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Numerical solution of linear and nonlinear Fredholm integral equations by using weighted meanvalue theorem
 Ahmet Altürk^{1}Email author
 Received: 5 June 2016
 Accepted: 3 November 2016
 Published: 14 November 2016
Abstract
Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semianalytical method that is based on weighted meanvalue theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.
Keywords
 Linear and nonlinear Fredholm integral equations
 Systems of Fredholm integral equations
 Systems of Fredholm integrodifferential equations
 Weighted mean value theorem
Mathematics Subject Classification
 Primary 45B05
 35B05
 Secondary 45G15
Introduction and preliminaries
Integral equations have numerous applications in virtually every branches of science. Many physical processes and mathematical models are usually governed by the integral equations. In particular, many initial and boundary value problems can easily be converted to integral equations. Since the subject has many potential application areas, it has attracted many researchers’ attentions from past to today. The literature is very rich of analytical and numerical techniques proposed for solving different kinds of integral equations.
The aim of this article is to propose a simple and effective method for obtaining solutions for a rather wide class of Fredholm integral equations of the second kind. In other words, I investigate linear and nonlinear Fredholm integral and integrodifferential equations of the second kind along with the systems of the mentioned classes of Fredholm equations. Before delving into the details of the proposed approach, the list of some available methods proposed by other researchers in the literature are given. The methods with a similar subject area are grouped together, such as wavelet methods (Lepik 2006, 2008; Alpert et al. 1990; Kajani et al. 2006), collocation methods (Zhongying et al. 2006; Maleknejad and Nedaiasl 2011; Jafarian et al. 2013), Adomian decomposition method (Adomian 1994; Wazwaz 1999), transform methods (Ezzati and Mokhtari 2012; Odibat 2008), homotopy perturbation method (Golbabai and Keramati 2008; Abbasbandy 2006), etc. There are also some excellent books from introductory to advanced level, such as Wazwaz (2011), Kress (2014), Rahman (2007) and Pipkin (1991).
The method that is introduced and investigated in this article is weighted integral meanvalue method (WMVM). The weighted meanvalue theorem are used and applied to the different kinds of Fredholm integral equations. As a result, a linear (or, nonlinear) system of algebraic equations are obtained. By solving these systems of equations, the desired solution for the integral equation will be reached. Elaborated examples are provided to show the applicability and validity of the proposed method.
Description of the method: weighted meanvalue method for integrals (WMVM)
Mean value theorems for both derivatives and integrals are very powerful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. Recently, some applications of the meanvalue theorem for solving different classes of Fredholm integral equations from one dimensional to higher dimensional have been introduced (Avazzadeh et al. 2011; Heydari et al. 2013; Li and Huang 2016). The results are promising and the method is very simple.
In this article, the weighted meanvalue theorem will be used to obtain solutions for a wide class of Fredholm integral equations. As it will be seen in the subsequent sections that under some mild conditions the weighted meanvalue theorem can be applied to Fredholm integral equations and significant results are obtained.
Theorem 1

Linear and nonlinear Fredholm integral equations of the second kind (“Solving linear and nonlinear Fredholm integral equations via WMVM” section)

Linear and nonlinear Fredholm integrodifferential equations of the second kind (“Solving Fredholm integrodifferential equations via WMVM” section)

Linear and nonlinear systems of Fredholm integral equations of the second kind (“Solving linear and nonlinear systems of Fredholm integral equations via WMVM” section)

Linear and nonlinear systems of Fredholm integrodifferential equations of the second kind (“Solving systems of Fredholm integrodifferential equations via WMVM” section)
I would like to point out that I do not aim for complete generality, but making simplifying assumptions that produce significant results. In Avazzadeh et al. (2011), the authors obtained significant results under the assumption that an application of the meanvalue theorem to Fredholm integral equations produces a number c rather than a function c(x). For some cases, this assumption produces an error in numerical solution (Zhong 2013). Throughout the paper I also assume \(c(x)=c.\)
Solving linear and nonlinear Fredholm integral equations via WMVM
Finally, substitute c and u(c) into (2) to get a solution.
Solving Fredholm integrodifferential equations via WMVM
Solving linear and nonlinear systems of Fredholm integral equations via WMVM
Solving systems of Fredholm integrodifferential equations via WMVM
Substituting \(c_1\) and \(c_3\) into first equation in (24) and \(c_2\) and \(c_4\) into second equation in (24), 4 equations will be obtained. Then, by substituting (23) and (24) into (22), 2 new equations will be obtained. Replacing x with \(c_1\) and \(c_3\) in the first equation and \(c_2\) and \(c_4\) in the second equation, there will be 4 more equations. Solving this nonlinear system of equations will give the desired solution.
Numerical results
In this section, numerical results are presented for various types of Fredholm integral equations mentioned in the previous sections. The results show the validity and efficiency of the method. It is important to note that all numerical computations are performed using Matlab software. For solving a nonlinear system of equations, the Matlab builtin functions use the Newton’s method with an initial guess or some modified versions of it. Since these methods are, in general, local, the initial guess plays a decisive role in obtaining solutions.
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Comparison and discussions
Example 7
As the final example, consider an equation for which the method introduced in Avazzadeh et al. (2011) does not provide a number \(c\in [0,1]\) when solving the nonlinear system of equations obtained after applying the method. This is shown by a geometric reasoning. it is also shown that applying WMVM will produce the exact solution.
Example 8
Conclusion
In this article, an effective method based on weighted meanvalue theorem for solving different types of Fredholm integral equations of the second kind, from linear to nonlinear equations and integrodifferential to the systems of equations involving them, is presented. The numerical and analytical solutions are conducted using Matlab. Thoroughly workedout examples are provided in order to show the accuracy and applicability of the presented approach.
Declarations
Acknowledgements
I would like to thank the editor and anonymous reviewers for their constructive comments and suggestions, which helped me to improve the manuscript.
Competing interests
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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