Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article
Special Issue

Nonlinear Problems: Analytical and Computational Approach with Applications

View this Special Issue

Research Article | Open Access

Volume 2012 |Article ID 401217 | 14 pages | https://doi.org/10.1155/2012/401217

The Optimal Homotopy Asymptotic Method for the Solution of Higher-Order Boundary Value Problems in Finite Domains

Academic Editor: Muhammad Aslam Noor
Received28 Jul 2011
Accepted12 Oct 2011
Published25 Dec 2011

Abstract

We solve some higher-order boundary value problems by the optimal homotopy asymptotic method (OHAM). The proposed method is capable to handle a wide variety of linear and nonlinear problems effectively. The numerical results given by OHAM are compared with the exact solutions and the solutions obtained by Adomian decomposition (ADM), variational iteration (VIM), homotopy perturbation (HPM), and variational iteration decomposition method (VIDM). The results show that the proposed method is more effective and reliable.

1. Introduction

In this paper, we consider a well-posed 𝑛th-order problem of the form𝑒(𝑛)ξ€·=πœ“π‘’,π‘’ξ…ž,…,𝑒(π‘›βˆ’1)ξ€Έ+πœ™(π‘Ÿ),π‘Ž<π‘Ÿ<𝑏,(1.1) with boundary conditions: 𝑒(π‘˜)(π‘Ž)=𝛼𝑖 and 𝑒(π‘˜)(𝑏)=𝛽𝑖, where π‘˜(<𝑛) is a nonnegative integer, 𝛼𝑖 and 𝛽𝑖 are real finite constants, and πœ™(π‘Ÿ) is a continuous function on [π‘Ž,𝑏].

Such types of problems have been investigated by many authors [1, 2] due to their mathematical importance and the potential for applications in hydrodynamic and hydromagnetic stability. Fifth-order boundary value problems arise in the mathematical modeling of viscoelastic flows. Sixth- and eighth-order differential equation govern physics of some hydrodynamic stability problems. When an infinite horizontal layer of fluid is heated from below and is subject to the action of rotation, instability sets in. When this instability is as ordinary convection, the ordinary differential equation is sixth order; when the instability sets in as overstability, it is modeled by an eighth-order ordinary differential equation. If an infinite horizontal layer of fluid is heated from below, with the supposition that a uniform magnetic field is also applied across the fluid in the same direction as gravity and the fluid is subject to the action of rotation, instability sets in. When instability sets in as ordinary convection, it is modeled by tenth-order boundary value problem.

So for the solution of these problems is concerned, many methods appeared in literature. The recent analytic methods are Adomian decomposition method (ADM) [3–5], variational iteration method (VIM) [6], homotopy perturbation method (HPM) [7–9], homotopy analysis method (HAM) [10, 11], differential transform method (DTM) [12], and so forth. Classical perturbation methods are based on the assumptions of small or large parameters, and they cannot produce a general form of an approximate solution. The nonperturbation methods like ADM and DTM can deal strongly with nonlinear problems, but the convergence region of their series solution is generally small. The HPM, which is an elegant combination of homotopy and perturbation technique, overcomes the restrictions of small or large parameters in the problems. It deals with a wide variety of nonlinear problems effectively. Recently, Marinca et al. [13–17] introduced OHAM for approximate solution of nonlinear problems of thin film flow of a fourth-grade fluid down a vertical cylinder. In their work, they have used this method to understand the behavior of nonlinear mechanical vibration of electrical machine. They also used the same method for the solution of nonlinear equations arising in the steady-state flow of a fourth-grade fluid past a porous plate and for the solution of nonlinear equation arising in heat transfer. This method is straight forward, reliable, and explicitly defined. It provides a convenient way to control the convergence of the series solution and allows adjustment of convergence region where it is needed.

Fifth- and sixth-order linear and nonlinear problems were solved by Wazwaz [18, 19], while using decomposition method. Noor et al. [20–25] investigated these type of problems using variational iteration method (VIM), homotopy perturbation method (HPM), and variational iteration decomposition method (VIDM). Modified variational iteration method (MVIM) and iterative method (ITM) were used by Mohyud-Din et al. [26, 27] for such type of problems. Kasi Viswanadham and Murali Krishna [28] used Quintic B-Spline Galerkin method for fifth-order boundary value problems. Siraj-ul-Islam et al. [29, 30] used numerical scheme for the solution of fifth- and sixth-order boundary value problems.

Recently, Ali et al. [31, 32] used OHAM for the solution of multipoint boundary value problems and twelfth-order boundary value problems. We use OHAM to find the approximate analytic solution of some higher-order BVPs. The results of OHAM are compared with those of exact solution, and the errors are compared with the existing results. This paper is organized as follows: Section 2 is devoted to the analysis of the proposed method. Some numerical examples are presented in Section 3. In Section  4, we concluded by discussing results of the numerical simulation using Mathematica.

2. Method Analysis for Two-Point Boundary Value Problems

Consider the differential equation ℒ𝑒=𝒩𝑒+πœ™,(2.1) along with boundary conditions: ℬ𝑒,πœ•π‘’ξ‚πœ•π‘Ÿ=0,(2.2) where β„’ is linear, 𝒩 is a nonlinear, and ℬ is a boundary operator. πœ™ is a known function which is continues for π‘ŸβˆΆπ‘ŸβˆˆΞ©. According to OHAM, we can construct a homotopy defined by[](1βˆ’π‘)β„’(𝑒(π‘Ÿ;𝑝))βˆ’π’½(𝑝)β„’(𝑒(π‘Ÿ;𝑝))βˆ’π’©(𝑒(π‘Ÿ;𝑝))βˆ’πœ™(π‘Ÿ)=0,(2.3) where π‘βˆˆ[0,1] is an embedding parameter, and 𝒽(𝑝) is a nonzero auxiliary function for 𝑝≠0 and β„Ž(0)=0. Equation (2.3) satisfies ℒ𝑒=0,for𝑝=0,ℒ𝑒=𝒩𝑒+πœ™,for𝑝=1.(2.4) The solution, 𝑒(π‘Ÿ,0)=𝑣0(π‘Ÿ), of ℒ𝑒=0 traces the solution curve 𝑒(π‘Ÿ) of (2.1), continuously as 𝑝 approaches to 1, where 𝑣0 is the solution of the zeroth-order problem, that will come in the next few lines.

The auxiliary function 𝒽(𝑝) is chosen in the form (it is a commonly used form) 𝒽(𝑝)=π‘šξ“π‘–=1𝑝𝑖𝐢𝑖,(2.5) where 𝐢𝑖:𝑖=1,2,…,π‘š are the convergence controlling constants which are to be determined. We will use this function unless otherwise stated. The auxiliary function can be chosen in a variety of ways, as reported by Marinca et al. [13–17]. We will use some other forms of 𝒽(𝑝) as well.

To get an approximate solution, we expand 𝑒(π‘Ÿ,𝑝) in Taylor’s series about 𝑝 in the following manner:𝑒(π‘Ÿ;𝑝)=𝑣0(π‘Ÿ)+βˆžξ“π‘š=1π‘£π‘šξ€·π‘Ÿ,𝐢1,𝐢2,…,πΆπ‘šξ€Έπ‘π‘š.(2.6) Substituting (2.5) and (2.6) into (2.3) and equating the coefficient of like powers of 𝑝, we obtain the following linear equations which can be integrated directly.

Zeroth-order problem: ℒ𝑣0𝑣=0,ℬ0,πœ•π‘£0ξ‚Άπœ•π‘›=0.(2.7)

First-order problem: ℒ𝑣1=ξ€·1+𝐢1ℒ𝑣0βˆ’πΆ1𝒩0𝑣0ξ€Έξ‚΅π‘£βˆ’πœ™,ℬ1,πœ•π‘£1ξ‚Άπœ•π‘›=0.(2.8)

Second-order problem: ℒ𝑣2=ξ€·1+𝐢1ℒ𝑣1βˆ’πΆ1𝒩1𝑣0,𝑣1ξ€Έ+𝐢2ℒ𝑣0βˆ’π’©0𝑣0ξ€Έξ‚΅π‘£βˆ’πœ™,ℬ2,πœ•π‘£2ξ‚Άπœ•π‘›=0.(2.9) Though we can construct higher-order problems easily, solutions upto the second-order problems are enough to produce excellent results.

If the series (2.6) is convergent at 𝑝=1, then the approximate solution in our case is,̃𝑒(π‘Ÿ)=𝑣(π‘Ÿ)=𝑣0(π‘Ÿ)+𝑣1ξ€·π‘Ÿ,𝐢1ξ€Έ+𝑣2ξ€·π‘Ÿ,𝐢1,𝐢2ξ€Έ.(2.10) By substituting (2.10) into (2.1), the resulting residual isβ„›ξ€·π‘Ÿ,𝐢1,𝐢2ξ€Έ=β„’(̃𝑒(π‘Ÿ))βˆ’π’©(̃𝑒(π‘Ÿ))βˆ’πœ™(π‘Ÿ).(2.11) If β„›=0, ̃𝑒 will be the exact solution. Otherwise, we minimize β„› over domain of the problem. To find the optimal values of 𝐢𝑖which minimizes β„›, many methods can be applied [13–17]. We follow two methods: the method of least squares and the Galerekin’s method. According to the method of least squares, we first construct the functionalπ’₯𝐢1,𝐢2ξ€Έ=ξ€œπ‘π‘Žβ„›2π‘‘π‘Ÿ,(2.12) and then minimizing it, we haveπœ•π’₯πœ•πΆ1=πœ•π’₯πœ•πΆ2=0.(2.13) According to the Galerekin’s method, we solve the following system for 𝐢1 and𝐢2:ξ€œπ‘π‘Žβ„›πœ•Μƒπ‘’πœ•πΆ1ξ€œπ‘‘π‘Ÿ=0,π‘π‘Žβ„›πœ•Μƒπ‘’πœ•πΆ2π‘‘π‘Ÿ=0.(2.14) Knowing 𝐢1 and 𝐢2, the approximate solution is well determined.

3. Some Numerical Examples

Example 3.1 (fifth-order linear). Consider the following problem: 𝑦(𝑣)(π‘₯)=π‘¦βˆ’15𝑒π‘₯βˆ’10π‘₯𝑒π‘₯,0<π‘₯<1,(3.1) with boundary conditions 𝑦(0)=0,π‘¦ξ…ž(0)=1,π‘¦ξ…žξ…ž(0)=0,𝑦(1)=0,π‘¦ξ…ž(1)=βˆ’π‘’.(3.2) The exact solution of this problem is 𝑦(π‘₯)=π‘₯(1βˆ’π‘₯)𝑒π‘₯.
We choose the auxiliary function as β„Ž(𝑝)=𝑝(𝐢1+𝐢2π‘₯). Plugging in this value in (2.3) of Section 2, we obtain the following linear problems which can be integrated directly.
Zeroth-order problem: 𝑦0(5)𝑦(π‘₯)=0,0(0)=0,π‘¦ξ…ž0(0)=0,𝑦0ξ…žξ…ž(0)=0,𝑦0(1)=0,π‘¦ξ…ž0(1)=βˆ’π‘’.(3.3)
First-order problem: 𝑦1(5)ξ€·π‘₯,𝐢1,𝐢2ξ€Έ=5𝑒π‘₯𝐢(3+2π‘₯)1+𝐢2π‘₯ξ€Έβˆ’ξ€·πΆ1+𝐢2π‘₯𝑦0𝑦(π‘₯),1(0)=0,π‘¦ξ…ž1(0)=0,𝑦1ξ…žξ…ž(0)=0,𝑦1(1)=0,π‘¦ξ…ž1(1)=0.(3.4)
Second-order problem: 𝑦2(5)ξ€·π‘₯,𝐢1,𝐢2ξ€Έ=ξ€·1+𝐢1+𝐢2π‘₯𝑦1(5)ξ€·π‘₯,𝐢1,𝐢2ξ€Έβˆ’ξ€·πΆ1+𝐢2π‘₯𝑦1ξ€·π‘₯,𝐢1,𝐢2ξ€Έ,𝑦2(0)=0,π‘¦ξ…ž2(0)=0,𝑦2ξ…žξ…ž(0)=0,𝑦2(1)=0,π‘¦ξ…ž2(1)=0.(3.5) Adding up the solutions of these problems, the second-order approximate solution, ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1ξ€·π‘₯,𝐢1,𝐢2ξ€Έ+𝑦2ξ€·π‘₯,𝐢1,𝐢2ξ€Έξ€·π‘₯+𝑂15ξ€Έ,(3.6) is determined by knowing the optimal values of the auxiliary constants, 𝐢1 and 𝐢2. Using Galerkin’s method, we obtain 𝐢1=βˆ’1.000245451, 𝐢2=0.000124615.
By considering these values, (3.6) becomes ̃𝑦(π‘₯)=π‘₯βˆ’0.5π‘₯3βˆ’0.333333π‘₯4βˆ’0.125π‘₯5βˆ’0.0333333π‘₯6βˆ’0.00694444π‘₯7βˆ’0.00119049π‘₯8βˆ’0.000173601π‘₯9βˆ’0.0000220495π‘₯10βˆ’2.48013Γ—10βˆ’6π‘₯11βˆ’2.50501Γ—10βˆ’7π‘₯12βˆ’2.27501Γ—10βˆ’8π‘₯13βˆ’2.0326Γ—10βˆ’9π‘₯14ξ€·π‘₯+𝑂15ξ€Έ.(3.7) Numerical results of the solution (3.7) are displayed in Table 1.


Exact sol.OHAM sol. (3.7) (ADM) (HPM) (VIM) (ITM) (VIHPM)

0.00.000000.00000.00000.00000.00000.00000.00000.0000
0.10.0994653830.099465383
0.20.1954244410.195424441
0.30.283470350.28347035
0.40.3580379270.358037927
0.50.4121803180.412180318
0.60.4373085120.437308512
0.70.4228880680.422888068
0.80.3560865480.356086548
0.90.2213642800.221364280
1.00.00000.0000.0000.0000.0000.0000.0000.000

= exact – approximate.

Example 3.2 (another fifth-order linear). Consider the following problem: 𝑦(5)(π‘₯)+π‘₯𝑦(π‘₯)=19π‘₯Cos(π‘₯)βˆ’2π‘₯3Cos(π‘₯)βˆ’41Sin(π‘₯)+2π‘₯2Sin(π‘₯),(3.8) with boundary conditions 𝑦𝑦(βˆ’1)=𝑦(1)=Cos(1),ξ…ž(βˆ’1)=π‘¦ξ…žπ‘¦(1)=βˆ’4Cos(1)+Sin(1),ξ…žξ…ž(βˆ’1)=3Cos(1)βˆ’8Sin(1).(3.9) Exact solution of this problem is 𝑦(π‘₯)=(2π‘₯2βˆ’1)Cos(π‘₯).
Considering the second-order solution ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1(π‘₯,𝐢1)+𝑦2(π‘₯,𝐢1,𝐢2)+𝑂(π‘₯13), we use the method of least squares to obtain 𝐢1=0.9940605306, 𝐢2=βˆ’3.9762851376.
Having these values, our solution in this case is ̃𝑦(π‘₯)=βˆ’0.999978+2.49992π‘₯2βˆ’1.04155π‘₯4+0.0846365π‘₯6βˆ’0.00276866π‘₯8+0.0000420239π‘₯10+3.38286Γ—10βˆ’7π‘₯12ξ€·π‘₯+𝑂13ξ€Έ.(3.10) Numerical results of the solution (3.10) are displayed in Table 2.


Exact sol.OHAM sol. (3.10) (3.10)

βˆ’1.00.5403023060.540302301
βˆ’0.80.1950778790.195077879
βˆ’0.6βˆ’0.231093972βˆ’0.231094010
βˆ’0.4βˆ’0.626321476βˆ’0.626321543
βˆ’0.2βˆ’0.901661252βˆ’0.901661334
0.0βˆ’1.000000000βˆ’1.000000086
0.2βˆ’0.901661252βˆ’0.901661334
0.4βˆ’0.626321476βˆ’0.626321543
0.6βˆ’0.231093972βˆ’0.231094010
0.80.1950778790.195077879
1.00.5403023060.540302301

= exact βˆ’ approximate.

Example 3.3 ([33] fifth-order nonlinear). Consider the following problem: 𝑦(𝑣)(π‘₯)=𝑦3(π‘₯)π‘’βˆ’π‘₯,0<π‘₯<1,(3.11) with boundary conditions 𝑦(0)=1,π‘¦ξ…ž(0)=1/2,π‘¦ξ…žξ…ž(0)=1/4,𝑦(1)=𝑒1/2,π‘¦ξ…ž1(1)=2𝑒1/2.(3.12) The exact solution for this problem is 𝑦(π‘₯)=𝑒π‘₯/2.
We consider the second-order solution, ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1(π‘₯,𝐢1)+𝑦2(π‘₯,𝐢1,𝐢2)+𝑂(π‘₯15).
Using Galerkin’s procedure in Section 2, we obtain the following values: 𝐢1=0.010868466,𝐢2=βˆ’0.029423113.(3.13) The second-order approximate solution is π‘₯̃𝑦(π‘₯)=1+2+π‘₯28+0.0205993π‘₯3+0.0030533π‘₯4+0.0000630671π‘₯5+5.2556Γ—10βˆ’6π‘₯6+3.754Γ—10βˆ’7π‘₯7+2.29401Γ—10βˆ’8π‘₯8+1.74892Γ—10βˆ’9π‘₯9βˆ’3.2692Γ—10βˆ’11π‘₯10βˆ’1.48596Γ—10βˆ’12π‘₯11βˆ’6.20243Γ—10βˆ’14π‘₯12βˆ’2.58495Γ—10βˆ’15π‘₯13+2.75287Γ—10βˆ’17π‘₯14ξ€·π‘₯+𝑂15ξ€Έ.(3.14) Numerical results of the solution (3.14) are displayed in Table 3.


Exact solutionOHAM sol. (3.14) (3.14) (DTM [33])

0.00.000000.00000.00000.0000
0.11.1051709181.105170919
0.21.2214027581.221402763
0.31.3498588081.349858818
0.41.4918246981.491824713
0.51.6487212711.648721287
0.61.8221188001.822118814
0.72.0137527072.013752717
0.82.2255409282.225540934
0.92.4596031112.459603112
1.02.7182818282.7182818240.0000.000

= exact – approximate.

Example 3.4 (sixth-order nonlinear). Consider the following problem: 𝑦(𝑣𝑖)(π‘₯)=𝑒π‘₯𝑦2(π‘₯),0<π‘₯<1,(3.15) with boundary conditions 𝑦(0)=1,π‘¦ξ…ž(0)=1,π‘¦ξ…žξ…ž(0)=1,𝑦(1)=π‘’βˆ’1,π‘¦ξ…ž(1)=βˆ’π‘’βˆ’1,π‘¦ξ…žξ…ž(1)=π‘’βˆ’1.(3.16) The exact solution is 𝑦(π‘₯)=π‘’βˆ’π‘₯.
For this problem, we take the auxiliary function β„Ž(𝑝)=𝑝(𝐢1+𝐢2π‘’βˆ’π‘₯), ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1ξ€·π‘₯,𝐢1,𝐢2ξ€Έ+𝑦2ξ€·π‘₯,𝐢1,𝐢2ξ€Έξ€·π‘₯+𝑂13ξ€Έ.(3.17) Using Galerkin’s method, we obtain 𝐢1=0.41243798998, 𝐢2=0.0014069149.
OHAM solution in this case is π‘₯̃𝑦(π‘₯)=1βˆ’0.999999994π‘₯+22βˆ’0.166666775π‘₯3+π‘₯424βˆ’0.008332465π‘₯5+0.001387441π‘₯6βˆ’0.000197784π‘₯7+0.000025071π‘₯8βˆ’3.002Γ—10βˆ’6π‘₯9+2.918Γ—10βˆ’7π‘₯10βˆ’6.7Γ—10βˆ’9π‘₯11βˆ’2.257Γ—10βˆ’9π‘₯12+1.806Γ—10βˆ’10π‘₯13βˆ’1.974Γ—10βˆ’11π‘₯14ξ€·π‘₯+𝑂15ξ€Έ.(3.18) Numerical results of the solution (3.18) are displayed in Table 4.


Exact solutionOHAM sol (3.18) (3.18) (ADM) (HPM) (VIM)

0.0110000
0.10.9048374180.904837418
0.20.8187307530.818730753
0.30.7408182210.740818221
0.40.6703200460.670320046
0.50.606530660.60653066
0.60.5488116360.548811636
0.70.4965853040.496585304
0.80.4493289640.449328965
0.90.406569660.40656966
1.00.3678794410.367879441

= exact βˆ’ approximate.

Example 3.5 (eighth-order nonlinear). Consider the following problem: 𝑦(𝑣𝑖𝑖𝑖)(π‘₯)=π‘’βˆ’π‘₯𝑦2(π‘₯),0<π‘₯<1,(3.19) with boundary conditions π‘¦ξ…ž(0)=1,π‘¦ξ…ž(0)=1,π‘¦ξ…žξ…ž(0)=1,π‘¦ξ…žξ…žξ…ž(0)=1,𝑦(1)=𝑒,π‘¦ξ…ž(1)=𝑒,π‘¦ξ…žξ…ž(1)=𝑒.(3.20) Considering the second-order solution ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1(π‘₯,𝐢1)+𝑦2(π‘₯,𝐢1,𝐢2)+𝑂(π‘₯13), the following values of the convergence controlling constants are obtained by using Galerkin’s method: 𝐢1=βˆ’1.451894673Γ—10βˆ’11,𝐢2=βˆ’0.000647581.(3.21) The approximate solution in this case is ̃𝑦(π‘₯)=1+π‘₯+0.5π‘₯2+0.166667π‘₯3+0.0416275π‘₯4+0.00884857π‘₯5+0.00117027π‘₯6+0.00033161π‘₯7+1.6061Γ—10βˆ’8π‘₯8+1.78456Γ—10βˆ’9π‘₯9+1.78456Γ—10βˆ’10π‘₯10+1.62233Γ—10βˆ’11π‘₯11+1.3494Γ—10βˆ’12π‘₯12ξ€·π‘₯+𝑂13ξ€Έ.(3.22) If the method of least squares is used to determine 𝐢’s, we have then 𝐢1=1.793Γ—10βˆ’8,𝐢2=βˆ’1.001347284.(3.23) The approximate solution in this case is ̃𝑦(π‘₯)=1+π‘₯+0.5π‘₯2+0.166667π‘₯3+0.0416667π‘₯4+0.00833313π‘₯5+0.00138918π‘₯6+0.000198233π‘₯7+0.000024835π‘₯8+2.75944Γ—10βˆ’6π‘₯9+2.75944Γ—10βˆ’7π‘₯10+2.50859Γ—10βˆ’8π‘₯11+2.08656Γ—10βˆ’9π‘₯12ξ€·π‘₯+𝑂13ξ€Έ.(3.24) Let us use the auxiliary function β„Ž(𝑝)=𝑝(𝐢1+𝐢2π‘’βˆ’π‘₯) and consider the second-order solution ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1ξ€·π‘₯,𝐢1,𝐢2ξ€Έ+𝑦2ξ€·π‘₯,𝐢1,𝐢2ξ€Έξ€·π‘₯+𝑂13ξ€Έ.(3.25) Using Galerkin’s method, we obtain 𝐢1=βˆ’0.9993171458, 𝐢2=βˆ’0.0012314995.
The OHAM solution in this case is π‘₯̃𝑦(π‘₯)=1+π‘₯+2+π‘₯2!33!+0.041666667π‘₯4+0.008333333π‘₯5+0.001388889π‘₯6+1.984Γ—10βˆ’4π‘₯7+2.480Γ—10βˆ’5π‘₯8+2.756Γ—10βˆ’6π‘₯9+2.756Γ—10βˆ’7π‘₯10+2.505Γ—10βˆ’8π‘₯11+2.088Γ—10βˆ’9π‘₯12ξ€·π‘₯+𝑂13ξ€Έ.(3.26) Numerical results of the solutions (3.22), (3.24), and (3.26) are displayed in Table 5.


ExactOHAM sol. (3.22) (3.24) (3.26) ([25])

0.0110000
0.11.1051709181.105170915
0.21.2214027581.221402732
0.31.3498588081.349858729
0.41.4918246981.491824562
0.51.6487212711.648721109
0.61.8221188001.822118662
0.72.0137527072.013752625
0.82.2255409282.225540900
0.92.4596031112.459603108
1.02.7182818282.718281828

= exact βˆ’ approximate.

Example 3.6 (nineth-order linear). Consider the following problem: 𝑦(9)(π‘₯)=𝑦(π‘₯)βˆ’9𝑒π‘₯,(3.27) with boundary conditions 𝑦(0)=1,π‘¦ξ…ž(0)=0,π‘¦ξ…žξ…ž(0)=βˆ’1,π‘¦ξ…žξ…žξ…ž(0)=βˆ’2,π‘¦ξ…žξ…žξ…žξ…ž(0)=βˆ’3,𝑦(1)=0,π‘¦ξ…ž(1)=βˆ’π‘’,π‘¦ξ…žξ…ž(1)=βˆ’2𝑒,π‘¦ξ…žξ…žξ…ž(1)=βˆ’3𝑒.(3.28) Exact solution is 𝑦(π‘₯)=(1βˆ’π‘₯)𝑒π‘₯.
For this linear problem, we take β„Ž(𝑝)=𝑝(𝐢1+𝐢2π‘₯), and according to the rest of the procedure of OHAM, the second-order solution, ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1(π‘₯,𝐢1,𝐢2)+𝑦2(π‘₯,𝐢1,𝐢2)+𝑂(π‘₯13), is determined by the values of 𝐢𝑖:𝑖=1,2. Following the Galerkin’s method, we obtain 𝐢1=βˆ’1, 𝐢2=0, for π‘Ž=0 and 𝑏=1.
The second-order approximate solution is π‘₯𝑦(π‘₯)=1βˆ’22βˆ’π‘₯33βˆ’π‘₯44βˆ’0.03333333333π‘₯5βˆ’0.0069444444π‘₯6βˆ’0.0011904762π‘₯7βˆ’0.0001736111π‘₯8βˆ’0.00002204586π‘₯9βˆ’2.48016Γ—10βˆ’6π‘₯10βˆ’2.50521Γ—10βˆ’7π‘₯11βˆ’2.29644Γ—10βˆ’8π‘₯12ξ€·π‘₯+𝑂13ξ€Έ.(3.29) Numerical results of the solution (3.29) are displayed in Table 6.


Exact OHAM sol. (3.29) ([27])

0.0110.00000.0000
0.10.9946538260.994653826
0.20.9771222060.977122206
0.30.9449011650.944901165
0.40.8950948180.895094818
0.50.8243606350.824360635
0.60.7288475200.728847520
0.70.6041258120.604125812
0.80.4451081860.445108186
0.90.2459603110.245960311
1.00.0000 0.000

= exact βˆ’ approximate.

Example 3.7 (tenth-order nonlinear). Consider the following problem:𝑦(𝑋)(π‘₯)=π‘’βˆ’π‘₯𝑦2(π‘₯),0<π‘₯<1,𝑦(0)=1,π‘¦ξ…ž(0)=1,π‘¦ξ…žξ…ž(0)=1,π‘¦ξ…žξ…žξ…ž(0)=1,𝑦(𝑖𝑣)(0)=1,𝑦(1)=𝑒,π‘¦ξ…ž(1)=𝑒,π‘¦ξ…žξ…ž(1)=𝑒,π‘¦ξ…žξ…žξ…ž(1)=𝑒,𝑦(𝑖𝑣)(1)=𝑒.(3.30) We consider the second-order solution ̃𝑦(π‘₯)=𝑦0(π‘₯)+𝑦1(π‘₯,𝐢1)+𝑦2(π‘₯,𝐢1,𝐢2)+𝑂(π‘₯13).
To find the values of 𝐢𝑖, we apply the Galarkin’s method. So solving the system ξ€œπ‘π‘Žπ‘…πœ•Μƒπ‘£πœ•πΆ1ξ€œπ‘‘π‘Ÿ=0,π‘π‘Žπ‘…πœ•Μƒπ‘£πœ•πΆ2π‘‘π‘Ÿ=0,(3.31) we have 𝐢1=0, 𝐢2=βˆ’1.023966086.
In this case, the approximate solution is π‘₯̃𝑦(π‘₯)=1+π‘₯+22+π‘₯36+π‘₯424+0.008333323π‘₯5+0.00138894π‘₯6+0.000198312π‘₯7+0.000024898π‘₯8+2.712Γ—10βˆ’6π‘₯9+2.8218Γ—10βˆ’7π‘₯10+2.5652Γ—10βˆ’8π‘₯11+2.1377Γ—10βˆ’9π‘₯12ξ€·π‘₯+𝑂13ξ€Έ.(3.32) Numerical results of the solution (3.32) are displayed in Table 7.


Exact OHAM sol. (3.32) ([27])

0.01100
0.11.1051709181.105170918
0.21.2214027581.221402758
0.31.3498588081.349858808
0.41.4918246981.491824698
0.51.6487212711.648721271
0.61.8221188001.822118800
0.72.0137527072.013752707
0.82.2255409282.225540928
0.92.4596031112.459603111
1.02.7182818282.718281828

= exact βˆ’ approximate.

4. Conclusions

In this paper, we have used OHAM to find the approximate analytic solution to higher-order two-point boundary value problems in finite domain. It is observed that the method is explicit, effective, and reliable. It works well for higher-order problems and represents the fastest convergence as well as a remarkable low error. The OHAM also provides us with a very simple way to control and adjust the convergence of the series solution using the auxiliary constants 𝐢𝑖’s which are optimally determined. Furthermore, by using different forms of the auxiliary function, more accuracy can be obtained. It has been also observed that for determining the optimal values of 𝐢’s, the performance of both the least squares and the Galerkin’s method is problem dependent. One can select one of these two which best suits the problem solution.

References

  1. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, UK, 1961.
  2. R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Teaneck, NJ, USA, 1986.
  3. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
  4. G. Adomian, β€œA review of the decomposition method and some recent results for nonlinear equations,” Computers & Mathematics with Applications, vol. 21, no. 5, pp. 101–127, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. A.-M. Wazwaz, β€œApproximate solutions to boundary value problems of higher order by the modified decomposition method,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 679–691, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. J.-H. He, β€œVariational approach to the sixth-order boundary value problems,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 537–538, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. J.-H. He, β€œHomotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. J.-H. He, β€œHomotopy perturbation method for solving boundary value problems,” Physics Letters. A, vol. 350, no. 1-2, pp. 87–88, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. H. E. Ji-Huan, β€œA Note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565–568, 2010. View at: Google Scholar
  10. S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  11. S. Liao, β€œNotes on the homotopy analysis method: some definitions and theorems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 983–997, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. J. K. Zhou, Deferential Transformation and its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
  13. V. Marinca, N. Herişanu, and I. Nemeş, β€œOptimal homotopy asymptotic method with application to thin film flow,” Central European Journal of Physics, vol. 6, no. 3, pp. 648–653, 2008. View at: Publisher Site | Google Scholar
  14. N. Herisanu, V. Marinca, T. Dordea, and G. Madescu, β€œA new analytical approach to nonlinear vibration of an electric machine,” Proceedings of the Romanian Academy, vol. 9, no. 3, 2008. View at: Google Scholar
  15. V. Marinca, N. Herişanu, C. Bota, and B. Marinca, β€œAn optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate,” Applied Mathematics Letters, vol. 22, no. 2, pp. 245–251, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. V. Marinca and N. Herişanu, β€œApplication of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 35, no. 6, pp. 710–715, 2008. View at: Publisher Site | Google Scholar
  17. V. Marinca and N. Herişanu, β€œOptimal homotopy perturbation method for strongly nonlinear differential equations,” Nonlinear Science Letters A, vol. 1, pp. 273–280, 2010. View at: Google Scholar
  18. A.-M. Wazwaz, β€œThe numerical solution of fifth-order boundary value problems by the decomposition method,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 259–270, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. A.-M. Wazwaz, β€œThe numerical solution of sixth-order boundary value problems by the modified decomposition method,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 311–325, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. M. Aslam Noor and S. T. Mohyud-Din, β€œAn efficient algorithm for solving fifth-order boundary value problems,” Mathematical and Computer Modelling, vol. 45, no. 7-8, pp. 954–964, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. M. A. Noor and S. T. Mohyud-Din, β€œVariational iteration method for fifth-order boundary value problems using He's polynomials,” Mathematical Problems in Engineering, vol. 2008, Article ID 954794, 12 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. M. Aslam Noor and S. T. Mohyud-Din, β€œVariational iteration technique for solving higher order boundary value problems,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1929–1942, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. M. A. Noor and S. T. Mohyud-Din, β€œA reliable approach for solving linear and nonlinear sixth-order boundary value problems,” International Journal of Computational and Applied Mathematics, vol. 2, no. 2, pp. 163–172, 2007. View at: Google Scholar
  24. M. A. Noor and S. T. Mohyud-Din, β€œHomotopy perturbation method for solving sixth-order boundary value problems,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2953–2972, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  25. M. A. Noor, S. T. Mohyud-Din, and M. Tahir, β€œVariational iteration decomposition method for solving eighth-order boundary value problems,” Differential Equations and Nonlinear Mechanics, vol. 2007, Article ID 19529, 16 pages, 2007. View at: Publisher Site | Google Scholar
  26. S. T. Mohyud-Din, A. Yildirim, and M. M. Hosseini, β€œAn iterative algorithm for fifth-order boundary value problems,” World Applied Sciences Journal, vol. 8, no. 5, pp. 531–535, 2010. View at: Google Scholar
  27. S. T. Mohyud-Din and A. Yildirim, β€œSolutions of tenth and ninth-order boundary value problems by modified variational iteration method,” Applications and Applied Mathematics, vol. 5, no. 1, pp. 11–25, 2010. View at: Google Scholar | Zentralblatt MATH
  28. K. N. S. Kasi Viswanadham and P. Murali Krishna, β€œQuintic B-Spline Galerkin method for fifth order boundary value problems,” ARPN Journal of Engineering and Applied Sciences, vol. 5, no. 2, 2010. View at: Google Scholar
  29. Siraj-ul-Islam and M. Azam Khan, β€œA numerical method based on polynomial sextic spline functions for the solution of special fifth-order boundary-value problems,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 356–361, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  30. Siraj-ul-Islam, I. A. Tirmizi, Fazal-i-Haq, and M. A. Khan, β€œNon-polynomial splines approach to the solution of sixth-order boundary-value problems,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 270–284, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  31. J. Ali, Siraj-ul-Islam, S. Islam, and G. Zaman, β€œThe solution of multipoint boundary value problems by the Optimal Homotopy Asymptotic Method,” Computers and Mathematics with Applications, vol. 59, no. 6, pp. 2000–2006, 2010. View at: Publisher Site | Google Scholar
  32. J. Ali, S. Islam, M. Tariq Rahim, and G. Zaman, β€œThe solution of special twelfth order boundary value problems by the optimal homotopy asymptotic method,” World Applied Sciences Journal, vol. 11, no. 3, pp. 371–378, 2010. View at: Google Scholar
  33. C. H. Che Hussin and A. Kiliçman, β€œOn the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method,” Mathematical Problems in Engineering, vol. 2011, Article ID 724927, 19 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Javed Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

1242Β Views | 688Β Downloads | 3Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles