homotopy perturbation method. He [7–10] developed the variational iteration method for solving nonlinear initial and boundary value problems. It is worth mentioning that the method was first considered by Inokuti et al. [11]. The basic motivation of this paper is to apply the variational iteration method to solve a system of integral equations. It is shown that the method provides the solution in a rapid convergent series. The variational iteration method has been shown [7–11] to solve effectively, easily and accurately a large class of linear and nonlinear, ordinary, partial, deterministic or stochastic differential equations with approximate solutions, which converge rapidly to accurate solutions. It is shown that fourth- and fifth-order boundary value problems are equivalent to the system of integral equations by using a suitable transformation. This alternative equivalent transformation plays an important and fundamental part in solving the boundary value problems. We show that the equivalent system of integral equations can be solved efficiently using the variational iteration method. This clearly indicates that the variational iteration technique may be considered as an alternative method for solving linear and nonlinear problems. 2. Variational iteration method To illustrate the basic concept of the technique, we consider the following general differential equation: Lu þ Nu ¼ gðxÞ;
ð3Þ
where L is a linear operator, N a nonlinear operator and g(x) is a forcing term. According to variational iteration method [7–11], we can construct a correct functional as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kðLun ðsÞ þ N ~ un ðsÞ gðsÞÞ ds; ð4Þ 0
where k is a Lagrange multiplier, which can be identified optimally via variational iteration method. The subscripts n denote the nth approximation, ~ un is considered as a restricted variation. i.e. d~un ¼ 0; (4) is called as a correct functional. The solution of the linear problems can be solved in a single iteration step due to the exact identification of the Lagrange multiplier. The principles of variational iteration method and its applicability for various kinds of differential equations are given [7–11,13]. For the sake of simplicity and to convey the idea of the technique, we consider the following system of differential equations: x0i ðtÞ ¼ fi ðt; xi Þ;
i ¼ 1; 2; 3; . . . ; n
ð5Þ
subject to the boundary conditions. xi ð0Þ ¼ ci ; i ¼ 1; 2; 3; . . . ; n. To solve the system by means of the variational iteration method, we rewrite system (5) in the following form: x0i ðtÞ ¼ fi ðxi Þ þ gi ðtÞ;
i ¼ 1; 2; 3; . . . ; n
subject to the boundary conditions. xi ð0Þ ¼ ci ; i ¼ 1; 2; 3; . . . ; n and gi is defined in (3).
ð6Þ
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
The correct functional for the nonlinear system (6) can be approximated as Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ xi ðtÞ ¼ x1 ðtÞ þ k1 ðx1 ðT Þ; f1 ð~x1 ðT Þ; ~x2 ðT Þ; . . . ; ~xðkÞ n ðT ÞÞ g1 ðT ÞÞ dT ; 0 Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ x2 ðtÞ ¼ x2 ðtÞ þ k2 ðx2 ðT Þ; f2 ð~x1 ðT Þ; ~x2 ðT Þ; . . . ; ~xðkÞ n ðT ÞÞ g2 ðT ÞÞ dT ; 0
.. . xnðkþ1Þ ðtÞ
¼
xðkÞ n ðtÞ
þ
Z
t
ðkÞ
1931
ð7Þ
ðkÞ
kn ðx0ðkÞ x1 ðT Þ; ~x2 ðT Þ; . . . ; ~xðkÞ n ðT Þ; fn ð~ n ðT ÞÞ gn ðT ÞÞ dT ; 0
where ki ¼ 1; i ¼ 1; 2; 3; . . . ; n are Lagrange multipliers, ~x1 ; ~x2 ; . . . ; ~xn denote the restricted variations. For ki ¼ 1; i ¼ 1; 2; 3; . . . ; n; we have the following iterative schemes: Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ x1 ðtÞ ¼ x1 ðtÞ ðx1 ðT Þ; f1 ðx1 ðT Þ; x2 ðT Þ; . . . ; xðkÞ n ðT ÞÞ g1 ðT ÞÞ dT ; 0 Z t ðkþ1Þ ðkÞ 0ðkÞ ðkÞ ðkÞ x2 ðtÞ ¼ x2 ðtÞ ðx2 ðT Þ; f2 ðx1 ðT Þ; x2 ðT Þ; . . . ; xðkÞ n ðT ÞÞ g2 ðT ÞÞ dT ; 0
.. . xnðkþ1Þ ðtÞ ¼ xðkÞ n ðtÞ
Z
t
ðkÞ
ð8Þ
ðkÞ
ðkÞ ðx0ðkÞ n ðT Þ; fn ðx1 ðT Þ; x2 ðT Þ; . . . ; xn ðT ÞÞ gn ðT ÞÞ dT : 0
If we start with the initial approximations xi ð0Þ ¼ ci ; i ¼ 1; 2; 3; . . . ; n then the approximations can be completely determined; finally we approximate the solution. ðkÞ ðnÞ xi ðtÞ ¼ Limk!1 xi ðtÞ by the nth term xi ðtÞ for i ¼ 1; 2; 3; . . . ; n. 3. Applications We first show that fourth- or fifth-order boundary value problem may be reformulated as a system of integral equations. We use the variational iteration method developed in Section 2 to solve the resultant system of integral equations. Example 3.1 [17]. Consider the following linear boundary value problem of fifth order: y ðvÞ ðxÞ ¼ y 15ex 10xex with boundary conditions yð0Þ ¼ 0; y 0 ð0Þ ¼ 1; y 00 ð0Þ ¼ 0;
ð9Þ yð1Þ ¼ 0;
y 0 ð1Þ ¼ e:
The exact solution of the problem is yðxÞ ¼ xð1 xÞex : Using the transformation dy ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ sðxÞ; dx
ds ¼ zðxÞ; dx
we rewrite the fifth-order boundary value problem (9) and (10) as a system of differential equations 8 dy > ¼ qðxÞ; > > dx > > dq > > < dx ¼ f ðxÞ; df ¼ sðxÞ; dx > > > ds > ¼ zðxÞ; > dx > > : dz ¼ y 15ex 10xex dx
with yð0Þ ¼ 0; qð0Þ ¼ 1; f ð0Þ ¼ 0; zð0Þ ¼ A; sð0Þ ¼ B.
ð10Þ
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The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ þ1; i ¼ 1; 2; 3; 4; 5. Rx 8 ðkþ1Þ y ðxÞ ¼ 0 þ 0 qðkÞ ðtÞ dt; > > > > Rx > ðkþ1Þ > ðxÞ ¼ 1 þ 0 f ðkÞ ðtÞ dt; > >q < Rx f ðkþ1Þ ðxÞ ¼ 0 þ 0 sðkÞ ðtÞ dt; > > Rx > > sðkþ1Þ ðxÞ ¼ A þ 0 zðkÞ ðtÞ dt; > > > > Rx : ðkþ1Þ z ðxÞ ¼ B þ 0 ð15et 10tet þ y ðkÞ ðtÞÞ dt; y ð0Þ ðxÞ ¼ 0;
qð0Þ ðxÞ ¼ 1;
f ð0Þ ðxÞ ¼ 0;
sð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: y ð1Þ ðxÞ ¼ þx; qð1Þ ðxÞ ¼ 1; f ð1Þ ðxÞ ¼ þAx; sð1Þ ðxÞ ¼ A þ Bx; zð1Þ ðxÞ ¼ B þ 5 5ex 10xex ; y ð2Þ ðxÞ ¼ þx; A 2 x; 2 B f ð2Þ ðxÞ ¼ Ax þ x2 ; 2 sð2Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex ; 1 zð2Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 ; 2 A 3 ð3Þ y ðxÞ ¼ x þ x ; 6 A B ð3Þ q ðxÞ ¼ 1 x2 þ x3 ; 2 6 B 5 f ð3Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex ; 2 2 x3 sð3Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ ; 6 1 2 ð3Þ x x z ðxÞ ¼ B þ 5 5e 10xe þ x ; 2 A 3 B 4 ð4Þ y ðxÞ ¼ x þ x þ x ; 6 24 A 2 B 3 5 5 ð4Þ q ðxÞ ¼ 1 x þ x 25 15x x2 þ x3 þ 25ex 10xex ; 2 6 2 6 B 5 x4 f ð4Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ ; 2 2 24 x3 ð4Þ x x s ðxÞ ¼ A þ Bx 5 þ 5x þ 5e 10xe þ ; 6 1 2 A 4 ð4Þ x x z ðxÞ ¼ B þ 5 5e 10xe þ x þ x ; 2 24 A B 15 5 5 y ð5Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex ; 6 24 2 6 24 qð2Þ ðxÞ ¼ 1
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
A 2 B 3 5 5 x5 x þ x 25 15x x2 þ x3 þ 25ex 10xex þ ; 2 6 2 6 120 B 5 f ð5Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex ; 2 2 x3 A 5 x; sð5Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 6 120 1 A B 5 x; zð5Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 A B 15 5 5 1 6 x; y ð6Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 A B 5 5 x5 ; qð6Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 B 5 A 6 x; f ð6Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 x3 A 5 B 6 x þ x; sð6Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 1 A B 5 25 x 45 35x x2 zð6Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 5 5 5 x þ 45ex 10xex ; x3 x4 þ 6 24 120 A B 15 5 5 1 6 x; y ð7Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 A B 5 5 x5 A 7 x; þ qð7Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 5040 B 5 A 6 B 7 x þ x; f ð7Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 x3 A 5 B 6 x þ x 55 45x sð7Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 x þ x þ 55ex 10xex ; x2 x3 x4 2 6 24 120 720 1 A B 5 25 x 45 35x x2 zð7Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 x þ 45ex 10xex þ x7 ; x x þ 6 24 120 5040 A B 15 5 5 1 6 A x þ x8 ; y ð8Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 40320 A B 5 5 x5 A 7 B x þ x8 ; þ qð8Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 40320 120 5040 B 5 A 6 B 7 x þ x f ð8Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 45 35 25 15 5 5 6 5 7 x x þ x þ 65ex 10xex ; 65 55x x2 x3 x4 2 6 24 120 720 5040 x3 A 5 B 6 x þ x 55 45x sð8Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 1 x þ x þ 55ex 10xex þ x8 ; x2 x3 x4 2 6 24 120 720 40320
qð5Þ ðxÞ ¼ 1
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1 A B 5 25 x 45 35x x2 zð8Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 x þ 45ex 10xex þ x7 ; x x þ 6 24 120 5040 A B 15 5 5 1 6 B x þ x9 ; y ð9Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex þ 6 24 2 6 24 720 362880 A B 5 5 x5 qð9Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 A 7 B 55 2 45 3 35 4 25 5 15 6 8 x þ x 75 65x x x x x x þ 5040 40320 2 6 24 120 720 5 7 5 x þ x8 þ 75ex 10xex ; 5040 40320 B 5 A 6 B 7 x þ x 65 55x f ð9Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 45 35 25 15 5 5 6 5 7 1 x x þ x þ 65ex 10xex þ x9 ; x2 x3 x4 2 6 24 120 720 5040 362880 x3 A 5 B 6 x þ x 55 45x sð9Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 1 x þ x þ 55ex 10xex þ x8 ; x2 x3 x4 2 6 24 120 720 40320 1 A B 5 25 x 45 35x x2 zð9Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 A x þ 45ex 10xex þ x7 þ x9 ; x x þ 6 24 120 5040 362880 A B 15 5 5 y ð10Þ ðxÞ ¼ x þ x3 þ x4 35 25x x2 x3 þ x4 þ 35ex 10xex 6 24 2 6 24 1 6 B 65 55 45 35 5 25 6 x þ x9 85 75x x2 x3 x4 x x þ 720 362880 2 6 24 120 720 15 7 5 5 x x8 þ x9 þ 85ex 10xex ; 5040 40320 362880 A B 5 5 x5 qð10Þ ðxÞ ¼ 1 x2 þ x3 25 15x x2 þ x3 þ 25ex 10xex þ 2 6 2 6 120 A 7 B 55 45 35 25 5 15 6 x þ x8 75 65x x2 x3 x4 x x þ 5040 40320 2 6 24 120 720 5 7 5 1 x þ x8 þ 75ex 10xex þ x10 ; 5040 40320 3628800 B 5 A 6 B 7 x þ x 65 55x f ð10Þ ðxÞ ¼ Ax þ x2 15 5x þ x2 þ 15ex 10xex þ 2 2 720 5040 45 35 25 15 5 5 6 5 7 1 x x þ x þ 65ex 10xex þ x9 ; x2 x3 x4 2 6 24 120 720 5040 362880 x3 A 5 B 6 x þ x 55 45x sð10Þ ðxÞ ¼ A þ Bx 5 þ 5x þ 5ex 10xex þ þ 720 6 120 35 25 15 5 5 5 6 1 A x þ x þ 55ex 10xex þ x8 þ x10 ; x2 x3 x4 2 6 24 120 720 40320 3628800 1 A B 5 25 x 45 35x x2 zð10Þ ðxÞ ¼ B þ 5 5ex 10xex þ x2 þ x4 þ 2 24 120 2 15 3 5 4 5 5 1 7 A B x x x þ 45e 10xe þ x þ x9 þ x10 : x x þ 6 24 120 5040 362880 3628800
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1935
Table 1 Error estimates x
Analytical solution yðxÞ ¼ xð1 xÞex
Errorsa (variational iteration)
Errorsa (B-spline)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.000000 0.099465382 0.195424441 0.283470349 0.358037927 0.412180317 0.437308512 0.422840684 0.356086548 0.221364279 0.000000
0.0000000 3E11 2E10 4E10 8E10 1.2E9 2E9 2.2E9 1.9E9 1.4E9 0.000
0.0000000 8.0E3 1.2E3 5.0E3 3.0E3 8.0E3 6.0E3 0.000 9.0E3 9.0E3 0.000000
a
Error = analytical solution numerical solution.
The solution is given as
1 3 1 4 1 5 1 6 1 7 1 1 1 1 8 x þ þ A x þ þ B x9 yðxÞ ¼ x þ Ax þ Bx x x 6 24 8 30 144 896 40320 72576 362880 1 1 1 A 1 x10 þ x11 x12 x12 x13 45360 403200 3991680 622702080 44478720 B 1 x14 þ Oðx15 Þ: 8717029120 544864320
Using the boundary conditions at x = 1, we have A ¼ 2:99999988;
B ¼ 8:00000054:
The series solution is given by yðxÞ ¼ x 0:49999998x3 0:33333355x4 0:125x5 0:03333333x6 0:006944444x7 0:0011904762x8 0:0001736111x9 0:0000220458x10 0:000002488x11 0:000000244x12 0:000000022x13 0:000000001x14 þ Oðx15 Þ; which is exactly the same as obtained in [17] by using the decomposition method and in [15] by using the homotopy perturbation method. Table 1 exhibits a comparison between the errors obtained by using the proposed variational iteration method [7–11] and by using the sixth degree B-spline method [2]. Examining this table closely shows the improvements obtained by using the proposed scheme. Higher accuracy can be obtained by evaluating more components of yðxÞ. Example 3.2 [17]. Consider the following nonlinear boundary value problem of fifth order: y ðvÞ ðxÞ ¼ ex y 2 ðxÞ
ð11Þ
with boundary conditions yð0Þ ¼ y 0 ð0Þ ¼ y 00 ð0Þ ¼ 1;
yð1Þ ¼ y 0 ð1Þ ¼ e:
The exact solution for this problem is yðxÞ ¼ ex : Using the transformation dy ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ sðxÞ; dx
ds ¼ zðxÞ; dx
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we rewrite the above fifth-order boundary value problem as a system of differential equations 8 dy > ¼ qðxÞ; > dx > > > dq > > < dx ¼ f ðxÞ; df ¼ sðxÞ; dx > > > ds > ¼ zðxÞ; > dx > > : dz ¼ ex y 2 ðxÞ dx
with yð0Þ ¼ qð0Þ ¼ f ð0Þ ¼ 1; sð0Þ ¼ A; zð0Þ ¼ B. The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ þ1; i ¼ 1; 2; 3; 4; 5. 8 Rx y ðkþ1Þ ðxÞ ¼ 1 þ 0 qðkÞ ðtÞ dt; > > > > ðkþ1Þ Rx > > ðxÞ ¼ 1 þ 0 f ðkÞ ðtÞ dt; > >q < Rx f ðkþ1Þ ðxÞ ¼ 1 þ 0 sðkÞ ðtÞ dt; > Rx > > > sðkþ1Þ ðxÞ ¼ A þ 0 zðkÞ ðtÞ dt; > > > > Rx : ðkþ1Þ ðxÞ ¼ B þ 0 et ðy ðkÞ ðtÞÞ2 dt; z y ð0Þ ðxÞ ¼ 1;
qð0Þ ðxÞ ¼ 1;
f ð0Þ ðxÞ ¼ 1;
sð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: y ð1Þ ðxÞ ¼ 1 þ x; qð1Þ ðxÞ ¼ 1 þ x; f ð1Þ ðxÞ ¼ 1 þ Ax; sð1Þ ðxÞ ¼ A þ Bx; zð1Þ ðxÞ ¼ B þ 1 ex ; x2 ; 2 A qð2Þ ðxÞ ¼ 1 þ x þ x2 ; 2 B ð2Þ f ðxÞ ¼ 1 þ Ax þ x2 ; 2 sð2Þ ðxÞ ¼ A þ Bx 1 þ x þ ex ; y ð2Þ ðxÞ ¼ 1 þ x þ
zð2Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex ; x2 A 3 þ x; 2 6 A B qð3Þ ðxÞ ¼ 1 þ x þ x2 þ x3 ; 2 6 B 1 f ð3Þ ðxÞ ¼ 1 þ Ax þ x2 þ 1 x þ x2 ex ; 2 2 ð3Þ x s ðxÞ ¼ A þ Bx 1 þ x þ e 6 þ 2x þ 6ex þ 4xex þ x2 ex ; y ð3Þ ðxÞ ¼ 1 þ x þ
1 zð3Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex þ 6 6ex 6xex 3x2 ex x3 ex x4 ex ; 4 x2 A 3 B 4 ð4Þ y ðxÞ ¼ 1 þ x þ þ x þ x ; 24 2 6 A B 1 1 qð4Þ ðxÞ ¼ 1 þ x þ x2 þ x3 1 þ x x2 þ x3 þ ex ; 2 6 2 6
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1937
B 2 1 x þ 1 x þ x2 ex þ 12 6x þ x2 12ex 6xex x2 ex ; 2 2 sð4Þ ðxÞ ¼ A þ Bx 1 þ x þ ex 6 þ 2x þ 6ex þ 4xex þ x2 ex 30 þ 6x þ 26ex 1 þ 24xex þ 9x2 ex þ 2x3 ex þ x4 ex ; 4 zð4Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex þ 6 6ex 6xex 3x2 ex x3 ex
f ð4Þ ðxÞ ¼ 1 þ Ax þ
1 A3 720 720ex 720xex 360x2 ex 120x3 ex 30x4 ex 6x5 ex ; x4 ex þ 4 216 x2 A 3 B 4 1 1 1 ð5Þ y ðxÞ ¼ 1 þ x þ þ x þ x þ 1 x þ x2 x3 þ x4 ex ; 24 2 6 24 2 6 A B 1 1 qð5Þ ðxÞ ¼ 1 þ x þ x2 þ x3 1 x x2 þ x3 þ ex 20 þ 12x 3x2 2 6 2 6 1 3 þ x þ 20ex þ 8xex þ x2 ex ; 3 B 1 ð5Þ f ðxÞ ¼ 1 þ Ax þ x2 þ 1 x þ x2 ex þ 12 6x þ x2 12ex 6xex 2 2 1 x2 ex þ 90 30x þ 3x2 90ex 60xex 18x2 ex 3x3 ex x4 ex ; 4 ð5Þ x x x 2 x s ðxÞ ¼ A þ Bx 1 þ x þ e 6 þ 2x þ 6e þ 4xe þ x e 30 þ 6x þ 30ex þ 24xex 1 A3 ð720 þ 720x 720ex þ 720xex þ 9x2 ex þ 2x3 ex þ x4 ex þ 4 216 þ 360x2 ex þ 120x3 ex þ 30x4 ex þ 6x5 ex Þ; zð5Þ ðxÞ ¼ B þ 1 ex þ 2 2ex 2xex x2 ex þ 6 6ex 6xex 3x2 ex x3 ex 1 A3 ½720 720ex 720xex 360x2 ex 120x3 ex 30x4 ex 6x5 ex x4 ex þ 4 216 B4 þ ð40320 40320ex 40320xex 20160x2 ex 6720x3 ex 1680x4 ex 576 336x5 ex 56x6 ex 8x7 ex x8 ex Þ: The solution is given as
1 2 1 3 1 4 1 5 1 6 1 7 1 1 x þ x þ x þ A yðxÞ ¼ 1 þ x þ x þ Ax þ Bx þ x8 2 6 24 120 720 5040 20160 40320 1 1 1 1 1 1 2 9 10 x þ A Aþ þ x þ x11 18144 362880 3628800 995840 997920 1900800 1 1 1 1 101 A2 þ A Bþ AB þ x12 Oðx13 Þ: 3421440 2280960 6842880 6842880 479001600
Using the boundary condition at x = 1, we have A ¼ 0:9999967742;
B ¼ 1:0000145020:
The series solution is yðxÞ ¼ 1 þ x þ 0:5x2 þ 0:166666236x3 þ 0:04166727092x4 þ 0:008333333333x5 þ 0:00138888888x6 þ 0:000198412x7 þ 0:00002480142729x8 þ 0:00005236x9 þ 0:000000275x10 0:00000898x11 0:000000064x12 þ Oðx13 Þ; which is exactly the same solution as obtained in [17] by using the decomposition method and in [15] by using the homotopy perturbation method.
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Table 2 Error estimates x
Analytical solution
Errorsa (variational iteration)
Errorsa (B-spline)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.000000000 1.105170918 1.221402758 1.349858808 1.491824698 1.648721271 1.822118800 2.013752707 2.225540928 2.459603111 2.718281828
0.0000 1.0E9 2.0E9 1.0E9 2.0E9 3.1E8 3.7E8 4.1E8 3.1E8 1.4E8 0.0000
0.0000 7.0E4 7.2E4 4.1E4 4.6E4 4.7E4 4.8E4 3.9E4 3.1E4 1.6E4 0.0000
a
Error = analytical solution numerical solution.
Table 2 shows the exact values, numerical solutions, and the errors obtained by using the variational iteration method [7–11] and by using the sixth degree B-spline method [2] for x ¼ 0:0; 0:1; 0:2; . . . ; 1:0: The table clearly indicates the improvements as compared with B-spline method. Example 3.3 [14]. We consider the following nonlinear boundary value problem of fourth order: uðivÞ ðxÞ ¼ u2 x10 þ 4x9 4x8 4x7 þ 8x6 4x4 þ 120x 48 with boundary conditions uð0Þ ¼ u0 ð0Þ ¼ 0; uð1Þ ¼ 1; u0 ð1Þ ¼ 1: The exact solution of this problem is
ð12Þ ð13Þ
uðxÞ ¼ x5 2x4 þ 2x2 : Using the transformation du ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ zðxÞ; dx
we rewrite the fourth-order boundary value problem (12) and (13) as a system of differential equations 8 du ¼ qðxÞ; > dx > > > dq < ¼ f ðxÞ; dx df > > dx ¼ zðxÞ; > > : dz ¼ u2 x10 þ 4x9 4x8 4x7 þ 8x6 4x4 þ 120x 48 dx with uð0Þ ¼ qð0Þ ¼ 0; f ð0Þ ¼ A; zð0Þ ¼ B. The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ 1; i ¼ 1; 2; 3; 4. 8 Rx uðkþ1Þ ðxÞ ¼ 0 0 qðkÞ ðtÞ dt; > > > Rx > ðkþ1Þ < q ðxÞ ¼ 0 0 f ðkÞ ðtÞ dt; Rx ðkþ1Þ > ðxÞ ¼ A 0 zðkÞ ðtÞ dt; >f > > Rx : ðkþ1Þ z ðxÞ ¼ B 0 ððuðkÞ ðtÞÞ2 t10 þ 4t9 4t8 4t7 þ 8t6 4t4 þ 120t 48Þ dt; uð0Þ ðxÞ ¼ 0;
qð0Þ ðxÞ ¼ 0;
f ð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: uð1Þ ðxÞ ¼ 0; qð1Þ ðxÞ ¼ Ax;
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
f ð1Þ ðxÞ ¼ A Bx; zð1Þ ðxÞ ¼ B þ uð2Þ ðxÞ ¼
x11 4 4 4 8 4 120 2 x þ 48x; x10 þ x9 þ x8 x7 þ x5 9 8 7 5 2 11 10
A 2 x; 2
B 2 x; 2 x12 2x11 2 1 x8 2x6 þ x10 x9 þ þ 20x3 24x2 ; f ð2Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 x11 4 4 4 8 4 120 2 x þ 48x; x10 þ x9 þ x8 x7 þ x5 zð2Þ ðxÞ ¼ B þ 9 8 7 5 2 11 10 A B uð3Þ ðxÞ ¼ x2 þ x3 ; 2! 3! B x13 x12 2x11 1 10 1 2 7 ð3Þ x x9 þ x 5x4 8x3 ; þ þ q ðxÞ ¼ Ax þ x2 þ 2 63 105 1716 330 495 180 x12 2x11 2 1 x8 2x6 þ x10 x9 þ þ 20x3 24x2 ; f ð3Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 x11 4 4 4 8 4 120 2 A2 x þ 48x x5 ; x10 þ x9 þ x8 x7 þ x5 zð3Þ ðxÞ ¼ B þ 9 8 7 5 2 11 10 20 A 2 B 3 x14 x13 1 12 1 11 1 10 1 8 ð4Þ x x þ x x þ x5 2x4 ; þ u ðxÞ ¼ x þ x 2! 3! 1980 630 420 24024 4290 2970 B x13 x12 2x11 1 10 1 2 7 x x9 þ x 5x4 8x3 ; þ þ qð4Þ ðxÞ ¼ Ax þ x2 þ 2 63 105 1716 330 495 180 x12 2x11 2 1 x8 2x6 A2 6 þ x10 x9 þ þ 20x3 24x2 þ x; f ð4Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 120 x11 4 4 4 8 4 120 2 A2 B2 7 x þ 48x x5 þ x10 þ x9 þ x8 x7 þ x5 x; zð4Þ ðxÞ ¼ B þ 9 8 7 5 2 11 10 20 252 14 13 A B x x 1 12 1 11 1 10 1 8 x x þ x x þ x5 2x4 ; þ uð5Þ ðxÞ ¼ x2 þ x3 2! 3! 1980 630 420 24024 4290 2970 B x13 x12 2x11 1 10 1 2 7 A 7 x x9 þ x 5x4 8x3 x; þ þ qð5Þ ðxÞ ¼ Ax þ x2 þ 2 63 105 840 1716 330 495 180 x12 2x11 2 1 x8 2x6 A2 6 B2 8 þ x10 x9 þ þ 20x3 24x2 þ x þ x; f ð5Þ ðxÞ ¼ A Bx 45 18 132 55 7 15 120 2016 A B x14 x13 1 12 1 11 1 10 1 8 A2 8 x x þ x x þ x5 2x4 þ þ x; uð6Þ ðxÞ ¼ x2 þ x3 2! 3! 1980 630 420 24024 4290 2970 6720 B x13 x12 2x11 1 10 1 2 7 x x9 þ x 5x4 þ þ qð6Þ ðxÞ ¼ Ax þ x2 þ 2 63 105 1716 330 495 180 A2 7 B2 9 x x; 8x3 840 18144 .. . qð2Þ ðxÞ ¼ Ax þ
The solution is uðxÞ ¼ uð0Þ ðxÞ þ uð1Þ ðxÞ þ uð2Þ ðxÞ þ uð3Þ ðxÞ þ ; A B uðxÞ ¼ x2 þ x3 2x4 þ x5 0:002380952380952x8 þ 0:001587301587301x10 2 6 0:000505050501x11 0:0003367x12 þ 0:0002331002331x13 0:000041625043x14 þ
A2 8 B x7 þ x þ 14112 6720
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Table 3 Error estimates x
Exact solution
Series solution
Errora
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0000000000 0.0198100000 0.0771200000 0.1662300000 0.2790400000 0.4062500000 0.5385599999 0.6678700000 0.7884800000 0.8982900000 1.0000000000
0.0000000000 0.0198099999 0.0771199999 0.1662299999 0.2790399999 0.4062499999 0.5385599999 0.6678699999 0.7884799999 0.8982899999 0.9999999999
0.0000E+00 4.579E16 1.5959E15 3.1641E15 4.7739E15 6.0507E15 6.6613E15 6.6613E15 5.2180E15 2.5535E15 3.3306E16
a
Error = Exact solution series solution.
Applying the boundary conditions at x ¼ 1, A ¼ 3:9999999999887983;
B ¼ 3:19339870128 1013 :
The series solution is uðxÞ ¼ 1:99999999993x2 þ 5:322233 1014 x3 2x4 þ x5 0:002380952380952x8 þ 7:04012 1017 x9 þ 4:03323 1017 x10 3:97902 1017 x11 8:94467 1018 x12 þ ; which is exactly the same as obtained in [14] by using the homotopy perturbation method. Table 3 shows the comparison between the exact solution and the series solution obtained by using the proposed algorithm. Higher accuracy can be obtained by evaluating more components of u(x). Example 3.4 [16]. We consider the following boundary value problem of fourth order: 1 uðivÞ ðxÞ ¼ ð1 þ cÞu00 ðxÞ cuðxÞ þ cx2 1 2 with boundary conditions uð0Þ ¼ 1 ¼ u0 ð0Þ;
uð1Þ ¼ 1:5 þ sinhð1Þ;
ð14Þ
u0 ð1Þ ¼ 1 þ coshð1Þ:
ð15Þ
The exact solution of this problem is 1 uðxÞ ¼ 1 þ x2 þ sinhðxÞ: 2 Using the transformation du ¼ qðxÞ; dx
dq ¼ f ðxÞ; dx
df ¼ zðxÞ; dx
we can rewrite the fourth-order boundary value problem as a system of differential equations 8 du ¼ qðxÞ; > dx > > > > < dq ¼ f ðxÞ; dx df > > ¼ zðxÞ; > dx > > : dz ¼ ð1 þ cÞu00 ðxÞ cuðxÞ þ 12 cx2 1 dx
with uð0Þ ¼ qð0Þ ¼ 1; f ð0Þ ¼ A; zð0Þ ¼ B. The above system of differential equations can be written as a system of integral equations with Lagrange multipliers ki ¼ 1; i ¼ 1; 2; 3; 4.
M. Aslam Noor, S.T. Mohyud-Din / Applied Mathematics and Computation 189 (2007) 1929–1942
1941
Rx 8 ðkþ1Þ u ðxÞ ¼ 1 0 qðkÞ ðtÞ dt; > > > R > < qðkþ1Þ ðxÞ ¼ 1 x f ðkÞ ðtÞ dt; 0 R x ðkÞ ðkþ1Þ > ðxÞ ¼ A 0 z ðtÞ dt; >f > > Rx : ðkþ1Þ ðxÞ ¼ B 0 ½ð1 þ cÞf ðkÞ ðtÞ cuðkÞ ðtÞ þ 12 ct2 1 dt; z uð0Þ ðxÞ ¼ 1;
qð0Þ ðxÞ ¼ 1;
f ð0Þ ðxÞ ¼ A;
zð0Þ ðxÞ ¼ B:
Consequently, we obtain the following approximations: uð1Þ ðxÞ ¼ 1 x; qð1Þ ðxÞ ¼ 1 Ax; f ð1Þ ðxÞ ¼ A Bx;
1 3 z ðxÞ ¼ B þ ð1 AÞx þ c ð1 AÞx x ; 6 ð1Þ
A 2 x; 2 B qð2Þ ðxÞ ¼ 1 Ax þ x2 ; 2
uð2Þ ðxÞ ¼ 1 x þ
x2 x2 1 c ð1 AÞ x4 ; 2 2 24 B 2 1 2 1 ð2Þ z ðxÞ ¼ B ð1 þ cÞ Ax x c x x þ cx3 x; 2 2 6
f ð2Þ ðxÞ ¼ A Bx þ ð1 AÞ
.. . Using essentially the technique of Example 3.3, we can find the other approximate values. Consequently, the approximate solution is given by A 2 B 3 1 A c A B 5 c 5 B x þ x x4 þ x4 x4 þ cx4 þ x x þ cx5 2 6 24 24 24 24 120 120 120 x6 A B c cx6 cx7 þ x8 þ 5040 20160 720 720
uðxÞ ¼ 1 þ x þ
Now using the boundary conditions at x = 1, we have A 1 þ gðcÞ; B 1 þ hðcÞ; where both gðcÞ and hðcÞ grow rapidly with c. In reality, they should go to zero as the number of terms in the series goes to infinity. Scott [16] has solved the problem (14) with boundary conditions (15) for very large values of c with the orthonormalization process. It has been shown in [16] that each time the solutions started to lose their linear independence, one has to perform orthonormalization. In fact, as c got bigger it required more normalization. We would like to mention [16] that Eq. (14) can be rewritten in the following equivalent form: ðivÞ 1 u ðxÞ u00 ðxÞ þ 1 c u00 ðxÞ uðxÞ þ x2 ¼ 0: 2 From this it follows that the solution of the second order system is also a solution of the fourth-order problem. Hence, the solution of (14) is independent of c. In view of these comments and facts, decomposition, homotopy, variational, tanh/cotanh and other like methods could be corrected by a similar process of dividing up the interval into pieces as suggested by Scott.
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Remark. We would like to point out the approximate solution obtained by the variational iteration technique is in good agreement with the exact solution for the small values of the parameter c and continuously depends on the parameter c. We have also solved the Example 3.4 by using the Adomian decomposition, Homotopy perturbation method and Differential transformation method, see [13] for full details. It turns out that all these methods give us the same approximate solution. This is natural since all these techniques are based on finding the solutions of the problems in terms of the series., no matter which technique one uses. Every method has some advantages over the others. Variational technique depends on finding the appropriate and suitable value of the parameter, which is easy to find than finding the derivatives of the Adomian polynomials. In the homotopy method, one usually express the solutions as series with unknown coefficients and substituting the series in the equation and equating the coefficients of the powers on both sides, which may be difficult in some cases. In such cases, variational iteration technique can be considered as an alternative and efficient method for solving linear and nonlinear problems. Acknowledgements We would like to express our deepest gratitude to Prof. M. Scott for his valuable and interesting suggestions and comments as well as providing us some special problems. The research of M. Aslam Noor is supported by the Higher Education Commission, Pakistan, through the research Grant No: I-28/HEC/HRD/2005/90; whereas the research of S. Tauseef Mohyud-Din is also supported by Higher Education Commission, Pakistan, through indigenous PhD scholarship scheme. References [1] R.P. Agarwal, Boundary-value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986. [2] H.N. Caglar, S.H. Caglar, E.E. Twizell, The numerical solution of fifth order boundary-value problems with sixth degree B-spline functions, Appl. Math. Lett. 12 (1999) 25–30. [3] M.M. Chawla, C.P. Katti, Finite difference methods for two-point boundary-value problems involving higher order differential equations, BIT 19 (1979) 27–33. [4] A.R. Davies, A. Karageoghis, T.N. Philips, Spectral Galerkin methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1988) 647–662. [5] E. Doedel, Finite difference methods for nonlinear two-point boundary-value problem, SIAM J. Numer. Anal. 16 (1979) 173–185. [6] D.J. Fyfe, Linear dependence relations connecting equal interval nth degree splines and their derivatives, J. Inst. Math. Appl. 7 (1971) 398–406. [7] J.H. He, Variational iteration method – a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech. 34 (1999) 699–708. [8] J.H. He, Variational method for autonomous ordinary differential equations, Appl. Math. Comput. 114 (2000) 115–123. [9] J.H. He, Variational theory for linear magneto–electro-elasticity, Int. J. Nonlinear Sci. Numer. Simul. 2 (4) (2001) 309–316. [10] J.H. He, Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fract. 19 (4) (2004) 847–851. [11] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: Variational Method in the Mech. of Solids, Pergamon Press, New York, 1978, pp. 156–162. [12] A. Karageoghis, T.N. Philips, A.R. Davies, Spectral collocation methods for the primary two-point boundary-value problems in modeling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1998) 805–813. [13] M. Aslam Noor, Some algorithms for solving boundary value problems, Preprint, 2006. [14] S.T. Moyud-Din, M. Aslam Noor, Homotopy perturbation method for solving fourth order boundary value problems, Math. Problems Eng. 2007 (2007) 1–15, Article ID 98602. [15] M. Aslam Noor, S.T. Mohyud-Din, An efficient algorithm for solving fifth order boundary value problems, Math. Comput. Model. 45 (2007) 954–964. [16] M. Scott, Special problems, private communication, 2006. [17] A.M. Wazwaz, The numerical solution of fifth-order boundary-value problems by Adomian decomposition method, J. Comput. Appl. Math. 136 (2001) 259–270.
Further reading [1] S. Momani, M. Aslam Noor, Numerical method of fourth-order fractional integro-differential equations, Appl. Math. Comput. 182 (2006) 754–760.