# Variational Iteration Method for Fifth-Order Boundary Value Problems

Authors: Muhammad Aslam Noor; Syed Tauseef Mohyud-Din
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

We apply the variational iteration method using He's polynomials (VIMHP) for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007). The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.


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## Body

## 1. Introduction

In this paper, we
general fifth-order boundary value problem(1.1)y(v)(x)=g(x)y+q(x) with
boundary conditions (1.2)y(a)=A1,y′(a)=A2,y″(a)=A3,y(b)=B1,y′(b)=B2, where y(x) and f(x,y) are real and as many times differentiable as
required for x∈[a,b]; and Ai, i=1,2,3 and Bi, i=1,2 are real finite constants. 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 [1–7]
and the references therein. Several numerical methods including spectral Galerkin
and collocation, decomposition, and sixth-order B-spline have been developed
for solving fifth-order boundary value problems, see [1, 7] and the references
therein. The use of spline function in the context of fifth-order boundary
value problems was studied by Fyfe [3], 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. He [8–18] developed the
variational iteration method and homotopy perturbation method for solving
linear, nonlinear, initial, and boundary value problems. It is worth mentioning
that the origin of variational iteration method can be traced back to Inokuti et al. [19], but the real potential of this technique was explored by
He [13–18]. Moreover, He
realized the physical significance of the variational iteration method, its
compatibility with the physical problems and applied this promising technique
to a wide class of linear and nonlinear, ordinary,
partial, deterministic, or stochastic differential equation; see [13–18]. The homotopy
perturbation method [8–12, 17] was also
developed by He by merging two techniques: the standard homotopy and the
perturbation. The homotopy perturbation method was formulated by taking the
full advantage of the standard homotopy and perturbation methods. The
variational iteration method and homotopy perturbation method have been applied
to a wide class of functional equations; see
[4, 5, 8–36] and
the references therein. In these methods, the solution is given in an infinite
series usually converging to an accurate solution, see [4, 5, 8–22, 25–36] and
the references therein. In a later work, Ghorbani [23, 24] splits the
nonlinear term into a series of polynomials calling them as He’s polynomials. Recently, Noor and Mohyud-Din used homotopy
perturbation, variational iteration, and the iterative methods [4–6] for
solving the fifth-order boundary value problems. The results are very
encouraging and reveal the complete reliability of the new algorithm.Inspired
and motivated by the ongoing research in this area, we use the variational
iteration method coupled with He’s polynomials for solving the fifth-order
boundary value problems in this paper. It is worth mentioning that the proposed
method is an elegant combination of variational iteration and the homotopy
perturbation methods and is mainly due to Ghorbani [23, 24]. The use of
He’s polynomials in the nonlinear term was first introduced by Ghorbani, see [23, 24]. The proposed algorithm provides the solution in a rapid convergent
series which may lead the solution in a closed form. In this technique, the correction
functional is developed [13–22] and the
Lagrange multipliers are calculated optimally via variational theory. The use
of Lagrange multipliers reduces the successive application of the integral
operator and the cumbersome of huge computational work, while still maintaining
a very high level of accuracy. Finally, He’s polynomials are introduced in the
correction functional and the comparison of like powers of p gives solutions of
various orders. The proposed iterative scheme takes full advantage of
variational iteration method and the homotopy perturbation method. It is worth
mentioning that the suggested method is applied without any discritization,
restrictive assumption, or transformation and is free from round off errors.
Unlike the method of separation of variables that requires initial and boundary
conditions, the method provides an analytical solution by using the initial
conditions only. The proposed method works efficiently and the results so far
are very encouraging and reliable. The fact that the proposed VIMHP solves
nonlinear problems without using Adomian’s polynomials can be considered as a
clear advantage of this method over the decomposition method. Several examples are given to verify the reliability
and efficiency of the algorithm.


## 2. Variational Iteration Method

To illustrate
the basic concept of the technique, we consider the following general
differential equation:(2.1)Lu+Nu=g(x), where L is a
linear operator, N a nonlinear operator, and g(x) is the forcing term. According to variational
iteration method [5, 13–22, 25, 32–36], we
can construct a correct functional as follows: (2.2)un+1(x)=un(x)+∫0xλ(Lun(s)+Nu˜n(s)−g(s))ds, where λ is a Lagrange multiplier [13–19], which can be
identified optimally via variational iteration method. The subscripts n denote the nth approximation, u˜n is considered as a restricted variation, that
is, δu˜n=0; and (2.2) 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 in [13–19]. In this
method, it is required first to determine the Lagrange multiplier λ optimally. The successive approximation un+1, n≥0 of the solution u will be readily
obtained upon using the determined Lagrange multiplier and any selective
function u0, consequently, the solution is given by (2.3)u=limn→∞un. For the convergence and error
estimates of variational iteration method, see Ramos [35].


## 3. Homotopy Perturbation Method

To explain the
homotopy perturbation method, we consider a general equation of the type(3.1)L(u)=0, where
L is an integral or differential operator. We define a convex homotopy H(u,p)
by (3.2)H(u,p)=(1−p)F(u)+pL(u), where F(u) is a
functional operator with known solutions v0, which can be obtained
easily. It is clear that for (3.3)H(u,p)=0, we have (3.4)H(u,0)=F(u),H(u,1)=L(u). This shows that H(u,p) continuously traces an implicitly defined
curve from a starting point H(v0,0) to a solution function H(f,1).
The embedding parameter monotonically increases from zero to unit as the
trivial problem F(u)=0 is continuously deforms the original problem L(u)=0. The embedding parameter p∈ (0, 1] can be considered as an expanding
parameter [3, 8–11, 23, 24]. The homotopy perturbation method uses the
homotopy parameter p as an expanding
parameter [4, 8–12, 23, 24, 27–31] to obtain (3.5)u=∑i=0∞piui=u0+pu1+p2u2+p3u3+⋯. If p→1, then (3.5) corresponds to (3.2) and becomes the
approximate solution of the form (3.6)f=limp→1u=∑i=0∞ui. It is well known that
series (3.6) is convergent for most of the cases and also the rate of convergence
depends upon L(u); see [4, 8–12, 23, 24, 27–31]. We assume
that (3.6) has a unique solution. The comparisons of like powers of p give solutions of various orders.


## 4. Variational Iteration Method Using He’s Polynomials (VIMHP)

To illustrate
the basic concept of the variational homotopy perturbation method, we consider
the following general differential equation:(4.1)Lu+Nu=g(x), where L is a linear operator, N a
nonlinear operator, and g(x) is the
forcing term. According to variational iteration method [5, 6, 13–22, 25, 26, 32–36], we can
construct a correct functional as follows: (4.2)un+1(x)=un(x)+∫0xλ(ξ)(Lun(ξ)+Nu˜n(ξ)−g(ξ))dξ, where λ is a Lagrange multiplier [13–19], which can
be identified optimally via variational iteration method. The subscripts n denote the nth approximation, u˜n is considered as a restricted variation, that
is, δu˜n=0; and (4.2) is called as a correct functional.
Now, we apply the homotopy perturbation method(4.3)∑n=0∞p(n)un=u0(x)+p∫0xλ(ξ)(∑n=0∞p(n)L(un)+∑n=0∞p(n)N(u˜n))dξ−∫0xλ(ξ)g(ξ)dξ,   which is the coupling of
variational iteration and He’s polynomials. The comparison of like powers of p gives solutions of various orders.


## 5. Numerical Applications

In this section,
we apply the variational iteration method using He’s polynomials (VIMHP) for
solving the fifth-order boundary value problems. We develop the correct
functional and calculate the Lagrange multipliers optimally via variational
theory, which reduces the successive application of the integral operator. The
selection of initial value is done carefully because the approximants are
heavily dependant upon initial value. He’s polynomials are introduced in the
correct functional and finally, the comparison of like powers ofp gives
solutions of various orders. Numerical results are very encouraging. For the
sake of comparison, we take the same examples as used in [4–7].Example 5.1  (See [4–7]).
Consider the following
nonlinear boundary value problem of fifth-order,(5.1)y(v)(x)=e−xy2(x) with boundary conditions (5.2)y(0)=y′(0)=y″(0)=1;y(1)=y′(1)=e. The exact solution for this problem
is (5.3)y(x)=ex. The correct functional for the
boundary value problem (5.1) and (5.2)  is given as (5.4)yn+1(x)=yn(x)+∫0xλ(s)(d5yndx5−e−xy˜n2(x))ds.  Making the correct functional
stationary, using λ=(1/4!)(s−x)4, as the Lagrange multiplier [13–18, 36], we get
the following iterative formula: (5.5)yn+1(x)=yn(x)+∫0x14!(s−x)4(d5yndx5−e−xy˜n2(x))ds,yn+1(x)=1+x+12!x2+13!Ax3+14!Bx4+∫0x14!(s−x)4(d5yndx5−e−xy˜n2(x))ds, where (5.6)A=y‴(0),B=y(iv)(0).Applying the variational iteration
method using He’s polynomials,(5.7)y0+py1+p2y2+⋯=1+x+12!x2+13!Ax3+14!Bx4+p∫0x14!(s−x)4(d5y0dx5+pd5y1dx5+p2d5y2dx5+p3d5y3dx5+⋯)ds−p∫0x14!(s−x)4e−x(y˜0+py˜1+p2y˜2+⋯)2ds. Comparing the coefficient of like
powers of p, consequently, we obtain the following approximants:(5.8)p(0):y0(x)=1,p(1):y1(x)=1+x2+(16A−16)x3+(124B+124)x4−e−x,p(2):y2(x)=1+x2+(16A−16)x3+(124B+124)x4−e−x+70A+140B+211116−(5758+40A+70B+(60+30A+70B)e−x)x+(1438+10A+15B−(12+5A+15B)e−x)x2−(3112+43A+53B+(43+13A+53B)e−x)x3+(524+112A+112B−(112+112B)e−x)x4×((132+70A+140B)e−x)+116e−2x,⋮The series
solution is given as(5.9)y(x)=1+x+12x2+16Ax3+124Bx4+1120x5+1720x6+15040x7+(120160A−140320)x8+(118144−1362880)x9+13628800x10+(1995840A2−1997920A+11900800)x11+(−13421440A2+12280960A−16842880B+16842880AB−101479001600)x12+O(x13).Imposing the
boundary conditions at x=1 and using y(1)=y′(1)=e leads to the following
system: (5.10)32863197120A+199609747900160B+14790016A2+16842880AB=e−1202243083479001600,285343570240A+6654713991680B+1498960A2+1570240AB=e−27292071330560. The solution of the above system gives
(5.11)A=0.9999967742,B=1.0000145020. Consequently, the series solution is given
as (5.12)y(x)=1+x+0.5x2+0.166666236x3+0.04166727092x4+0.008333333333x5+0.00138888888x6+0.000198412x7+0.00002480142729x8+0.00005236x9+0.000000275x10−0.00000898x11−0.000000064x12+O(x13), which is in full agreement
with [4–7].Table1 shows the exact values and the
errors obtained by using the homotopy perturbation method (HPM) [4],
variational iteration method (VIM) [5], decomposition method (ADM) [7], the
sixth degree B-spline method [1], iterative method (ITM) [6], and the
variational iteration method using He’s polynomials (VIMHP) for x=0.0,0.1,0.2,…,1.0. The table clearly indicates the improvements
as compared with B-spline method. Higher accuracy can be obtained by evaluating
more components of y(x).Table 1
Error estimates.
xExact solution*ErrorsHPMB-splineVIMADMITMVIMHP0.01.0000000000.0000.00000.0000.0000.0000.0000.11.1051709181.0E–9–7.0E–41.0E–91.0E–91.0E–91.0E–90.21.2214027582.0E–9–7.2E–42.0E–92.0E–92.0E–92.0E–90.31.3498588081.0E–84.1E–41.0E–81.0E–81.0E–81.0E–80.41.4918246982.0E–84.6E–42.0E–82.0E–82.0E–82.0E–80.51.6487212713.1E–84.7E–43.1E–83.1E–83.1E–83.1E–80.61.8221188003.7E–84.8E–43.7E–83.7E–83.7E–83.7E–80.72.0137527074.1E–83.9E–44.1E–84.1E–84.1E–84.1E–80.82.2255409283.1E–83.1E–43.1E–83.1E–83.1E–83.1E–80.92.4596031111.4E–81.6E–41.4E–81.4E–81.4E–81.4E–81.02.7182818280.0000.0000.0000.0000.0000.000*Error = exact solution − series solution.Remark 5.2.
The numerical
results clearly indicate that the results obtained by HPM, VIM, ADM, ITM, and
the proposed VIMHP are the same. Moreover, it shows the improvements as compare
to B-spline method.Example 5.3  (See [4–7]).
Consider
the following linear boundary value problem of fifth-order:(5.13)y(v)(x)=y−15ex−10xex with boundary conditions (5.14)y(0)=0,y′(0)=1,y″(0)=0;y(1)=0,y′(1)=−e. The exact solution of the problem
is (5.15)y(x)=x(1−x)ex. The correct
functional for the boundary value problem (5.13) and (5.14) is given as (5.16)yn+1(x)=yn(x)+∫0xλ(s)(d5yndx5−(y˜n(x)−15ex−10xex))ds. Making the correct functional
stationary, using λ=(1/4!)(s−x)4, as the Lagrange multiplier [13–18, 36], we get
the following iterative formula: (5.17)yn+1(x)=yn(x)+∫0x14!(s−x)4(d5yndx5−(y˜n(x)−15ex−10xex))ds,yn+1(x)=x+13!Ax3+14!Bx4+∫0x14!(s−x)4(d5yndx5−(y˜n(x)−15ex−10xex))ds, where (5.18)A=y‴(0),B=y(iv)(0). Applying the variational iteration
method using He’s polynomials, (5.19)y0+py1+p2y2+⋯=x+13!Ax3+14!Bx4+p∫0x14!(s−x)4(d5y0dx5+pd5y1dx5+p2d5y2dx5+p3d5y3dx5+⋯)ds+p∫0x14!(s−x)4(15ex+10xex−(y˜0+py˜1+p2y˜2+⋯))ds. Comparing the coefficient of like
powers of p, consequently, we obtain the following approximants: (5.20)p(0):y0(x)=−35−24x−153x2+(16A−56)x3+(524+124B)x4+(35−10x)ex,p(1):y1(x)=−120−99x−40x2+(16A−10)x3+(−4024+124B)x4−724x5−130x6−1336x7+(140320A−18064)x8+(1362880B+172576)x9+(120−20x)ex,p(2):y2(x)=−255−224x−1952x2+(16A−552)x3+(−13524+124B)x4−x5−631440x6−261008x7+(140320A−128064)x8+(1362880B−872576)x9−1103680x10−1166320x11−131933440x12+(255−30x)ex+⋯,⋮ The series solution
is given as (5.21)y(x)=x+16Ax3+124Bx4−18x5−130x6−1144x7+(−1896+140320A)x8+(−172576+1362880B)x9−145360x10+1403200x11−13991680x12−A622702080x12−144478720x13−(B8717029120−1544864320)x14+O(x15). Imposing the
boundary conditions at x=1 and using y(1)=y′(1)=−e leads to the following system: (5.22)148284463889574400A+15121362880B=−648723077778377600,239595841479001600A+672140320B=−e−346812729937600. The solution of the above system
gives (5.23)A=−2.99999988,B=−8.00000054.
The series
solution is given by(5.24)y(x)=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+O(x15), which is in full
agreement with [4–7].Table2 shows the exact values and the
errors obtained by using the homotopy perturbation method (HPM) [4],
variational iteration method (VIM) [5], decomposition method (ADM) [7], the
sixth degree B-spline method [1], iterative method (ITM) [6], and the variational
iteration method using He’s polynomials (VIMHP) for x=0.0,0.1,0.2,…,1.0. The table clearly indicates the improvements
as compared with B-spline method. Higher accuracy can be obtained by evaluating
more components of y(x).Table 2
Error estimates.
xExact solution*ErrorsHPMB-splineVIMADMITMVIMHP0.00.0000000.00000.00000.00000.00000.00000.00000.10.099465–3E–11–8.0E–3–3E–11–3E–11–3E–11–3E–110.20.195424–2E–10–1.2 E–3–2E–10–2E–10–2E–10–2E–100.30.283470–4E–10–5.0 E–3–4E–10–4E–10–4E–10–4E–100.40.358038–8E–103.0 E–3–8E–10–8E–10–8E–10–8E–100.50.412180–1.2E–98.0 E–3–1.2E–9–1.2E–9–1.2E–9–1.2E–90.60.437309–2E–96.0 E–3–2E–9–2E–9–2E–9–2E–90.70.422888–2.2E–9–0.000–2.2E–9–2.2E–9–2.2E–9–2.2E–90.80.356087–1.9E–99.0 E–3–1.9E–9–1.9E–9–1.9E–9–1.9E–90.90.221364–1.4E–9–9.0E–3–1.4E–9–1.4E–9–1.4E–9–1.4E–91.00.0000000.0000.0000000.0000.0000.0000.000*Error = exact solution − series solution.Remark 5.4.
The numerical
results clearly indicate that the results obtained by HPM, VIM, ADM, ITM, and
the proposed VIMHP are the same. Moreover, it shows the improvements as compare
to B-spline method.


## 6. Conclusions

In this paper, we applied the variational iteration
method using He’s polynomials (VIMHP) for finding the solution of boundary
value problems of fifth-order. The method is applied in a direct way without
using linearization, transformation, discretization, or restrictive assumptions. It may be
concluded that VIMHP is very powerful and efficient in finding the analytical
solutions for a wide class of boundary value problems. The method gives more
realistic series solutions that converge very rapidly in physical problems. It
is worth mentioning that
the method is capable of reducing the volume of the computational work as compared
to the classical methods while still maintaining the high accuracy of the
numerical result, the size reduction amounts to the improvement of performance
of approach. The fact that the VIMHP solves nonlinear problems without using
Adomian’s polynomials is a clear advantage of this technique over the
decomposition method.



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