DOI : 10.17577/IJERTV13IS060089
- Open Access
- Authors : Danish Ul Islam, Dr. Anjali Shirivastava
- Paper ID : IJERTV13IS060089
- Volume & Issue : Volume 13, Issue 06 (June 2024)
- Published (First Online): 21-06-2024
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Solving Fredholm Integral Equations by Numerical Methods Adomian Decomposition Method and Variational Iteration Method
Danish Ul Islam
Research Scholar, Department of Mathematics, Vikram University, Ujjain 405010, Madhya Pradesh, India
Dr. Anjali Shirivastava
Assistant Professor, Department of Mathematics, Vikram University, Ujjain 405010, Madhya Pradesh, India
ABSTRACT
In this research paper, numerical methods Adomian Decomposition Method and Variational Iteration Method is used to solve integral equations .Numerical examples illustrate the accuracy of the Adomian Decomposition Method and Variational Iteration Method.
KEYWORDS: Integral Equations, Fredholm integral equation, Adomian Decomposition Method, Variational Iteration Method
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ADOMIAN DECOMPOSITION METHOD
The Adomian Decomposition method (ADM) is very powerful method which considers the approximate solution of a non-linear equation as an infinite series which actually converges to the exact solution in this paper, ADM is proposed to solve some first order, second order and third order differential equations and integral equations . The Adomian Decomposition method (ADM) was firstly introduced by George Adomian in 1981. This method has been applied to solve differential equations and integral equations of linear and non-linear problem in Mathematics, Physics, Biology and Chemistry up to know a large number of research paper have been published to show the feasibility of the decomposition method.
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Adomian Decomposition Method for Solving Fredholm Integral Equations
The Adomian decomposition method includes decomposing the unknown function () of any equation into a sum of an infinite number of constituents well-defined by the decomposition series
=0
Equivalently
() =
()
() = 0() + 1() + 2() + 3() +
When the components (), 0 will be resolved. The Adomian decomposition method concerns itself with discover the constituents
0(), 1(), 2()
From above we can found recurrence relation as
() = () + (, )( ())
=0
=0
Or equivalently
0() + 1() + 2() + = () + (, )[0() + 1 () + ]
The zeroth component 0() is acknowledged by all terms that are not comprised under the integral sign. This means that the constituents (), 0 of the unknown function ()are totally resolved by the recurrence relation
Equivalently
0() = (), +1() = (, ) (), 0
0() = (),
1() = (, )0 (),
2() = (, )1 (),
3() = (, )2 (),
and so on are the other constituents. Thus, the constituents 0 (), 1(), 2(), . are resolved totally.
-
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VARIATIONAL ITERATION METHOD
Variational iteration method is a strong and power device for solving various kinds of linear and non-linear functional equations. It was presented by Ji-Huan He in1998 and has been used by many mathematicians and engineers to resolve many kinds of functional equations, for instance wave equation, hyperbolic differential equations, Telegraph equation, non-linear chemistry problems, Cauchy reaction diffusion problem, and several other equations. Variational iteration method (VIM) is used to resolve integral equations. By this method we can resolve many non-linear equations. The following general non-linear system is measured to demonstrate the basic idea of this method
[()] + [()] = ()is a linear operator, is a non-linear operator and () is a continuous function. The elementary character of the method is to shape a correction functional for the system, which states
+1() = () + ()[() + () ()]
0
Langragian multiplier which can be acknowledged optimally via Varaitional theory, is the nth approximate solution and signifies the limited difference
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NUMERICAL EXAMPLES
Example 1. Consider the Fredholm integral equation given as
0
() = + 1(2 + 2)()
BY ADOMIAN DECOMPOSITIOIN METHOD
0() =
() = 1(2 + 2)
1 0
0
= 1 22 + 3
= 2 +
3 4
() = 1[(2 + 2) 2 + ]
2 0 3 4
= 2 + 31
6
() = 1
240
2
2 2
31
3 0 [( +
) + ]
6 240
= 1 23
3122 4
313
0 [
+ + +
6 240 6
]
240
= 21464 + 1890
And so on
17280
28800
() =1.44479167 + 0.5847221892
BY VARIATIONAL ITERATION METHOD
0() =
() = + 1(2 + 2)
1 0
= + 2 +
3 4
= 2 + 5
3 4
() = + 1(2 + 2) 2 + 5
2 0
= +
1 32
3
522
4
4
53
0 ( 3 + 4 + 3 +
= 2 + 331
4
2 240
() = + 1(2 + 2) 2 + 331
3
= +
0
1 33122
2
3313
240
4
23
0 [
+
240
+ +
240 2
]
2
= 33682 + 1387
5760 960
There are various noise terms appearing in the iteration and we will obtain the solution given as
() = 0.5847222222 + 1.44479167
Values
0.1
0.2
0.3
0.4
0.5
ADM
0.150326389
0.312347222
0.486062498
0.671472218
0.868576382
VIM
0.150326389
0.312347223
0.486062501
0.671472218
0.868576391
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
ADM
0 0.2 0.4 0.6
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
VIM
0 0.2 0.4 0.6
Example 2. Consider the Fredholm integral equation given as
1
() = 1 + 1 ( + 22)()
1 1
BY ADOMIAN DECOMPOSITION METHOD
0() = 1
() = 1 ( + 22)1
1 1
= 1 ( + 2 1 2 )
1 1
= 22
3
() = 1 ( + 22) ()
2 1 1
= 1 ( + 22) 22
1 3
= 1 (23 + 224)
1 3 3
= 42
15
() = 1 ( + 22) ()
3 1 2
= 1 ( + 22) 42
1 15
= 1 (43 + 424)
And so on
1 15 15
= 82
75
() = 1 + 1.042
BY VARIATIONAL ITERATION METHOD
0() = 1
() = 1 + 1 ( + 22)1
1 1
= 1 + 22
3
() = 1 + 1 ( + 22)1
2 1
= 1 + 1 ( + 22 + 22 + 224)1
1 3 3
= 1 + 142
15
() = 1 + 1 ( + 22)1 + 422
3 1 45
= 1 + 1 ( + 423 + 22 + 4224)
1 45 45
= 1 + 822
75
There are various noise terms appearing in the iteration and we will obtain the solution given as
() = 1 + 1.093333332
Values
0.1
0.2
0.3
0.4
0.5
ADM
1.0104
1.0416
1.0936
1.1664
1.26
VIM
1.01093333
1.0437332
1.09839997
1.17493328
1.2733325
1.2
1.15
1.1
1.05
1
ADM
0 0.2 0.4 0.6
1.2
1.18
1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
VIM
0 0.1 0.2 0.3 0.4 0.5 0.6
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CONCLUSION
In this paper, ADM and VIM were successfully applied to solve the Fredholm integral equations. VIM and ADM are very powerful and operative method for finding the solution for wide classes of problems. It is worth noting that these numerical methods are quick and converge approximately at one point.
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REFERENCES
-
Fawziah, A.M., Kirtiwant, P.G. and Priyanka, P. A. The approximate solution of Fredholm integral equation by decomposition method and its modification, international journal of mathematics and its application, 6(2) (2018), 327-334.
-
Rabbani, M. and Jamali, R., Solving nonlinear system of mixed Volterra -Fredholm integral equations by using Variational Iteration method, The Journal of Mathematics and Computer Science, 5(4) (2012), 280-287.
-
Ray, S.S. and Sahu, P.K., Numerical methods for solving Fredholm integral equations of the second kind, Abstract and Applied Analysis, 8(5) 2013, 1-17.
-
Rehman,M., Integral equations and there applications, Dalhousi university Canada, (2007).
-
Ul Islam, D and Ul Haq.I , Study of some integral equations arising in applications of science and engineering, JETIR, 6(6) 2019, 714-718.
-
Kaliyappan, M. and Hariharan, S., Solving non-linear differential equations using Adomian decomposition method through sagemath,
International Journal of Innovative Technology and Exploring Engineering, 8(6) (2019), 510
-
Li, W. and Pang, Y., Application of Adomian decomposition method to non-linear system, Advances in Difference Equations, 67(1) (2019), 33-35.
-
Noor,M.A, and Shah,F.A., Variational Iteration Technique for Constructing Methods for Solving Non-linear Equations, Mathematical Sciences Letter, 4(2) (2014), 145-152.
-
He, J. H. and Wu, X. H Variational iteration method (new development and applications) Computer and Mathematics with Applications, 54(7-8) (2007), 881-894
-
Abaoub, A., Shkheam, A. S. and Zail, S. M The Adomian decomposition method of Volterra integral equation of second kind American journal of applied mathematics, 6(4) (2018), 142-148.
-
S.Shakeri,R.Saadati, S.M.Vaezpour,J.Vahidi Variational Iteration method for solving integral equations Journal of applied science, 9(4)
(2009), 799-800
-
Singh, I and Ul Islam, D Hybrid numerical methods for solving Volterra integral equation of the first kind International Journal of Advanced Research In engineering and technology, 11(10) (2020), 694-705.
-
R.N.Parjapati, R.Mohan, P. Kumar Numerical Solution of Generalized Abels Integral Equation by Variational Iteration Method American
Journal of Computational Mathematics, (2012), 312-315.
-
M.Hasan,A.Matin Solving Nonlinear Integral Equations by using Adomian Decomposition Method Journal of applied and computational
mathematics, 2017, 2-6.
-
Yasir Abeid Husain, M.A.Bashir The Application Of Adomian Decomposition Technique To Volterra Integral Type Of Equations
International Journal of Scientific and Research Publications, 9(3) (2019), 117-120.
-
Jafar Biazar, Mehdi Gholami Porshokouhi, Behzad Ghanbari,Mohammad Gholami Porshokouhi Numerical solution of functional integral
equations by the variational iteration method Journal of Computational and Applied Mathematics., (2011), 2581-2585.
-
Afroza Shirin1, Md. Shafiqul Islam Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials University of Dhaka, Dhaka 1000, Bangladesh, 1-9.