International Journal of Scientific & Engineering Research Volume 2, Issue 9, September2011 1
ISSN 22295518
Comparison of Variational Iteration
Decomposition Method with Optimal Homotopy Asymptotic of Higher Order Boundary Value Problems
Mukesh Grover, Dr. Arun Kumar Tomer
Asymptotic method, Ordinary Differential Equations, Error Estimates.
—————————— ——————————
In literature different techniques are available for the numerical solution of twelfth order boundary value problems.
In this Paper, we consider the general 12th order boundary value
problems of the type: y12(x) +f(x) y(x) =g(x), x [a, b] (1)
With boundary conditions:
y(a) =a1 y(b) = b1
analytical solutions to many linear and nonlinear problems arising in engineering and science, such as nonlinear oscillators with discontinuities [3], nonlinear Volterra –Fredholm integral equations [11], Twelfth order differential equations have several important applications in engineering. Solution of linear and nonlinear boundary value problems of twelfthorder was implemented by Wazwaz using Adomian decomposition method. Chandrasekhar [9] 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
y(1)(a) = a
y(2)(a) = a y(3)(a) = a y(4)(a) = a y(5)(a) = a
y(1) (b) = b
y(2) (b) = b y(3) (b) = b y(4)( ) = b y(5)(b) = b
same direction as gravity, instability will occur. Several researchers developed numerical techniques for solving twelfth order differential equations. The Adomian Decomposition Method [1, 4], the Differential Transform Method [15], the Variational Iteration Method, the successive iteration, the splines [5, 6], the Homotopy Perturbation Method [7], the Homotopy Analysis Method etc Recently Vasile Marinca et al. [10,12,14] introduced OHAM for approximate solution of nonlinear problems of thin film flow of a fourth grade fluid down a vertical cylinder. OHAM is straight forward, reliable and it does not need
Where a i , b j, here i , j =1,2,3,4,5,6 are finite real constants and
the functions f(x) and g(x) are continuous on [a, b]. The
motivation of this problem is to extend Optimal Homotopy Asymptotic Method to solve linear and nonlinear twelfth order boundary value problems. We also compared the results obtained from these techniques with the available exact solution in different literatures. Some properties of solutions of a given differential equation may be determined without finding their exact form in especially in nonlinear behavior. If as self contained formula for the solution is not available, the solution may be numerically approximated using computers. To overcome these difficulties, a modified form of the variational method called Variational Iteration Decomposition Method. VIDM has since then been effectively utilized in obtaining approximate
to look for h curves like VIDM. Moreover, this method provides a convenient way to control the convergence of the series solution. Most recently, Javed Ali et al. used OHAM for the solutions of multipoint boundary value problems. The results of OHAM presented in this work are compared with those of exact solution VIDM.
To illustrate the basis concept of the technique, we consider the following general differential equation
L1u + Nu =g(x) (2) Where L is a linear operator, N a nonlinear operator and g(x) is
IJSER © 2011 http://www.ijser.org
International Journal of Scientific & Engineering Research Volume 2, Issue 9, September2011 2
ISSN 22295518
the in homogenous term. According to variational iteration method, we can construct a correct functional as follows
un+1(x) = un(x) + (L un (s) + N ) (3)
where is a Lagrange multiplier [14 to 18],which can be identified optimally via variational iteration method. The subscripts n denote the nth approximation, is considered as a restricted variation. i.e. =0. The relation (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.
Now we recall basic principles of the Adomian decomposition method [10, 12] for solving differential equations. Consider the general equation T u = g, where T represents a general nonlinear differential operator involving both linear and nonlinear terms. The linear term is decomposed into L + R where L is easily invertible and R is the reminder of the linear operator. For convenience, L may be taken as the highest order derivation. Thus the equation may be written as
Lu + R (u) + Nu = g(x) (4) where Nu represents the nonlinear terms. From (3) we have Lu = g  R (u) – Nu (5)
Since L is invertible the equivalent expression is
u =L1 g –L1 R (u) –L1 N (u) (6)
A solution u can be expressed as following series
The Bm’s are given as, There appears to be no welldefined method for constructing a definitive set of polynomials for arbitrary F, but rather slightly different approaches are used for different specific functions. One possible set of polynomials is given by
B0 = F(u0) , B1 = (xx1)
B2 =(xx2) + B3= (xx3)
+(xx1)(xx2) +
……………………………………………
can be used to construct Adomian polynomials, when F(u0) is a nonlinear function Put the value of equation (8) and (9) in equation (3), we get
Consequently, with a suitable u0 we can write, Put one by one
j=1, 2, 3, …….. in above expression.
with reasonable u0 which may be identified with respect to the definition of L−1 , g and u , n>0 is to be determined. The
u (x) =  L1
u (x) =  L1
R (u ) – L1 B R (u ) – L1 B
nonlinear term Nu will be decomposed by the infinite series of
Adomian polynomials
……………
1 1
un+1(x) =  L
R (un) – L Bn
where Bn’s are obtained by writing
Here λ is a parameter introduced for convenience. From (9) and
(10) we have
Here
To illustrate the basis concept of the Variational iteration decomposition method, we consider the following general differential equation (1). According to variational iteration method [11], we can construct a correct functional (2), we define the solution by the series
and the nonlinear term
Where Bn are the Adomian polynomials and can be generated for all type of nonlinearities according to the algorithm developed in [13] which yields the following
IJSER © 2011 http://www.ijser.org
International Journal of Scientific & Engineering Research Volume 2, Issue 9, September2011 3
ISSN 22295518
y : [1λ(11)
(s) ]s=x =0 .
Hence, we obtain the following iterative scheme
The method is called Variational iterative decomposition method.
y12(x) = x y(x)x3ex 23xex 120ex (14)
with following conditions:
The Lagrange multiplier, therefore, can identify as follows:
λ
Making the correct functional stationary, using =
x)11 Lagrange multiplier [16 to 20], Substituting the identified
multiplier into above Eq. we have the following iteration formula:
yn+1(x) = x+
Exact solution is y (x) = x (1 x) ex
The correct functional for the boundary value problem is given as
Where
A= , B = , C= ,
To find the optimal s , calculation variation with respect to y
n, we have the following stationary conditions:
(m)
D= , E= , F=
Using the Variational iterative decomposition method, we get, yn+1(x) = x+
yn : λ
y(m1)
(s) =0 ,
Where B
are Adomian polynomials for nonlinear operator N(y)
y(m2)
n : [λ(s)]s=x =0 ,
n : [λ’(s)]s=x =0 ,
.
.
.
.
m
= x y(x) and can be generated for all type of nonlinearities
according to the algorithm which yields the following
B0 = xy0(x),
B1=y1 ,
y : [1λ(m1)
(s) ]s=x =0 (15)
B2 =
But in above Eq. (14) value of m is 12, put all these values of m in Eq.(15) and get follows:
(12)
…………… (16)
yn : λ
y(11)
(s) =0 ,
From the above relation, we find y0(x), y1(x),……..and get the series solution as follow:
n : [λ(s)]s=x =0 ,
n : [λ’(s)]s=x =0 ,
.
.
.
.
y(x) = x    + + +
+ + +

IJSER © 2011 http://www.ijser.org
International Journal of Scientific & Engineering Research Volume 2, Issue 9, September2011 4
ISSN 22295518
+…………………
The coefficients A, B, C, D, E, F, G can be obtained using the boundary conditions at x =1,
A = 23.9999985, B =35.000057, C= 47.998961,
D =63.0108031, E =79.9359481, F= 99.17376631.
The series solution can, thus, be written as
y12(x) = x y(x)x3ex 23xex 120ex
with following conditions:
Exact solution is y (x) = x (1 x) ex
We construct the following zeroth and firstorder problems.
y12(x) =x    + 
(x) = x y(x)x3ex
23xex
120ex
+…………
 0.000173641 –
+
with following conditions
FirstOrder Problem
(x) = (1+ C1) (120 + 23x + x ) e
+ C x y (x) + (1 + C ) (x)
1 0 1
With same above boundaries’ conditions Solutions to these problems are given by Equations. (20) and (21) respectively
y0(x) = (280800 +280800 e
221760x 58920 exx
2 x 2
3 x 3 4
85800x +4320e x
21600x 120 e x 3960x 
5 6 6
560x +197720040x +72737161ex
836896800x
+307877125ex7+ 1451896600ex8
1282301040x9+
9 10
10 11
471732190ex + 5741817x
11
211229665ex
103986080x
+38254341ex )
(20)
y1 (x, C1) = C1
(216060216060ex+ (176352 39708 ex) x+ (71045
2723 ex) x2+ (18795+ 84 ex) x3 + (3663  ex) x4+ 558.833x5+
69.1417x6+7.07738x7+0.603671x8+0.0425263+0.00237265x10+0.
0000892391x113.75782× 107x 13 4.23959 × 108x 14 
3.28063109x 15 2.06473 1010x 161.11334 × 1011x 17 5.24805 ×
1012x 18 2.17518 × 1014x 197.89181 × 1016x 20 2.46619× 1017x
21  6.40591 ×1019x 22 1.27589 ×1020x 23 1.55619 1022x 24)
(21)
Considering the OHAM firstorder solution, Y app (x, C1) = y0(x) + y1 (x, C1)
(22)
and using Eq.(18) with a = 0.5 and b = 1, we get C1 = 
0.00260417. Using this value the firstorder solution (22) is well
determined.
IJSER © 2011
International Journal of Scientific & Engineering Research Volume 2, Issue 9, September2011 5
ISSN 22295518
In this paper, the Comparison of the results obtained by the Homotopy perturbation method and optimal homotopy asymptotic method of Twelfth order boundary value problems. The numerical results in the Tables [1.11.2], show that the optimal homotopy asymptotic method provides highly accurate numerical results as compared to Homotopy perturbation method. It can be concluded that optimal homotopy asymptotic method is a highly efficient method for solving 12th order boundary value problems arising in various fields of engineering and science.
[1] Adomian, G., 1991. A review of the decomposition method and some recent results for Nonlinear equations. Comput. Math. Appl., 101127.
[2] He, J.H. (2004). The Homotopy Perturbation Method for Non Linear Oscillators with Discontinuities. Applied Maths. Computer. 287 –292.
[3] He, J.H., 2003. Variational approach to the sixth order boundary value problems. Applied Math. Computer, 537538.
[4] Wazwaz, A.M., 2000. Approximate solutions to boundary value problems of higher order By the modified decompositionmethod.Computer.Math.Appl.679691
[5] Siddique, S.S. and E.H. Twizell, 1997. Spline solutions of linear twelfthorder boundary Value Problems Computer Math. Appl., 371390.
[6] Siddique, S.S. and Ghazala Akram, 2008. Solutions of 12th order boundary value problems Using nonpolynomial spline technique. Appl. Math.Comput.,559571.
[7] He, J.H., 2006. Homotopy perturbation methodfor solving boundary value problems.8788.
[8] Ghotbi, Abdoul R., Barari, A. and Ganji, D. D, (2008). Solving RatioDependent Predator Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method. Math. Prob.Eng., 18.
[9] Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, Reprinted: Dover Books, New York, 1981.
[10] Marinca, V., N. Herisanu and I. Nemes, 2008.Optimal homotopy asymptotic method with Application to thin film flow. Cent. Eur. J. Phys. 648653.
[11] Ghasemi, M., Tavassoli Kajani,M. and Babolian, E. (2007). Numerical Solution of the Nonlinear Volterra–Fredholm Integral Equations by using Homotopy Perturbation Method. Appl. Math. Computer 446449.
[12] Marinca, V. and N. Herisanu, 2008. An optimal homotopy asymptotic method for solving Nonlinear equations arising in
heat transfer. Int. Comm. Heat and Mass Tran. 710715.
[13] S. H. Mirmoradi, S. Ghanbarpour, I. Hosseinpour, A. Barari, (2009). Application of Homotopy perturbation method and variational iteration method to a nonlinear fourth order Boundary value problem. Int. J.Math.Analysis., 11111119.
[14] Marinca, V., N. Herisanu, C. Bota and B. Marinca,2009. An optimal homotopy asymptoticMethod applied to the steady flow of fourthgrade fluid pasta porous plate. App.Math.24551
[15] Islam, S.U., S. Haq and J. Ali, 2009. Numerical solution of special 12thorder boundaryValue Problems using differential transform method. Comm. Nonl. Sc. Nu. Sim.11321138.
[16] J.H. He. Variational iteration method, a kind of nonlinear analytical techniques, some Examples inter. j. Nonlinear Mech., (1999):699708
[17] J.H. He. Variational iteration method, new development and applications, Computer math.Appl.54 (2007):881894
[18] J.H. He. Variational iteration method, some resent results and new interpretations. J. Computer math.Appl.54 (2007):881
894
[19] J.H. He. Variational iteration method, a kind of nonlinear analytical techniques, some Examples inter. j. Nonlinear Mech., (1999):699708
[20] M.A. Noor , S.T. MohyudDin: a new approach to solving fifth order Boundary value Problems. , inter. j. Nonlinear Science,(2009):143148
Mukesh Grover is currently pursuing Ph.D degree program from Dravidian University, Kuppam, A.P. 517425. I am assistant Professor in Mathematics in Department of Applied Sciences. Giani Zail Singh College of Engg. And Technology, Bathinda, India. My current research interest is in Numerical Analysis.
Email: grover.mukesh@yahoo.com
IJSER © 2011 http://www.ijser.org