Numerical solution of linear integral equations system using the Bernstein collocation method
 Ahmad Jafarian,
 Safa A Measoomy Nia,
 Alireza K Golmankhaneh,
 Dumitru Baleanu
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Abstract
Since in some application mathematical problems finding the analytical solution is too complicated, in recent years a lot of attention has been devoted by researchers to find the numerical solution of this equations. In this paper, an application of the Bernstein polynomials expansion method is applied to solve linear second kind Fredholm and Volterra integral equations systems. This work reduces the integral equations system to a linear system in generalized case such that the solution of the resulting system yields the unknown Bernstein coefficients of the solutions. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.
 Tricomi FG: Integral Equations. Dover, New York; 1982.
 Lan X: Variational iteration method for solving integral equations. Comput. Math. Appl. 2007, 54:1071–1078. CrossRef
 Babolian E, Sadeghi Goghary S, Abbasbandy S: Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method. Appl. Math. Comput. 2005, 161:733–744. CrossRef
 Liao SJ: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton; 2003. CrossRef
 Abbasbandy S: Numerical solution of integral equation: homotopy perturbation method and Adomian’s decomposition method. Appl. Math. Comput. 2006, 173:493–500. CrossRef
 Kanwal RP, Liu KC: A Taylor expansion approach for solving integral equations. Int. J. Math. Educ. Sci. Technol. 1989, 2:411–414. CrossRef
 Maleknejad K, Aghazadeh N: Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylorseries expansion method. Appl. Math. Comput. 2005, 161:915–922. CrossRef
 Nas S, Yalcynbas S, Sezer M: A Taylor polynomial approach for solving highorder linear Fredholm integrodifferential equations. Int. J. Math. Educ. Sci. Technol. 2000, 31:213–225. CrossRef
 Babolian E, Masouri Z, HatamzadehVarmazyar S: A direct method for numerically solving integral equations system using orthogonal triangular functions. Int. J. Ind. Math. 2009, 2:135–145.
 Jafari H, Hosseinzadeh H, Mohamadzadeh S: Numerical solution of system of linear integral equations by using Legendre wavelets. Int. J. Open Probl. Comput. Sci. Math. 2010, 5:63–71.
 Bhatti MI, Bracken P: Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 2007. doi:10.1016/j.cam.2006.05.002
 Farouki RT, Goodman TNT: On the optimal stability of the Bernstein basis. Math. Comput. 1996,65(216):1553–1566. CrossRef
 Yousefi SA, Behroozifar M: Operational matrices of Bernstein polynomials and their applications. Int. J. Syst. Sci. 2010,41(6):709–716. CrossRef
 Bhatta DD, Bhatti MI: Numerical solution of KdV equation using modified Bernstein polynomials. Appl. Math. Comput. 2006, 174:1255–1268. CrossRef
 Mandal BN, Bhattachary S: Numerical solution of some classes of integral equations using Bernstein polynomials. Appl. Math. Comput. 2007, 191:1707–1716. CrossRef
 Bhattacharya S, Mandal BN: Use of Bernstein polynomials in numerical solution of Volterra integral equations. Appl. Math. Sci. 2008, 6:1773–1787.
 Maleknejad K, Hashemizadeh E, Ezzati R: A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Commun. Nonlinear Sci. Numer. Simul. 2011, 161:647–655. CrossRef
 Jafarian A, Measoomy NS: Utilizing feedback neural network approach for solving linear Fredholm integral equations system. Appl. Math. Model. 2012. doi:10.1016/j.apm.2012.09.029
 Hochstadt H: Integral Equations. Wiley, New York; 1973.
 Title
 Numerical solution of linear integral equations system using the Bernstein collocation method
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Advances in Difference Equations
2013:123
 Online Date
 May 2013
 DOI
 10.1186/168718472013123
 Online ISSN
 16871847
 Publisher
 Springer International Publishing
 Additional Links
 Topics
 Authors

 Ahmad Jafarian ^{(1)}
 Safa A Measoomy Nia ^{(1)}
 Alireza K Golmankhaneh ^{(2)}
 Dumitru Baleanu ^{(3)} ^{(4)} ^{(5)}
 Author Affiliations

 1. Department of Mathematics, Islamic Azad University, Urmia Branch, Urmia, Iran
 2. Department of Physics, Islamic Azad University, Urmia Branch, Urmia, Iran
 3. Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Çankaya University, Balgat, Ankara, 0630, Turkey
 4. Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia
 5. Institute of Space Sciences, P.O. Box MG23, Magurele, Bucharest, 76900, Romania