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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 796286, 7 pages
http://dx.doi.org/10.1155/2014/796286
Research Article

On Bernstein Polynomials Method to the System of Abel Integral Equations

1Department of Mathematics, Islamic Azad University, Urmia Branch, P.O. BOX 969, Urmia, Iran
2Department of Physics, Islamic Azad University, Urmia Branch, P.O. BOX 969, Urmia, Iran
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey
5Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania

Received 20 November 2013; Revised 3 March 2014; Accepted 4 March 2014; Published 7 May 2014

Academic Editor: Hossein Jafari

Copyright © 2014 A. Jafarian 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.

Linked References

  1. M. Khan and M. A. Gondal, “A reliable treatment of Abel's second kind singular integral equations,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1666–1670, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. L. Bougoffa, R. C. Rach, and A. Mennouni, “A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1785–1793, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. M. Abdulkawi, Z. K. Eshkuvatov, and N. M. A. Nik Long, “A note on the numerical solution of singular integral equations of cauchy type,” World Academy of Science, Engineering and Technology, vol. 3, no. 9, pp. 893–896, 2009. View at Google Scholar
  4. S. Banerjea and B. Dutta, “Solution of a singular integral equation and its application to water wave problems,” Journal of the Indian Institute of Science, vol. 86, no. 3, pp. 265–278, 2006. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. J. Helsing, “A fast and stable solver for singular integral equations on piecewise smooth curves,” SIAM Journal on Scientific Computing, vol. 33, no. 1, pp. 153–174, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. K. Maleknejad and A. Salimi Shamloo, “Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 500–505, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. L. Huang, Y. Huang, and X. Li, “Approximate solution of Abel integral equation,” Computers and Mathematics with Applications, vol. 56, no. 7, pp. 1748–1757, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. M. S. Akel and H. S. Hussein, “Numerical treatment of solving singular integral equations by using Sinc approximations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3565–3573, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. J. Biazar and H. Ebrahimi, “A new technique for systems of Abel-Volterra integral equations,” International Journal of the Physical Sciences, vol. 7, no. 1, pp. 89–99, 2012. View at Google Scholar
  10. M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, and C. Cattani, “Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 1, pp. 37–48, 2014. View at Publisher · View at Google Scholar
  11. S. A. Yousefi and M. Behroozifar, “Operational matrices of Bernstein polynomials and their applications,” International Journal of Systems Science, vol. 41, no. 6, pp. 709–716, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. M. Alipour and D. Rostamy, “Solving nonlinear fractional differential equations by Bernstein polynomials operational matrices,” The Journal of Mathematics and Computer Science, vol. 5, no. 3, pp. 185–196, 2012. View at Google Scholar
  13. D. Rostamy, M. Alipour, H. Jafari, and D. Baleanu, “Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis,” Romanian Reports in Physics, vol. 65, no. 2, pp. 334–349, 2013. View at Google Scholar
  14. D. Baleanu, M. Alipour, and H. Jafari, “The Bernstein operational matrices for solving the fractional quadratic riccati differential equations with the Riemann-Liouville derivative,” Abstract and Applied Analysis, vol. 2013, Article ID 461970, 7 pages, 2013. View at Publisher · View at Google Scholar
  15. M. Alipour and D. Baleanu, “Approximate analytical solution for nonlinear system of fractional differential equations by BPs operational matrices,” Advances in Mathematical Physics, vol. 2013, Article ID 954015, 9 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. Alipour, D. Rostamy, and D. Baleanu, “Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices,” Journal of Vibration and Control, vol. 19, no. 16, pp. 2523–2540, 2013. View at Publisher · View at Google Scholar
  17. M. Alipour and D. Rostamy, “BPs operational matrices for solving time varying fractional optimal control problems,” The Journal of Mathematics and Computer Science, vol. 6, pp. 292–304, 2013. View at Google Scholar
  18. A. Jafarian, S. Measoomy Nia, A. K. Golmankhaneh, and D. Baleanu, “Numerical solution of linear integral equations system using the Bernstein collocation method,” Advances in Difference Equations, vol. 2013, article 123, 2013. View at Publisher · View at Google Scholar
  19. V. K. Singh, R. K. Pandey, and O. P. Singh, “New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials,” Applied Mathematical Sciences, vol. 3, no. 5, pp. 241–255, 2009. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. O. P. Singh, V. K. Singh, and R. K. Pandey, “A stable numerical inversion of Abel's integral equation using almost Bernstein operational matrix,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 111, no. 1, pp. 245–252, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Dixit, O. P. Singh, and S. Kumar, “A stable numerical inversion of generalized Abels integral equation,” Applied Numerical Mathematics, vol. 62, no. 5, pp. 567–579, 2012. View at Publisher · View at Google Scholar · View at Scopus
  22. O. R. Işik, M. Sezer, and Z. Güney, “Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 7009–7020, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing, China; Springer, Berlin, Germany, 2011.
  24. B. N. Mandal and S. Bhattacharya, “Numerical solution of some classes of integral equations using Bernstein polynomials,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1707–1716, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus