Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem
© The Author(s) 2016
Received: 5 June 2016
Accepted: 3 November 2016
Published: 14 November 2016
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 semi-analytical method that is based on weighted mean-value 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.
Mathematics Subject Classification
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 integro-differential 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 mean-value method (WMVM). The weighted mean-value 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 mean-value 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 mean-value 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 mean-value 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 mean-value theorem can be applied to Fredholm integral equations and significant results are obtained.
Linear and nonlinear Fredholm integral equations of the second kind (“Solving linear and nonlinear Fredholm integral equations via WMVM” section)
Linear and nonlinear Fredholm integro-differential equations of the second kind (“Solving Fredholm integro-differential 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 integro-differential equations of the second kind (“Solving systems of Fredholm integro-differential 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 mean-value 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 integro-differential equations via WMVM
Solving linear and nonlinear systems of Fredholm integral equations via WMVM
Solving systems of Fredholm integro-differential 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.
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 non-linear system of equations, the Matlab built-in 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.
Comparison and discussions
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.
In this article, an effective method based on weighted mean-value theorem for solving different types of Fredholm integral equations of the second kind, from linear to nonlinear equations and integro-differential to the systems of equations involving them, is presented. The numerical and analytical solutions are conducted using Matlab. Thoroughly worked-out examples are provided in order to show the accuracy and applicability of the presented approach.
I would like to thank the editor and anonymous reviewers for their constructive comments and suggestions, which helped me to improve the manuscript.
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.
- Abbasbandy S (2006) Numerical solution of integral equation: homotopy perturbation method and Adomians? Decomposition method. Appl Math Comput 173:493–500MathSciNetMATHGoogle Scholar
- Adomian G (1994) Solving Frontier problems of physics: the decomposition method. Springer, Amsterdam View ArticleMATHGoogle Scholar
- Alpert B, Beylkin G, Coifman R, Rokhlin V (1990) Wavelets for the fast solution of second-kind integral equations. http://www.cs.yale.edu/publications/techreports/tr837. Accessed 10 April 2016
- Apostol TM (1967) Calculus, vol 1. Wiley, New YorkMATHGoogle Scholar
- Avazzadeh Z, Heydari M, Loghmani GB (2011) Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem. Appl Math Model 35:2374–2383MathSciNetView ArticleMATHGoogle Scholar
- Babolian E, Biazar J, Vahidi AR (2004) The decomposition method applied to systems of Fredholm integral equations of the second kind. Appl Math Comput 148:443–452MathSciNetMATHGoogle Scholar
- Ezzati R, Mokhtari F (2012) Numerical solution of Fredholm integral equations of the second kind by using fuzzy transforms. Int J Phys Sci 7:1578–1583View ArticleGoogle Scholar
- Golbabai A, Keramati B (2008) Modified homotopy perturbation method for solving Fredholm integral equations. Chaos Solitons Fractals 37:1528–1537ADSMathSciNetView ArticleMATHGoogle Scholar
- Heydari M, Avazzadeh Z, Navabpour H, Loghmani GB (2013) Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems. Appl Math Model 37:432–442MathSciNetGoogle Scholar
- Jafarian A, Nia SAM, Golmankhaneh AK, Baleanu D (2013) Numerical solution of linear integral equations system using the Bernstein collocation method. Adv Differ Equ 2013:123MathSciNetView ArticleGoogle Scholar
- Kajani MT, Ghasemi M, Babolian E (2006) Numerical solution of linear integro-differential equation by using sine–cosine wavelets. Appl Math Comput 180:569–574MathSciNetMATHGoogle Scholar
- Kress R (2014) Linear integral equations. Springer, New YorkView ArticleMATHGoogle Scholar
- Lepik Ü (2006) Haar wavelet method for nonlinear integro-differential equations. Appl Math Comput 176:324–333MathSciNetMATHGoogle Scholar
- Lepik Ü (2008) Solving integral and differential equations by the aid of non-uniform Haar wavelets. Appl Math Comput 198:326–332MathSciNetMATHGoogle Scholar
- Li H, Huang J (2016) A novel approach to solve nonlinear Fredholm integral equations of the second kind. SpringerPlus 5:154. doi:10.1186/s40064-016-1810-8 View ArticlePubMedPubMed CentralGoogle Scholar
- Maleknejad K, Nedaiasl K (2011) Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations. Comput Math Appl 62:3292–3303MathSciNetView ArticleMATHGoogle Scholar
- Mikaeilvand N, Noeiaghdam S (2014) Mean value theorem for integrals and its application on numerically solving of Fredholm integral equation of second kind with Toeplitz plus Hankel Kernel. Int J Ind Math 6(4):351–360Google Scholar
- Odibat ZM (2008) Differential transform method for solving Volterra integral equation with separable kernels. Math Comput Model 48:1144–1149MathSciNetView ArticleMATHGoogle Scholar
- Pipkin AC (1991) A course on integral equations. Springer, New YorkView ArticleMATHGoogle Scholar
- Rahman M (2007) Integral equations and their applications. WIT Press, Southampton, BostonMATHGoogle Scholar
- Wazwaz AM (1999) A reliable modification of the Adomian decomposition method. Appl Math Comput 102:77–86MathSciNetMATHGoogle Scholar
- Wazwaz AM (2011) Linear and nonlinear integral equations. Springer, BerlinView ArticleMATHGoogle Scholar
- Zhong XC (2013) Note on the integral mean value method for Fredholm integral equations of the second kind. Appl Math Model 37:8645–8650MathSciNetView ArticleGoogle Scholar
- Zhongying C, Micchelli CA, Xu Y (2006) Fast collocation methods for second kind integral equations. SIAM J Numer Anal 40(1):344–375MathSciNetMATHGoogle Scholar