Homotopy Perturbation Method for Solving Twelfth Order Boundary Value Problems

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163

Homotopy Perturbation Method for Solving Twelfth Order Boundary Value Problems

H.Mirmoradi^*, H. Mazaheripour, S.Ghanbarpour, A.Barari Department of Civil Engineering, University of Mazandaran, Babol, Iran

ABSTRACT

Approximate analytical solutions are obtained using homotopy perturbation method (HPM), which is a coupling of traditional perturbation method and the homotopy method. Two examples are presented to illustrate the effectiveness of HPM for solving twelfth order boundary value problems. In order to achieve adequately precise results with HPM, it is usually required to calculate at least two statements of the p- terms. However, it was shown in the numerical examples that highly accurate results were obtained by calculating only one p-term of the series, revealing the effectiveness of the HPM solution. It was concluded that HPM is a powerful tool for solving high-order boundary value problems arising in various fields of engineering and science.

KEYWORDS: Twelfth order boundary value problems, Approximate analytical solution, Homotopy perturbation method

1. Introduction

Many important phenomena occurring in various fields of engineering and science are frequently modeled through linear and nonlinear differential equations. However, it is still very difficult to obtain closed form solutions for most models of real–life problems. Approximate analytical solutions for solving such problems have been gaining popularity in the recent years. The perturbation method – a numerical method used for solving linear and nonlinear problems - is based on assuming a small parameter. The majority of nonlinear problems however, especially those having strong nonlinearity, have no small parameters at all and the approximate solutions obtained by the perturbation methods, in most cases, are valid only for small values of the small parameter. Generally, the perturbation solutions are valid as long as the scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion on which the small parameter should exist. Thus, it is essential to check the validity of the approximations numerically and/or experimentally. To overcome these difficulties, a modified form of the perturbation method called homotopy perturbation method (HPM) was introduced by He [1], followed by a thorough description of the method where it was verified that the method is, in fact, a coupling between the traditional perturbation method and the homotopy method [2]. HPM has since then been effectively utilized in obtaining approximate analytical solutions to many linear and nonlinear problems arising in engineering and science, such as nonlinear oscillators with discontinuities [3], nonlinear wave equations [4,5], Burgers and coupled Burgers’ equations [6], general boundary value problems [7], Helmholtz equation and fifth order KDV equation [8], Axisymmetric flow over a stretching sheet [9], thin film flow of a fourth grade fluid down a vertical cylinder [10], nonlinear Volterra –Fredholm integral equations [11], nonlinear fourth order boundary value problem [12], Ratio-Dependent Predator-Prey System with Constant Effort Harvesting [13].

Twelfth order differential equations have several important applications in engineering.

Chandrasekhar [14] showed that when an infinite horizontal layer of fluid is put into rotation and simultaneously subjected to heat from below and a uniform magnetic field across the fluid in the same

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direction as gravity, instability will occur. It was then shown that if instability sets in as overstability, the problem is governed by a twelfth order differential boundary value equation. Such problems arise in geophysics when studying core fluid adjacent to the core-mantle boundary. Fearn and Richardson [15] used a cylindrical model and a twelfth order boundary value problem to study the combined effects of a stably stratified layer and a spatially varying magnetic field on thermally driven convection. Sachdev et al. [16]

studied flow in a paraboloidal basin or an eddy bounded by a free surface using simplified equations of a twelfth degree nonlinear ordinary differential equation. In plate deflection theory, several researchers considered the problem of transverse shear and normal strain and stress effects on antisymmetric bending of isotropic plates by deriving twelfth order partial differential equations [17-20]. Vekua [21] showed that twelfth order differential equations also arise in studying elastic shells having variable thickness.

Several researchers developed numerical techniques for solving twelfth order differential equations.

For example, Twizell et al. [22] developed finite difference methods for solving high-order eigenvalue problems arising in thermal instability; Islam et al. [23] used differential transform method; Siddiqi and Twizell [24] used polynomial splines of twelfth degree; Wazwaz [25] used the modified decomposition method. As stated above, HPM will be used herein to solve twelfth degree boundary value problem of the general form:

[ ]

(12)

0 1

(1) (1)

0 1

(2) (2)

0 1

(3) (3)

0 1

(4) (4)

0 1

(5) (5)

0 1

( ) ( ) ( ) ( ), , ,

( ) , ( ) ,

y x f x y x g x x a b

y a y b

α α

γ γ

δ δ

ν ν

ξ ξ

ω ω

+ = ∈

= =

(1)

where

α

i,

γ

i,

δ

i,

ν

i,

ξ

i and

ω

i; i = 0, 1 are finite real constants and the functions f(x) and g(x) are continuous on [a,b].

2. Basic idea of Homotopy-perturbation method

To explain this method, let us consider the following function:

( ) ( ) 0,

A u − f r = r ∈Ω

(2)

with the boundary conditions of:

( , u ) 0, ,

B u r

n

∂ = ∈Γ

∂

⁽³⁾

where

A

,

B

,

f r ( )

and

Γ

are a general differential operator, a boundary operator, a known analytical function and the boundary of the domain

Ω

, respectively. Generally speaking, the operator

A

can be divided into a linear part

L

and a nonlinear part

N u ( )

. Eq. (2) may therefore be written as:

( ) ( ) ( ) 0

L u + N u − f r =

(4)

By the homotopy technique, we construct a homotopy

(3)

165 ( , ) : [0,1]

v r p Ω× → R

which satisfies:

[

0

] [ ]

( , ) (1 ) ( ) ( ) ( ) ( ) 0,

[0,1], ,

H v p p L v L u p A v f r

p r

= − − + − =

∈ ∈Ω

⁽⁵⁾

or

[ ]

0 0

( , ) ( ) ( ) ( ) ( ) ( ) 0

H v p = L v − L u + pL u + p N v − f r =

(6)

where

p ∈ [0,1]

is an embedding parameter, while

u

₀ is an initial approximation of Eq.(2), which satisfies the boundary conditions. Obviously, from Eqs. (5) and (6) we will have:

( , 0) ( ) ( ) 0,

H v = L v − L u 0 =

₍₇₎

( ,1) ( ) ( ) 0,

H v = A v − f r =

(8)

The changing process of p from zero to unity is just that of

v r p ( , )

from

u

₀ to

u r ( )

. In topology, this is called deformation, while

L v ( ) − L u ( )

₀ and

A v ( ) − f r ( )

are called homotopy. According to the HPM, we can first use the embedding parameter p as a “small parameter”, and assume that the solutions of Eqs. (5) and (6) can be written as a power series in p:

2

0 1 2

...

v = v + pv + p v +

(9)

Setting

p = 1

yields in the approximate solution of Eq. (9) to:

0 1 2

lim 1

u v v v v

= p = + + +

→ L

₍₁₀₎

The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantages. The series (10) is convergent for most cases. However, the convergent rate depends on the nonlinear operator

( )

A v

. Moreover, He [1] made the following suggestions:

(1) The second derivative of

N v ( )

with respect to v must be small because the parameter may be relatively large, i.e.

p → 1

.

(2) The norm of ¹

N

L v

−

∂

must be smaller than one so that the series converges.

3. Numerical examples

In order to illustrate the ability of HPM in solving twelfth order problems, we consider the following two examples herein:

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166

Example 1. For

^x ^∈ [ ] ^0,1

, consider the following boundary value problem

(12) 3

(1) (1)

(2) (2)

(3) (3)

(4) (4)

(5) (5)

( ) ( ) (120 23 ) ,

(0) 0, (1) 0

(0) 1, (1) ,

(0) 0, (1) 4 ,

(0) 3, (1) 9 ,

(0) 8, (1) 16

(0) 15, (1) 25 ,

y x x y x x x e

x

y y

y y e

+ = − + +

= =

⎧ ⎪ = = −

⎪ ⎪ = = −

⎪ ⎨

= − = −

⎪ ⎪ = − = −

⎩

(11)

The analytic solution of the above differential system is

( ) (1 ) ,

^x

y x = x − x e

(12)

In order to solve Eq. (11) by the HPM, a homotopy can be constructed as follows:

12 12

12 12 0

12

3 12

( , ) (1 )( ( ) )

( ( ) ( ) (120 23 ) 0,

( )

)

^x

d d

H v p p v x v

dx dx

p d v x xv x x x

dx

x

e

= − − +

+ + + + =

(13)

Substituting

v = v

₀

+ pv

₁

+ ...

in to Eq. (13) and rearranging the resultant equation based on powers of p- terms, one has:

12 0

: d ( ) 0,

p v x

dx =

⁽¹⁴⁾

12

1 3

1 0

: ( d

12

( )) 120

^x

( )

^x

23

^x

0,

p v x e xv x e x xe

dx + + + + =

⁽¹⁵⁾

12 2

2 1

: ( 12

( )) ( ) 0,

p

d

v x xv x

dx + =

(16)

With the following conditions:

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167

0 0

2 2

0 0

2 2

3 3

0 0

3 3

4 4

0 0

4 4

5 5

0 0

5 5

(0) 0, (1) 0,

(0) 1, (1) ,

(0) 0, (1) 4 ,

(0) 3, (1) 9

(0) 8, (1) 16

(0) 15, (1) 25

(0) 0, (1) 0,

(0) 0,

i i

i

v v

d d

v v e

dx dx

d d

v v e

dx dx

d d

v v e

dx dx

d d

v v e

dx dx

d d

v v e

dx dx

v v

d d

dx v d

= =

⎧ ⎪

⎪ = = −

⎪ ⎪

⎪ = = −

⎪⎪ ⎨

= − = −

⎪ ⎪

⎪ = − = −

⎪ ⎪

⎪ = − = −

⎪⎩

= =

=

2 2

3 3

4 4

5 5

(1) 0,

(0) 0, (1) 0 ,

(0) 0, (1) 0 1, 2,...

(0) 0, (1) 0

i

i i

x v

d d

v v

dx dx

d d

v v i

dx dx

d d

v v

dx dx

d d

v v

dx dx

⎧ ⎪

⎪ =

⎪ ⎪

⎪ = =

⎪⎪ ⎨ = = =

⎪ ⎪

⎪ = =

⎪ ⎪

⎪ = =

⎪⎩

(17)

With the effective initial approximation for

v

₀ from the conditions (17), solution of Eq. (14) may be written as follows:

6 11 5 10 4 9

0

3 8 3 7 6 5

4 3

( ) 4.584 10 10 10

10 10

1.625 1.816

1.184 6.946 0.0333 0.125

0.3333 0.5

v x

x x x

x x x x

x x x

− − −

− −

= − × × ×

× ×

− − −

− − − −

− +

(18)

In the same manner, the rest of components were obtained using the Maple package.

According to the HPM, we can conclude that:

( ) lim ( ) ⁰( ) ¹( ) ... , 1

u x v x v x v x

p

= = + +

→ ⁽¹⁹⁾

Therefore, substituting the value of

v x

₀

( )

from Eq. (18) in to Eq. (19) yields:

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168

6 11 5 10 4 9

3 8 3 7 6 5

4 3

( ) 4.584 10 10 10

10 10

1.625 1.816

1.184 6.946 0.0333 0.125

0.3333 0.5

u x

x x x

x x x x

x x x

− − −

− −

= − × × ×

× ×

− − −

− − − −

− +

(20)

Comparison of the approximate solutions with exact solution is tabulated in Table. 1 along with the errors of the HPM, showing a remarkable agreement. It is noteworthy that even higher accuracy could be achieved by calculating second and third terms from Eqs. (15) and (16).

Table 1

Comparison between exact solution of Eq. (11) and the solution from HPM

Example 2. For

^x ^{∈ −} [ ^1,1 ]

, consider the following boundary value problem

(12)

(1) (1)

(2) (2)

(3) (3)

(4) (4)

(5) (5)

( ) ( ) 12(2 cos( ) 11sin( )), ( 1) (1) 0

( 1) (1) 2 sin(1),

( 1) (1) 4 cos(1) 2 sin(1), ( 1) (1) 6 cos(1) 6 sin(1), ( 1) (1) 8 cos(1) 12 sin(1), ( 1) (1) 20 cos

y x y x x x x

y y

− = − +

− = =

− = − = − −

− = = −

− = − = +

− = = − (1) 10 sin(1),

⎧ ⎪

⎪ ⎪⎪

⎨ ⎪

⎪ ⎪

⎪ +

⎩

(21)

x Exact solution HPM Error of HPM

0 0.0000000000 0.0000000000 0.000000E+000 0.1 0.0994653826 0.0994653826 3.000000E-011 0.2 0.1954244413 0.1954244413 0.000000E+000 0.3 0.2834703497 0.2834703496 -1.000000E-010 0.4 0.3580379275 0.3580379277 2.000000E-010 0.5 0.4121803178 0.4121803189 1.100000E-009 0.6 0.4373085120 0.4373085164 4.400000E-009 0.7 0.4228880685 0.4228880820 1.350000E-008 0.8 0.3560865485 0.3560865853 3.680000E-008 0.9 0.2213642800 0.2213643701 9.010000E-008 1.0 0.0000000000 2.02700E-007 2.027000E-007

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169

The analytic solution of the above differential system is

( ) (

2

1) sin

y x = x − x

(22)

A homotopy can be constructed as follows:

12 12

12 12 0

12

( , ) (1 )( ( ) )

( ( ) ( ) 24 cos( ) 132 sin( )) 0, d d

( )

H v p p v x v

dx dx

p d v x v x x x x

dx

= − −

x

+

− + + =

(23)

Substituting

v = v

₀

+ pv

₁

+ ...

in to Eq. (23) and rearranging the resultant equation based on powers of p- terms, one has:

12 0

: d ( ) 0,

p v x

dx =

⁽²⁴⁾

12 1

1 0

: ( d

12

( )) 24 cos( ) ( ) 132 sin( ) 0,

p v x x x v x x

dx + − + =

⁽²⁵⁾

12 2

2 1

: ( 12

( )) ( ) 0,

p

d

v x v x

dx − =

(26)

With the following conditions(see next page):

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170

with the effective initial approximation for

v

₀ from the conditions (27) and solution of Eq. (24) may be written as follows:

6 11 4 9 3 7

0

1 5 3

4 3

( ) 2.632 10 10 10

10

2.008 8.531

1.749 1.6666 0.99999

0.3333 0.5

v x

x x x

− − −

−

= × × ×

×

− + −

− −

− +

(28)

In the same manner, the rest of components were obtained using the Maple package.

According to the HPM, we can conclude that:

( ) lim ( ) ⁰( ) ¹( ) ... , 1

u x v x v x v x

p

= = + +

→ (29)

Therefore, substituting the value of

v x

₀

( )

from Eq. (28) in to Eq. (29) yields:

6 11 4 9 3 7

1 5 3

( ) 2.632 10 10 10

10

2.008 8.531

1.749 1.6666 0.99999

u x

x x x

− − −

−

= × × ×

×

− + −

− −

(30)

Comparison of the approximate solutions with exact solution is tabulated in Table. 2 along with errors of HPM, revealing the high accuracy of the results from HPM. Once again, as stated in example 1, it is obvious that higher accuracy could be obtained without any difficulty.

2 2

0 0

2 2

3 3

0 0

3 3

4 4

0 0

4 4

5 5

0 0

5 5

( 1) (1) 4 c o s (1) 2 s in (1)

( 1) (1) 6 c o s (1) 6 s in (1)

( 1) (1) 8 c o s (1) 1 2 s in (1)

( 1) (1) 2 0 c o s (1) 1 0 s in (1

d d

v v

d x d x

d d

v v

d x d x

d d

v v

d x d x

d d

v v

d x d x

− = − = − −

− = = −

− = − = +

− = = − +

2 2

3 3

4 4

5 5

)

( 1) 0 , (1) 0 ,

( 1) 0 , (1) 0 1, 2 , ...

( 1) 0 , (1) 0

i i

v v

d d

v v

d x d x

d d

v v

d x d x

d d

v v i

d x d x

d d

v v

d x d x

d d

v v

d x d x

⎪⎪

⎪⎪⎨

⎪⎪

⎪⎩

− = =

⎧⎪

⎪ − = =

⎪⎪

⎪ − = =

⎪⎪⎨ − = = =

⎪⎪

⎪ − = =

⎪⎪

⎪ − = =

⎪⎩

(27)

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171

Table 2

Comparison between exact solution of Eq. (21) and the solution from HPM

4. Conclusion

In this paper, solution of twelfth order boundary value problem has been given using homotopy perturbation method. The numerical examples considered revealed that homotopy perturbation method is both accurate and effective for solving twelfth order boundary value problems, since highly accurate results were obtained in both examples by calculating a single p-term. It can be concluded that homotopy perturbation method is a highly efficient method for solving high-order boundary value problems arising in various fields of engineering and science.

References

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x Exact solution HPM Error of HPM

-1 0.000000000 -1.60000E-009 -1.60000E-009

-0.8 0.258248192 0.2582481925 -2.00000E-010

-0.6 0.361371183 0.3613711820 -1.00000E-009

-0.4 0.327111407 0.3271114041 -3.00000E-009

-0.2 0.190722557 0.1907225537 -3.90000E-009

0 0.000000000 0.0000000000 0.00000E+000

0.2 -0.190722557 -0.1907225537 3.30000E-009

0.4 -0.327111407 -0.3271114041 2.90000E-009

0.6 -0.361371183 -0.3613711820 1.00000E-009

0.8 -0.258248192 -0.2582481925 -5.00000E-010

1.0 0.000000000 1.60000E-009 1.60000E-009

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