Homotopy Analysis Method for Second-Order Boundary Value Problems of Integro-differential Equations

7/30/2019 Homotopy Analysis Method for Second-Order Boundary Value Problems of Integro-differential Equations

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Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2012, Article ID 365792, 18 pagesdoi:10.1155/2012/365792

Research ArticleHomotopy Analysis Method forSecond-Order Boundary Value Problems of Integrodifferential Equations

Ahmad El-Ajou, 1 Omar Abu Arqub, 1 and Shaher Momani 2

1 Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan2

Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, JordanCorrespondence should be addressed to Omar Abu Arqub, [email protected]

Received 1 April 2012; Accepted 19 June 2012

Academic Editor: Gabriele Bonanno

Copyright q 2012 Ahmad El-Ajou et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In this paper, series solution of second-order integrodi ff erential equations with boundary condi-tions of theFredholm and Volterra types by means of thehomotopy analysis method is considered.The new approach provides the solution in the form of a rapidly convergent series with easily

computable components using symbolic computation software. The homotopy analysis methodprovides us with a simple way to adjust and control the convergence region of the innite seriessolution by introducing an auxiliary parameter. The proposed technique is applied to a few testexamples to illustrate the accuracy, e fficiency, and applicability of the method. The results revealthat the method is very e ff ective, straightforward, and simple.

1. Introduction

Integrodi ff erential equations IDEs are often involved in the mathematical formulation of physical and engineering phenomena. IDEs can be encountered in various elds of sciencesuch as physics, chemistry, biology, and engineering. These kinds of equations can also befound in numerous applications, such as electromagnetic, plasma physics, elasticity, uiddynamics, oscillation theory, polymer rheology, chemical kinetics, biomechanics, and controltheory 15 . Since it is usually impossible to obtain the closed-form solutions to second-order boundary value problems of IDEs met in practice, these problems must be attacked byvarious approximate and numerical methods.

The purpose of this paper is to extend the application of the homotopy analysismethod HAM to provide symbolic approximate solution for the second-order boundary


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2 Discrete Dynamics in Nature and Society

value problems of IDEs of the following Fredholm type:

u x

b

a

K x, s f u s , u s ds g u x , u x h x 0, a s, x b, 1.1

and the following Volterra type:

u x x

aK x, s f u s , u s ds g u x , u x h x 0, a s, x b, 1.2

subject to the boundary conditions

u a 1, u b 2, 1.3

where a, b, 1, 2 are real nite constants, u is an unknown function to be determined, Kand h are continuous functions on a, b a, b and a, b , respectively, and f, g are nonlinearcontinuous functions of u, u . For details about the existence and uniqueness of solutions forsuch problems, see 68 . However, we assume that 1.1 and 1.2 subject to the boundaryconditions 1.3 have a unique analytic solution on the given interval.

The numerical solvability of second-order IDEs with boundary conditions of theFredholm and Volterra types and other related equations has been pursued by severalauthors. To mention a few, in 9 , the authors have discussed the Legendre polynomialsmethod for solving Fredholm equation u x

ba K x, s u s ds f x p x u x q x u x

0. Furthermore, the compact nite di ff erence method is carried out in 10 for the Volterra IDEu x

xa K x, s u s ds g x, u x 0 and the Fredholm IDE u x

ba K x, s u s ds

g x, u x 0. Recently, the monotone iterative sequences method for solving Volterraequation u x

x

0K x, s f u s ds g x 0 is proposed in 11 .

The HAM, which is proposed by Liao 1217 , is eff ectively and easily used tosolve some classes of linear and nonlinear problems without linearization, perturbation, ordiscretization. The HAM is based on the homotopy, a basic concept in topology. The auxiliaryparameter is introduced to construct the so-called zero-order deformation equation. Thus,unlike all previous analytic techniques, the HAM provides us with a family of solutionexpressions in auxiliary parameter . As a result, the convergence region and rate of solutionseries are dependent upon the auxiliary parameter and thus can be greatly enlarged bymeans of choosing a proper value of . This provides us with a convenient way to adjust andcontrol convergence region and rate of solution series given by the HAM.

In the recent years, extensive work has been done using HAM, which provides analyti-cal approximations for linear and nonlinear equations. This method has been implemented inseveral functional equations, such as nonlinear water waves 15 , unsteady boundary-layerows 16 , solitary waves with discontinuity 17 , Klein-Gordon equation 18 , projectilemotion with the quadratic resistance law 19 , systems of fractional di ff erential equations20 , nonlinear fractional di ff erential equations 21, 22 , systems of fractional algebraic-

diff erential equations 23 , fractional SIR model 24 , singular Volterra integral equation 25 ,charged particle motion for certain congurations of oscillating magnetic elds 26 , Volterrapopulation growth model 27 , systems of fractional IDEs 28 , and high-order IDEs 29 .

The outline of the paper is as follows. In the next section, the basic idea of the HAMis introduced. In Section 3, we utilize the statement of the HAM for solving second-order


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Discrete Dynamics in Nature and Society 3

IDEs with boundary conditions of the Fredholm and Volterra types. In Section 4, numericalexamples are given to illustrate the capability of HAM. This paper ends in Section 5 withsome concluding remarks.

2. Homotopy Analysis Method

The principles of the HAM and its applicability for various kinds of functional equations aregiven in 1228 . For convenience of the reader, we will present a review of the HAM 1217then we will extend the HAM to construct a symbolic approximate solution to second-orderIDEs with boundary conditions of the Fredholm and Volterra types. To achieve our goal, weconsider the following nonlinear functional equation:

N u x 0, x a, 2.1

where N is a nonlinear operator and u x is an unknown function of independent variable x.Let u0 x denote an initial guess of the exact solution of 2.1 , / 0 an auxiliaryparameter, H x / 0 an auxiliary function, and an auxiliary linear operator with theproperty f x 0 when f x 0. The auxiliary parameter , the auxiliary functionH x , and the auxiliary linear operator play important roles within the HAM 12 . As wewill see later, choosing suitable values of the auxiliary parameter will help us to adjust andcontrol the convergence region of the series solution.

Liao 1217 constructs, using q 0, 1 as an embedding parameter, the so-calledzero-order deformation equation as

1 q x ; q u0 x q H x N x ; q , 2.2

where x ; q is the solution of 2.1 which depends on , H x , , u0 x , and q. When q 0,the zero-order deformation equation 2.2 becomes

x ; 0 u0 x , 2.3

and when q 1, since / 0 and H x / 0, the zero-order deformation equation 2.2 reducesto

N x ; 1 0. 2.4

So, x ; 1 is exactly the solution of the nonlinear equation 2.1 . Thus, according to2.3 and 2.4 , as q increases from 0 to 1, the solution x ; q varies continuously from the

initial approximation u0 x to the exact solution u x .Dene the so-called mth-order deformation derivatives

um x 1

m!m x ; q

qmq 0

, 2.5


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expanding x; q in Taylor series with respect to the embedding parameter q, using 2.3 and2.5 , we obtain

x ; q u0 x

m 1um x qm

. 2.6

Assume that the auxiliary parameter , the auxiliary function H x , the initial approx-imation u0 x , and the auxiliary linearoperator are properly chosen so that the power series2.6 of x ; q converges at q 1. Then, we have under these assumptions the series solution

u x u0 x m 1 um x .Dene the vector un {u0 x , u1 x , u2 x , . . . , un x }. Diff erentiating equation 2.2

m-times with respect to embedding parameter q, then setting q 0 and dividing them by m!.Using 2.5 , we have the so-called mth-order deformation equation as

um x mum1 x H x Rm um1 x , m 1, 2, . . . , n , 2.7

where

Rm um1 x1

m 1 !m1N x ; q

qm1 q 0, 2.8

m0, m 1,1, m > 1.

2.9

For any given nonlinear operator , the term Rm um1 x can be easily expressed by 2.8 . Thus, we can gain u0 x , u1 x , u2 x , . . . by means of solving the linear high-orderdeformation equation 2.7 one after the other in order. The mth-order approximation of u xis given by u x m1k 0 uk x .

3. Solving Second-Order Boundary Value Problems of IDEs by HAM

Throughout this section, we will utilize the construction for the IDEs of the Fredholm type inorder not to increase the length of the paper without the loss of generality for the remainingtype. However, similar construction can be implemented for the Volterra type.

Accordingly, we extend the application of the HAM to solve the second-order IDEs of the Fredholm type

N u x : u x b

aK x, s f u s , u s ds g u x , u x h x 0, a s, x b,

3.1

subject to the boundary conditions 1.3 .First of all, we assume that 3.1 satises the initial condition u a , where the

unknown constant can be later determined by substituting the boundary condition u b 2 into the obtained solution. After that, we chose the initial guess u0 such that the initial


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conditions u0 a 1 and u0 a are satised. That is, the initial guess will be of the formu0 u0 x; . Finally, we take the auxiliary linear operator d2/dx 2 which satises theproperty c1 c2t 0, where c1, c2 are constants of integrations.

The so-called zero-order deformation equation will be dened as

1 qd2

dx 2 x ; q, u0 x ; q H x N x ; q, , 3.2

where

N x ; q, d2 x; q,

dx 2

b

aK x, s f s; q, , s; q, ds g x ; q, , x ; q, h x ,

3.3

when q 0, 3.2 reduces to x ; q, u0 x ; . In this case, the so-called mth-orderdeformation equation can be constructed as

d2

dx 2um x ; mum1 x ; H x Rm um1 x ; . 3.4

Operating the operator J 2, the inverse operator of d2/dx 2 to both sides of 3.4 , then, themth-order deformation equation will have the following form:

um x ; mum1 x; xa x H Rm um1 ; d, 3.5where

Rm um1 x; d2um1 x ;

dx 2

b

aK x, s

1m 1 !

m1 f s; q, , s; q, qm1 q 0

ds

1m 1 !

m

1

g x ; q, , x ; q, qm1 q 0

h x 1 m ,

3.6

and the mth-order approximation of u x can be given as

u x m1

k 0uk x ; . 3.7


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If we substitute the boundary condition u b 2 into 3.7 , then we obtain an equa-tion of the variable . And so, the approximate solution for 3.1 and 1.3 will be completelyconstructed. For simplicity, in this paper, we assume that the value of the auxiliary functionis kept at H x 1.

4. Numerical Results and Discussion

HAM provides an analytical approximate solution in terms of an innite power series.However, there is a practical need to evaluate this solution and to obtain numerical valuesfrom the innite power series. The consequent series truncation and the practical procedureare conducted to accomplish this task. In this section, we consider four examples todemonstrate the performance and e fficiency of the present technique. Through this paper allthe symbolic and numerical computations are performed by using Mathematica 7.0 softwarepackage.

To show the accuracy of the present method for our problems, we report three types

of error. The rst one is the residual error, Re, dened as

Re x :d2

dx 2umHAM x

b

aK x, s f umHAM s ,

dds

umHAM s ds

g umHAM x ,d

dxumHAM x h x ,

4.1

for the Fredholm type and similarly for the Volterra type, while the exact, Ex, and relative, Rl,errors are dened, respectively, by

Ex x : uExact x umHAM x ,

Rl x :uExact x umHAM x

|uExact x |,

4.2

where s, x a, b , umHAM is the mth-order approximation of u x obtained by the HAM, and

uExact is the exact solution.

Example 4.1. Consider the following linear Fredholm IDE:

u x 1

1ses cos xu s ds 2u x x 0, 1 s, x 1, 4.3


u 1 1, u 1 1. 4.4


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The exact solution is u x c1e 2x c2e 2x c3 cos x 0.5x , where c1 0.4206 . . ., c2

0.3545 . . ., and c3 0.2662 . . ..According to 3.6 and 2.5 , we have

Rm um1 x ; d2um1 x ;

dx 2

1

1ses cos x um1 s; ds 2um1 x ; x 1 m .

4.5

Assuming the initial approximation of 4.3 has the form u0 x x 1 1, which satisesthe initial conditions u0 1 1 and u0 1 . Consequently, the rst few terms of theHAM series solution for 4.3 according to these initial conditions are as follows:

u1 x 0.31668 0.5311 1.88088x 1.12044x 1 x2

0.16667 0.33333 x3 0.73576 0.14313 cos x ,4.6

u2 x u1 x

0.033333 2 60.81144 6.54379 21.76335 17.39749 x

20.49956 14.06698 x2 23.8088 1.20436 x3 5 1 x4

0.5 x5 81.0036 0.77298 cos x .4.7

To determine the value of , we must chose a value to the auxiliary parameter . Itwas proved that if we set 1, then we have the Adomian decomposition solution whichis a special case of the HAM solution 30, 31 . Therefore, 1 is available value for . Now,substitute 1 and the boundary condition at x 1 into the 10th order approximation of u x to get 1.4818348333.

Figure 1 shows the 10th-order approximate solution of 4.3 and 4.4 obtained byHAM at 1 together with the exact solution. It is clear from the gure that theapproximate solution is in high agreement with the exact solution.

Since the value of has been identied previously, the valid region for the values of can be obtained by the so-called -curve which correspond to the horizontal line as shown inFigure 2 . Thus, the valid region of is 1.5 < < 0.8.

Figure 3 shows a comparison between the exact and residual errors obtained from the10th-order approximation of u x for 4.3 and 4.4 at 1, while Figures 4 and 5 showrespectively how the exact and residual errors change with varying the values of .


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0.5 1

0.5

1

x

u (x )

1 0 .5

0 .5

1

Figure 1: Solutions of 4.3 and 4.4 . Dotted line: the 10th-order approximate HAM solution at 1 andsolid line: the exact solution.

0.2

0.4

0.6

0.8

1

1.2

1.4

u

( 0 )

, u

( 0 )

1 .6 1 .4 1 .2 1 0 .8 0 .6 0 .4

Figure 2: The -curves of u 0 and u 0 which are corresponding to the 10th-order approximate HAMsolution of 4.3 and 4.4 . Dotted line: u 0 and solid line: u 0 .

Example 4.2. Consider the following nonlinear Fredholm IDE:

u x 1

0x s 2eu s ds u x

2 32

x2 53

x712

0, 0 s, x 1, 4.8


u 0 0, u 1 ln 2 . 4.9

The exact solution is u x ln x 1 .


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0.5 1

E r r o r

1 0 .5x

8

6

4

2

10

10 6

Figure 3: Errors of 4.3 and 4.4 at 1. Dotted line: the exact error and solid line: the residual error.

0.5 1

7

6

4

2

1

3

5

1 0 .5

x

E x

( x

)

10 6

Figure 4: Exact error of 4.3 and 4.4 at diff erent values of . Dash-dotted line: 1.1, dotted line: 1, and solid line: 0.9.

5

1 0 .5x

0.5 1

R

e ( x

)

10 6

20

15

10

Figure 5: Residual error of 4.3 and 4.4 at diff erent values of . Dash-dotted line: 1.1, dotted line: 1, and solid line: 0.9.


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According to 3.6 and 2.5 , we have

Rm

um1

x ; d2

dx 2u

m1x ;

3

2x2

5

3x

7

121

m

1

0x s 2

1m 1 !

m1 exp s; q, qm1 q 0

ds

m1

i 0

ddx

u i x; d

dxum1i x; .

4.10

Choose the initial guess approximation as u0 x kx , which satises the initial conditionsu0 0 0 and u0 0 . Then, we have the following approximations terms of the HAMseries solution for 4.8 :

u1 xhx 2

72372 72e 72e 24x 24ex 36e2

24ex 2 6x22 6ex22 213 20x 3 9x23 365 ,4.11

u2 x u1 xh2x2

2160847520 95040e 47520e2 97200 66960e 30240e2

5760x 11520ex 5760e2x 360002 24480e219440e22 10800x 2 6480ex 2 4320e2x 2 360x22360e2x22 720ex22 75603 12240e3 12240e234800x 3 960ex 3 3600e2x 3 540x23 180ex23

360e2x23 3780e4 5580e24 1260x 4 900ex 4

2400e2x 4 300x24 120ex24 360e2x24 151205

16770e5 1620e25 600ex 5 900e2x 5 105x25

75ex25 180e2x25 15600e6 270e26 4320x 6

4540ex 6 180e2x 6 30ex26 45e2x26 7710e74420ex 7 720x27 745ex27 6308 2160e8

600x 8 1800ex 8 270x28 540ex28 72x38

72ex38 540e9 420x 9 360ex 9 300x29

90ex29 108x39 108010 720x 11 .4.12


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0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

u ( x

)

Figure 6: Solutions of 4.8 and 4.9 . Dotted line: the 5th-order approximate HAM solution at 1 andsolid line: the exact solution.

0.2

0.4

0.6

0.8

1

1.2

1.4

1 0 .8 0 .6 0 .4 0 .2 0

u

( 1 )

, u

( 1 )


As in the previous example, we determine an introductory value of the constant bysubstituting 1 and the boundary condition at x 1 into the obtained HAM solution.So, the 5th-order approximation of u x gives 0.9507338904. Figure 6 shows the 5th-orderapproximate solution of 4.8 and 4.9 obtained by HAM at 1 together with the exactsolution, while Figure 7 shows the -curve corresponding to the 5th-order approximate HAMsolution. It is clear from Figure 7 that the valid region of is 0.7 < < 0.4.

Our next goal is to show how the auxiliary parameter aff ects the approximatesolutions. However, it is evident from Figure 6 that both curves corresponding to the 5th-order approximate HAM solution and the exact solution are not coinciding in general at

1. To improve the solution, let us compute other values of at diff erent values of .Table 1 shows the values of the constant at diff erent values of the auxiliary parameter .

Representation the exact solution and the 5th-order approximate HAM solution of 4.8 and 4.9 at diff erent values of mentioned in Table 1 shows that all these solutions

are approximately identical. However, Figure 8 shows the correspondence between the exactsolution and the 5th-order approximate HAM solution of 4.8 and 4.9 at 0.6. Fromthe last mentioned gure, we see that we can achieve a good approximation with the exactsolution by using a few terms in HAM.


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Table 1: The values of at diff erent values of .

0.8 0.9973778496

0.6 0.9987971735

0.4 0.9834454136

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

u ( x

)

Figure 8: Solutions of 4.8 and 4.9 . Dotted line: the 5th-order approximate HAM solution at 0.6and solid line: the exact solution.

Figure 9 shows a comparison between the exact errors of 4.8 and 4.9 at diff erentvalues of corresponding to the 5th-order approximate HAM solution. It is to be noted thatthe accuracy of certain node at 0.6 is more convincing than the other values of .Example 4.3. Consider the following linear Volterra IDE:

u x x

0es sin xu s ds u x

12

ex sin2x sin x 0, 0 s, x 2

, 4.13


u 0 1, u 2

1. 4.14

The exact solution is u x sin x cos x .According to 3.6 and 2.5 , we have

Rm um1 x ; d2

dx 2um1 x ;

x

0sin xe s d

dsum1 s; ds um1 x;

12

ex sin2x sin x 1 m .4.15


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0.2 0.4 0.6 0.8 1

5

x

E x

( x

)

10 6

20

15

10

Figure 9: Exact error of 4.8 and 4.9 at diff erent values of . Dash-dotted line: 0.4, dotted line: 0.6, and solid line: 0.8.

Choose the initial guess approximation as u0 x x 1, which satises the initial conditionsu0 0 1 and u0 0 . Using the iteration formula 3.5 , we can directly obtain thefollowing approximations terms of the HAM series solution for 4.13 :

u1 x 1

150 12 75 120x 75x 75x2 25x 3 75ex cos x

12ex cos2x 150sin x 150 sin x 9ex sin2x ,4.16

u2 x u1 x1

1200 2 2159856

4225300ex

5376x65 552x

2 160x3 50x4 2349

2

15e2x

300ex 1080x 300x

2 100x

3 10x

5

350

e2x 375 8ex 25 177 cos2x465169

e2x cos3x 750sin x

3e2x sin x 600ex sin x 840 sin x1800ex sin x600exx sin x

3e2x cos x 3 20ex 12 10x 25 20x 5x2 5 sin x4068

25ex sin2x 84ex sin2x

405169

e2x sin3x .

4.17

The rest of components of the iteration formula 3.5 can be obtained in a similar way.To nd the value of , substitute 1 and the boundary condition at x / 2 into

the 5th-order approximation of u x to get 1.0000000668. After that, substituting the valueof in the 5th-order approximation of u x and plotting the -curve, we can obtain the validregion of which is 1.3 < < 0.7 as shown in Figure 10 .

The graph of the 5th-order approximate HAM solution of 4.13 and 4.14 at 1together with the exact solution are depicted in Figure 11 . It is evident from the gure that the5th-order approximate HAM solution agrees with the exact solution in the interval 0, / 2 .


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0.5

0.6

0.7

0.8

0.9

1

1.1

2 1 .5 1 0 .5

u

( 0 )

, u

(

0 )


0.5 1 1.5

0.5

1

1

0 .5

x u

( x

)

Figure 11: Solutions of 4.13 and 4.14 . Dotted line: the 5th-order approximate HAM solution at

1

and solid line: the exact solution.

In Table 2 , the exact, relative, and residual errors have been calculated for various xin 0, / 2 to measure the extent of agreement between the 5th-order approximate HAMsolution at 1 and the exact solution. From the table, it can be seen that the HAM providesus with the accurate approximate solution of 4.13 and 4.14 .

Example 4.4. Consider the following nonlinear Volterra IDE:

u x

x

0

u s 2dsx2 sinh x

14

sinh2 x 0, 0 s, x 1, 4.18


u 0 0, u 1 sinh 1 . 4.19

The exact solution is u x sinh x.


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Table 2: Exact, relative, and residual errors of 4.13 and 4.14 at 1 and 1.0000000668.x Exact error Relative error Residual Error0.1 4.9737841010 5.5562411010 1.112207 109

0.2 2.4261731011

3.1049171011

5.782878109

0.3 3.2022721012 4.8532781012 5.4409011070.4 1.2449611011 2.3417251011 2.5720901060.5 1.0305321010 2.5882551010 8.4200421060.6 5.4993661010 2.109517109 2.2022341050.7 2.259143109 1.872873108 4.9648771050.8 7.674839109 3.716740107 1.0088861040.9 2.263322108 1.399558107 1.9002791041.0 5.985469108 1.987414107 3.378840104... ... ... .../ 2 4.370067106 0.370067106 4.598268103

According to 3.6 and 2.5 , we have

Rm um1 x ; d2

dx 2um1 x ;

x2 sinh x

14

sinh2 x 1 m

x

0

m1

j 0u j s; um1j s;

ds.

4.20

Assume that the initial approximation of 4.18 has the form u0 x x , which satises the

initial conditions u0 0 0 and u0 0 . Using the iteration formula 3.5 , we can directlyobtain the following approximations terms of the HAM series solution for 4.18 subject tothese initial conditions:

u1 x1

120 x 135 10x2 2x42 15 8 cosh x sinh x , 4.21

u2 x u1 x1

3360 2 3780x 280x3 13545x 126x5

83

x7

56x5229

x93 6720x cosh x 1052

x cosh2 x 3360sinh x

20160 sinh x 210sinh2 x315

4 sinh2 x .4.22

Similarly to the previous discussion, one can substitute 1 and the boundary condition atx 1 into the 5th-order approximation of u x to get 1.0000000063. Figure 12 shows theagreement between the 5th-order approximate HAM solution of 4.18 and 4.19 at 1and the exact solution.


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0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

x

u ( x

)

Figure 12: Solutions of 4.18 and 4.19 . Dotted line: the 5th-order approximate HAM solution at 1and solid line: the exact solution.

0.5

1

1.5

2 1 .5 0 .5 0

u

( 1 )

, u

( 1 )

1


Since the value of has been identied previously, the valid region for the values of can be obtained by the so-called -curve which correspond to the horizontal line as shown inFigure 13 . Thus, it is clear from the gure that the valid region of is 1.2 < < 0.7.

The detailed data of u x for 4.18 and 4.19 that includes the exact nodal values, the5th-order approximate HAM solution nodal values, and the exact error are given in Table 3 .It is clear that the accuracy obtained using HAM is in advanced by using only ve termsapproximations.

5. Conclusion

The main concern of this work has been to propose an e fficient algorithm for the solution of second-order IDEs with boundary conditions of the Fredholm and Volterra types. The goalhas been achieved by extending the HAM to solve this class of boundary value problems.We can conclude that the HAM is a powerful and e fficient technique in nding approximatesolutions for linear and nonlinear boundary value problems of second-order IDEs of di ff erenttypes. The proposed algorithm produced a rapidly convergent series by choosing suitable


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Table 3: Numerical results of 4.18 and 4.19 at 1 and 1.0000000063.x Exact solution HAM solution Exact error0.1 0.1001667500 0.1001667500 5.7604641014

0.2 0.2013360025 0.2013360025 2.119789 1013

0.3 0.3045202934 0.3045202934 5.76793810130.4 0.4107523258 0.4107523258 8.99766410140.5 0.5210953055 0.5210953055 2.40155110130.6 0.6366535821 0.6366535821 1.91963110120.7 0.7585837018 0.7585837018 3.09993110110.8 0.8881059822 0.8881059820 2.17412010100.9 1.0265167257 1.0265167244 1.2825351091.0 1.1752011936 1.1752011876 6.091291109

values of the auxiliary parameter . After computing several approximants and using the boundary conditions at the boundary points, we can easily determine the approximatesolution.

There are two important points to make here. First, HAM provides us with a simpleway to adjust and control the convergence region of the series solution by introducing theauxiliary parameter . Second, the results obtained by HAM are very e ff ective and conve-nient in linear and nonlinear cases with less computational work. This conrms our belief that the e fficiency of our technique gives it much wider applicability for general classes of linear and nonlinear problems.

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Homotopy Analysis Method for Second-Order Boundary Value Problems of Integro-differential Equations