Journal of Applied Mathematics

Volume 2013, Article ID 682537, 5 pages

http://dx.doi.org/10.1155/2013/682537

## New Perturbation Iteration Solutions for Fredholm and Volterra Integral Equations

^{1}Department of Mechanical Engineering, Celal Bayar University, Muradiye, 45140 Manisa, Turkey^{2}Department of Mathematics, Faculty of Science, Nevşehir University, 50300 Nevşehir, Turkey

Received 29 December 2012; Accepted 22 April 2013

Academic Editor: Livija Cveticanin

Copyright © 2013 İhsan Timuçin Dolapçı et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, recently developed perturbation iteration method is used to solve Fredholm and Volterra integral equations. The study shows that the new method can be applied to both types of integral equations. Some numerical examples are given, and results are compared with other studies to illustrate the efficiency of the method.

#### 1. Introduction

As one of the most important subjects of mathematics, differential and integral equations are widely used to model a variety of physical problems. Perturbation methods have been used in search of approximate analytical solutions for over a century [1–3]. Algebraic equations, integral-differential equations, and difference equations could be solved by these techniques approximately.

However, a major difficulty in the implementation of perturbation methods is the requirement of a small parameter or inserting a small artificial parameter in the equation. Solutions obtained by these methods are therefore restricted by a validity range of physical parameters. To eliminate the small parameter assumption in regular perturbation analysis, iteration techniques are incorporated with perturbations. Many attempts in this issue appear in the literature recently [4–13].

Recently, a new perturbation-iteration algorithm has been developed by Pakdemirli and his coworkers [14–16]. A preliminary study of developing root finding algorithms systematically [17–19] finally led to generalization of the method to differential equations also [14–16]. An iterative scheme is constituted over the perturbation expansion in the new technique. The method has been successfully implemented to first-order equations [15] and Bratu-type second-order equations [14].

In this paper, this new technique is applied to integral equations for the first time. Fredholm and Volterra integral equations are considered, where , , and is the unknown function to be determined. Results are compared with some other studies.

#### 2. Overview of the Method

In the present paper, the simplest perturbation-iteration algorithm is used by taking one correction term in the perturbation expansion and correction terms of only first derivatives in the Taylor series expansion, that is, , [14–16]. Consider the Volterra integral equation that has the form of where and is the artificially introduced perturbation parameter. In this method, we use only one correction term in the perturbation expansion:

Substituting (5) into (3) and expanding in a Taylor series with first-order derivatives only yield or All derivatives are evaluated at .

Starting with the initial condition , first has been calculated by the help of (7). Then we substitute into (5) to find . Iteration process is repeated using (7) and (5) until we obtain a satisfactory result.

#### 3. Application

*Example 1. *Consider the Fredholm integral equation of the second kind
with exact solution
Equation (8) can be rewritten in the following form:
where is a small parameter. The terms in (7) are
Note that introducing the small parameter as a coefficient of the integral term simplifies (7) and makes it solvable. For this specific example (7) reads

When applying the iteration formula (5), we select an initial guess appropriate to the boundary condition and at each step we determine coefficients from the boundary condition. Starting with the initial function
and using the formula, the approximate solutions at each step are

Higher iterations are not given here for brevity. Using a symbolic manipulation software, iterations could be calculated up to any order. In Table 1, some of our iterations are compared with the exact solution and the error between the exact solution, and are given which are of order .

*Example 2. *Consider the following integral equation:
The exact solution of the problem is
Equation (15) can be rewritten in the following form:
where is a small artificial parameter. The terms in (7) are
Equation (7) reduces to

Choosing the initial guess
and using the formula, the approximate solutions at each step are

Higher iterations are not given for brevity. In Table 2, some of our iterations are compared with the exact solution, and the errors between the exact solution and are given which are of order .

*Example 3. *Consider the equation
with the exact solution
Equation (22) is rewritten in the following form:
where is an artificially introduced small parameter. The terms in (7) are
Equation (7) reduces to
Choosing the initial guess
and using the formula, the approximate solutions at each step are
Higher iterations are not given for brevity. In Table 3, some of our iterations are compared with the exact solution, and the errors between the exact solution and are given which are of order .

*Example 4. *Consider the following integral equation:
The exact solution of the problem is
Equation (29) is rewritten in the following form:
and proceeding in a similar way yields the following iteration algorithm:
One may select the initial guess as . The successive approximations are

Higher iterations are not given for brevity. In Table 4, some of our iterations are compared with the exact solution, and the errors between the exact solution and are given which are of order .

#### 4. Conclusion

In this paper, we have applied the newly developed Perturbation Iteration Algorithm to some Fredholm and Volterra type integral equations for the first time. Numerical results show that method is an effective perturbation-iteration technique producing successful analytical results for integral equations.

#### References

- A. H. Nayfeh,
*Perturbation Methods*, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1973. View at Zentralblatt MATH · View at MathSciNet - D. W. Jordan and P. Smith,
*Nonlinear Ordinary Differential Equations*, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, New York, NY, USA, 2nd edition, 1987. View at Zentralblatt MATH · View at MathSciNet - A. V. Skorokhod, F. C. Hoppensteadt, and H. Salehi,
*Random Perturbation Methods with Applications in Science and Engineering*, vol. 150 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 2002. View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Iteration perturbation method for strongly nonlinear oscillations,”
*Journal of Vibration and Control*, vol. 7, no. 5, pp. 631–642, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. E. Mickens, “Iteration procedure for determining approximate solutions to nonlinear oscillator equations,”
*Journal of Sound and Vibration*, vol. 116, no. 1, pp. 185–187, 1987. View at Publisher · View at Google Scholar · View at MathSciNet - R. E. Mickens, “A generalized iteration procedure for calculating approximations to periodic solutions of ‘truly nonlinear oscillators’,”
*Journal of Sound and Vibration*, vol. 287, no. 4-5, pp. 1045–1051, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. E. Mickens, “Iteration method solutions for conservative and limit-cycle ${x}^{1/3}$ force oscillators,”
*Journal of Sound and Vibration*, vol. 292, no. 3–5, pp. 964–968, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Cooper and R. E. Mickens, “Generalized harmonic balance/numerical method for determining analytical approximations to the periodic solutions of the ${x}^{4/3}$ potential,”
*Journal of Sound and Vibration*, vol. 250, no. 5, pp. 951–954, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Hu and Z.-G. Xiong, “Oscillations in an ${x}^{(2m+2)/(2n+1)}$ potential,”
*Journal of Sound and Vibration*, vol. 259, no. 4, pp. 977–980, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Homotopy perturbation method with an auxiliary term,”
*Abstract and Applied Analysis*, Article ID 857612, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Q. Wang and J. H. He, “Nonlinear oscillator with discontinuity by parameter-expansion method,”
*Chaos, Solitons and Fractals*, vol. 35, no. 4, pp. 688–691, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Iqbal and A. Javed, “Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden type equation,”
*Applied Mathematics and Computation*, vol. 217, no. 19, pp. 7753–7761, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Iqbal, M. Idrees, A. M. Siddiqui, and A. R. Ansari, “Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method,”
*Applied Mathematics and Computation*, vol. 216, no. 10, pp. 2898–2909, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Aksoy and M. Pakdemirli, “New perturbation-iteration solutions for Bratu-type equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 8, pp. 2802–2808, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Pakdemirli, Y. Aksoy, and H. Boyacı, “A new perturbation-iteration approach for first order differential equations,”
*Mathematical and Computational Applications*, vol. 16, no. 4, pp. 890–899, 2011. View at Google Scholar · View at MathSciNet - Y. Aksoy, M. Pakdemirli, S. Abbasbandy, and H. Boyacı, “New perturbation-iteration solutions for nonlinear heat tranfer equations,”
*International Journal of Numerical Methods for Heat and Fluid Flow*, vol. 22, no. 7, 2012. View at Google Scholar - M. Pakdemirli and H. Boyacı, “Generation of root finding algorithms via perturbation theory and some formulas,”
*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 783–788, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Pakdemirli, H. Boyacı, and H. A. Yurtsever, “Perturbative derivation and comparisons of root-finding algorithms with fourth order derivatives,”
*Mathematical and Computational Applications*, vol. 12, no. 2, pp. 117–124, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Pakdemirli, H. Boyacı, and H. A. Yurtsever, “A root-finding algorithm with fifth order derivatives,”
*Mathematical and Computational Applications*, vol. 13, no. 2, pp. 123–128, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet