Abstract: In this study, He's variational iteration method (VIM) is applied to singularly perturbed Volterra integral equations (VIEs). To show the efficiency of the ...
World Applied Sciences Journal 22 (11): 1657-1661, 2013 ISSN 1818-4952 © IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.22.11.714
He's Variational Iteration Method for Solving the Singularly Perturbed Volterra Integral Equations 1
1
Nurettin Do an, 2Vedat Suat Ertürk and 3Shaher Momani
Gazi University, Technology Faculty, Computer Engineering Department, 06500 Teknikokullar, Ankara, Turkey 2 Department of Mathematics, Faculty of Arts and Sciences, Ondokuz May s University, 55139, Samsun, Turkey 3 Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan
Abstract: In this study, He's variational iteration method (VIM) is applied to singularly perturbed Volterra integral equations (VIEs). To show the efficiency of the method, some singularly perturbed Volterra integral equations are solved as numerical examples. Numerical results show that the variational iteration method is very effective and convenient for solving a large number of singularly perturbed problems with high accuracy. Key words: Singularly perturbed Volterra integro-differential equations Singularly perturbed problems Variational iteration method INTRODUCTION In this paper we consider the singularly perturbed Volterra integral equations(VIEs) [1-4]. x
∫
y ( x= ) g ( x) + K ( x, t , y (t )), 0 ≤ t ≤ X
(1.1)
0
where is a small parameter satisfying 0 < 0 are when the properties of the solution with incompatible with those when = 0. For > 0, (1.1) is an integral equation of the second kind which typically is well posed whenever K is sufficiently well behaved. When = 0, (1.1) reduced to an integral equation of the first kind whose solution may well be incompatible with the case for > 0. The interest here is in those problems which do imply such an incompatibility in the behavior of y near x=0. This suggests the existence of boundary layer near the origin where the solution undergoes a rapid transition [1-4].
Volterra integral equations Approximate solutions
Analytical methods commonly used to solve linear and nonlinear equations are very restricted and numerical techniques involving discretization of the variables on the other hand gives rise to rounding off errors. The aim of our study is to introduce He’s variational iteration method [6-11] as an alternative to existing methods in solving the singularly perturbed Volterra integral equations (VIEs). According to the authors knowledge this paper represents the first application of the variational iteration method to the singularly perturbed Volterra integral equations (VIEs). Recently introduced variational iteration method by He [6-9], which gives rapidly convergent successive approximations of the exact solution if such a solution exists, has proven successful in deriving analytical solutions of linear and nonlinear differential equations. Mohyud-Din [13-15] used this method with He’s polynomials for solving nonlinear problems. This method is preferable over numerical methods as it is free from rounding off errors and neither requires large computer power/memory. To illustrate the basic idea of the method, we consider the following general differential equation:
Corresponding Author: N. Do an, Gazi University, Technology Faculty, Computer Engineering Department, 06500 Teknikokullar, Ankara, Turkey. Tel: +903122028572; Fax: +903122120059.
1657
World Appl. Sci. J., 22 (11): 1657-1661, 2013
(
Lu + Nu = g(x)
y ( x ) = x + 1 − exp − x
where L is a linear operator, N is a non-linear operator and g(x) is an inhomogeneous term. According to the variational iteration method, we can write a following correct functional as: yn +1 ( x ) =yn ( x ) +
x
∫
Using He’s variational iteration method, we obtain the following iteration formula
y0(x) = 0. By the iteration formula and the initial approximation, we have the following first iteration solution:
[7, 8, 12] has applied this method for
y1 ( x= ) y1= ( x)
0
which has the exact solution
x
∫ 1 + t − y0 ( t ) dt
1 x2 x+
We have the following iterations as follows y2 ( x ) =
Example 1: Let us first consider the following singularly perturbed Volterra Integral Equation [2, 16]:
∫ 1 + t − y (t ) dt
1
0
Applications and Numerical Results: In this section, to demonstrate the effectiveness of the proposed algorithm, the VIM is applied to the singularly perturbed Volterra integral equations (VIEs).
x
∫ 1 + t − yn ( t ) dt
According to the variational iteration method, we assume an initial approximation in the form
obtaining analytical solutions of integro differential equations. The paper has been organized as follows. Section 2 gives illustrative examples of VIM. The discussion of our results is given in Section 3.
y ( x) =
x
0
{Lyn ( t ) + Nyn ( t ) − g ( t )} dt
Here is a general Lagrangian multiplier, which can be identified optimally via the variational theory. The second term on the right is called the correction and y n is considered as a restricted variation, i.e., He
1
yn +1 ( x= )
0
y n = 0.
) − (1 − exp ( − x ) )
2 x3 x ( −1 + 1 x− + 6 2
x3 (1 − x4 1 y3 ( x ) = x + + 24 2 6 2
)
) + x 2 ( −1 + ) 2
and so on, in this manner the rest of the iterations can be obtained. The 18th iteration is obtained as
x18 ( −1 + ) x17 ( − 1) x19 x− + + + 17 17 16 1216451008832000 6402373705728000 355687428096000 16 15 14 13 x ( −1 + ) x (1 − ) x ( −1 + ) x (1 − ) + + + + 15 14 13 12 20922789888000 1307674368000 87178291200 6227020800 1 . y18 ( x ) = x12 −1 + 10 9 8 7 11 x − 1 1 1 1 1 x x x x − + − − + − ( ) + ( ) + ( )+ ( ) + ( ) + ( ) + 479001600 11 39916800 10 3628800 9 362880 8 40320 7 5040 6 x 6 ( −1 + ) x5 (1 − ) x 4 ( −1 + ) x3 (1 − ) x 2 ( −1 + ) + + + + 5 4 3 2 2 120 24 6 720
Comparison of numercial results for 18th iteration with the exact solution for = 1, = 0.75, = 0.5 and = 0.25 are shown in Table 1. From the numerical results in Table 1, it is clear that the approximate solutions are in high agreement with the exact solutions and the solutions continuously depend on the parameter . It is evident that the efficiency of this approach can be dramatically enhanced by computing further terms or further components of y(x). 1658
World Appl. Sci. J., 22 (11): 1657-1661, 2013 Table 1: Numerical results compared to exact solution for Example 1
x
= 1.0 --------------------------y(x)Approx y(x)Exact
= 0.75 -------------------------------------y(x)Approx y(x)Exact
= 0.5 -------------------------------------y(x)Approx y(x)Exact
= 0.25 --------------------------------------y(x)Approx y(x)Exact
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.000000 0.131207 0.258518 0.382420 0.503338 0.621646 0.737668 0.851690 0.963962 1.074700 1.184100
0.000000 0.190635 0.364840 0.525594 0.675336 0.816060 0.949403 1.076700 1.199050 1.317350 1.432330
0.000000 0.347260 0.613003 0.824104 0.998578 1.148500 1.281960 1.404390 1.519430 1.629510 1.736260
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.000000 0.131207 0.258518 0.382420 0.503338 0.621646 0.737668 0.851690 0.963962 1.074700 1.184100
0.000000 0.190635 0.364840 0.525594 0.675336 0.816060 0.949403 1.076700 1.199050 1.317350 1.432330
0.000000 0.347260 0.613003 0.824104 0.998578 1.148500 1.281960 1.404390 1.519430 1.629510 1.736260
Example 2: Now we consider the following singularly perturbed Volterra Integral Equation [2, 16, 17]: x
∫ (1 + x − t ) 1 + t − y ( t ) dt
y ( x= )
0
which has the exact solution y ( x) = x + 1 +
1 − 1
2
Where the parameters = 1
1 −1 + 1 − 4 2
(
),
2
1
1 − 1 + exp ( 1x ) −
and
= 1
2
1 −1+
1 exp (
2x
)
are defined as
1 −1 − 1 − 4 2
(
).
Using He’s variational iteration method, we obtain the following iteration formula 1
yn +1 (= x)
x
∫ (1 + x − t ) 1 + t − yn ( t ) dt, 0
According to the variational iteration method, we assume an initial approximation in the form y0(x) = 0. By the iteration formula and the initial approximation, we have the following first few terms of the iteration solution: y1 ( = x)
1
x
dt ∫ (1 + x − t ) 1 + t − y0 ( t ) = 0
1 x3 x + x2 + , 6
y2 ( x ) =
1 x 3 x 2 x3 x 4 − − x + x2 + − , 6 2 3 24
y3 ( = x)
x3 ( −1 + )(1 + x + 2 1 x5 x (1 + x ) − − 30 2 6 2
) + x 4 ( −2 + x + 3 ) + x 2 ( −1 − x + x ) 24
2
1659
2
World Appl. Sci. J., 22 (11): 1657-1661, 2013 Table 2: Numerical results compared to exact solution for Example 2
x
= 1.0 -------------------------------------------y(x)Approx y(x)Exact
= 0.75 ------------------------------------------y(x)Approx y(x)Exact
= 0.5 --------------------------------------------y(x)Approx y(x)Exact
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.104829 0.218600 0.340519 0.468267 0.601563 0.738600 0.877829 1.017600 1.156162 1.291667
0.0 0.137400 0.281363 0.429400 0.578844 0.726852 0.870400 1.006289 1.131141 1.241400 1.333333
0.0 0.198983 0.391733 0.571650 0.731733 0.864583 0.962400 1.016983 1.019733 0.961650 0.833333
0.0 0.104833 0.218669 0.340519 0.469413 0.604405 0.744584 0.889072 1.037037 1.187692 1.340300
0.0 0.137510 0.282340 0.433015 0.588175 0.746572 0.907074 1.068664 1.230437 1.391602 1.551470
and so on, in this manner the rest of the iterations can be obtained. Comparison of numercial results for 2 th iteration with the exact solution for = 1, = 0.75 and = 0.5, are shown in Table 2.
y2 ( x ) =
Example 3: Now we consider the following singularly perturbed Volterra Integral Equation [2, 16]: x
∫e
y ( x) =
x −t
0
y 2 ( t ) − 1 dt ,
(3.1)
which has the exact solution 2(1 − e
y= ( x)
( − 1)e
x
x
)
= , + +1
1
4+
2
.
1
yn +1 ( x ) =
∫e
x −t
0
According to the variational iteration method, we assume an initial approximation in the form
y -0.2
x −t
0
=
1− e
x
0.8
1
x
-1 Fig. 1: Plots of Eq.(3.1) when = 1. Exact solution(___); VIM solution (---). y
-0.4
∫e
0.6
-0.8
By the iteration formula and the initial approximation, we have the following first iteration solution: x
0.4
-0.6
-0.2
1
0.2
-0.4
y0(x) = 0
y1 ( x ) =
2
and so on, in this manner the rest of the iterations can be obtained. The evolution results for the exact solution (3.2) and th 4 iteration solution obtained using variational iteration method, for different values of , are shown in Figs. 1-3.
(3.2)
yn 2 ( t ) − 1 dt.
2 − 2e x + 2 x + x 2
. . .
Using He’s variational iteration method, we obtain the following iteration formula x
1 − ex −
0.0 0.199683 0.397589 0.592269 0.782594 0.967719 1.147046 1.320191 1.486949 1.647272 1.801234
0.2
0.4
0.6
0.8
1
x
-0.6 -0.8
y 2 t − 1 dt 0 ( )
-1
= 0.75. Exact Fig. 2: Plots of Eq.(3.1) when solution(___); VIM solution (---) 1660
World Appl. Sci. J., 22 (11): 1657-1661, 2013 y 0.2
0.4
0.6
0.8
1
7.
x
-0.2
8.
-0.4 -0.6
9.
-0.8 -1
Fig. 3: Plots of Eq.(3.1) when = 0.5. Exact solution(___); VIM solution (---)
10.
CONCLUSION In this study, He's variational iteration method is successfully applied to singularly perturbed Volterra integral equations. A symbolic calculation software package, MATHEMATICA is used for all calculations. The results are presented in comparison with the exact solutions. It is observed that the method is applicable to this type of integral equations. The method gives rapidly converging series solutions. The results obtained reveal out that VIM is a quite, powerful solution technique.
11.
12.
13.
REFERENCES 1.
2.
3.
4.
5.
6.
Alnasr, M.H., 1997. Numerical treatment of singularly perturbed Volterra integral equations, Ph.D. Thesis, Ain Shams University, Egypt. Alnasr, M.H., 2000. Modified Multilag Methods For Singularly Perturbed Volterra Integral Equations, Int. J.Comput. Math., 75(1): 221-233. Lange, C.G. and D.R. Smith, 1988. Singular perturbation analysis of integral equations, Stud. Appl. Math., 79(11): 1-63. Angell, J.S. and W.E. Olmstead, 1987. Singularly perturbed Volterra integral equations SIAM J. Appl. Math., 47(1): 1-14. Brunner, H. and P.J. Van der Houwen, 1986. The Numerical Solution of Volterra Equations, CWI. Monographs, vol. 3, North-Holland Publishing, Amsterdam. He, J.H., 2007. Variational iteration method -Some recent results and new interpretations, J. Comput. Appl. Math., 207(1): 3-17.
14.
15.
16.
17.
1661
He, J.H., 2004. Variational principles for some nonlinear partial differential equations with variable coefficients, CHAOS SOLITON FRACT, 19(4): 847-851. He, J.H., 2000. Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114: 115-123. He, J.H., 2006. Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20(10): 1141-1199. Do an, N., M. Al-Quran, V.S. Ertürk and S. Momani, 2010. Variational Iteration Method For Solving Singularly Perturbed Two-Point Boundary Value Problems, International Journal of Pure and Applied Mathematics, 58(1): 11-19. Al-Quran, M. and N. Do an, 2010. Variational Iteration Method For Solving Two-Parameter Singularly Perturbed Two Point Boundary Value Problem, Applications and Applied Mathematics: An International Journal (AAM), 5(1): 81-95. Wang, S.Q. and J.H. He, 2007. Variational iteration method for solving integro-differential equations, PHYS LETT A, 367(3): 188-191. Mohyud-Din, S.T., 2009. Solving heat and wave like equations using He’s polynomials, Mathematical Problems in Engineering (2009); Article ID 427516, IF = 0.545. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, 2009. Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering (2009); Article ID 234849, 25 pages, doi:10.1155/2009/234849, IF = 0.545. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, 2009. Solving second-order singular problems using He’s polynomials, World Applied Sciences Journal, 6(6): 769-775. Alnasr, M.H. and S. Momani, 2008. Application of Homotopy Perturbation Method To Singularly Perturbed Volterra Integral Equations, JAS, 8(6): 1073-1078. Ibrahim, M.A.K. and M.H. Alnasr, 1998. Quadrature methods with regularization technique for singularly perturbed Volterra integral equations, Int. J.Comput. Math., 68(3-4): 285-299.
