Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 7498136, 10 pages
https://doi.org/10.1155/2017/7498136
Research Article

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran 14515, Iran
2Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Correspondence should be addressed to M. Ali Fariborzi Araghi; moc.liamg@ihgara.izrobiraf

Received 11 November 2016; Revised 7 February 2017; Accepted 8 February 2017; Published 16 March 2017

Academic Editor: Haipeng Peng

Copyright © 2017 Sedigheh Farzaneh Javan 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. K. Meetz and W. Engi, Electromagnetische Felder, Springer, Berlin, Germany, 1979.
  2. W. Thirring, Lehrbuch der Mathematischen Physik, vol. 2, Springer, Vienna, Austria, 1978. View at Publisher · View at Google Scholar
  3. K. F. Warnick, R. H. Selfridge, and D. V. Arnold, “Teaching electromagnetic field theory using differential forms,” IEEE Transactions on Education, vol. 40, no. 1, pp. 53–68, 1997. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 34, no. 3, pp. 380–387, 2007. View at Publisher · View at Google Scholar · View at Scopus
  5. H. Vosughi, E. Shivanian, and S. Abbasbandy, “A new analytical technique to solve Volterra's integral equations,” Mathematical Methods in the Applied Sciences, vol. 34, no. 10, pp. 1243–1253, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. J.-P. Kauthen, “A survey of singularly perturbed Volterra equations,” Applied Numerical Mathematics. An IMACS Journal, vol. 24, no. 2-3, pp. 95–114, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. A. Kilbas and M. Saigo, “On solution of nonlinear Abel-Volterra integral equation,” Journal of Mathematical Analysis and Applications, vol. 229, no. 1, pp. 41–60, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. C. Minggen and D. Zhongxing, “On the best operator of interpolation,” Mathematica Numerica Sinica, vol. 8, no. 2, pp. 209–216, 1986. View at Google Scholar
  9. C. Minggen and L. Yingzhen, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2009.
  10. S. Abbasbandy, B. Azarnavid, and M. S. Alhuthali, “A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems,” Journal of Computational and Applied Mathematics, vol. 279, pp. 293–305, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. N. Shawagfeh, O. Abu Arqub, and S. Momani, “Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method,” Journal of Computational Analysis and Applications, vol. 16, no. 4, pp. 750–762, 2014. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. F. Geng and M. Cui, “Solving a nonlinear system of second order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1167–1181, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. B. Azarnavid, F. Parvaneh, and S. Abbasbandy, “Picard-reproducing kernel Hilbert space method for solving generalized singular nonlinear Lane-Emden type equations,” Mathematical Modelling and Analysis, vol. 20, no. 6, pp. 754–767, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. O. A. Arqub, M. Al-Smadi, and N. Shawagfeh, “Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 8938–8948, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. L. Yang and M. Cui, “New algorithm for a class of nonlinear integro-differential equations in the reproducing kernel space,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 942–960, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. O. Abu Arqub, M. Al-Smadi, and S. Momani, “Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 839836, 16 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Bushnaq, B. Maayah, S. Momani, and A. Alsaedi, “A reproducing kernel Hilbert space method for solving systems of fractional integrodifferential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 103016, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. M. Inc, A. Akgül, and F. Geng, “Reproducing kernel Hilbert space method for solving Bratu's problem,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 38, no. 1, pp. 271–287, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A.-M. Wazwaz, A First Course in Integral Equations, World Scientific, 1997. View at MathSciNet
  20. S. Javadi, E. Babolian, and E. Moradi, “New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations,” Communications in Numerical Analysis, vol. 2014, Article ID cna-00205, 7 pages, 2014. View at Publisher · View at Google Scholar
  21. W. Jiang and Z. Chen, “Solving a system of linear Volterra integral equations using the new reproducing kernel method,” Applied Mathematics and Computation, vol. 219, no. 20, pp. 10225–10230, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. A.-M. Wazwaz, Linear and Nonlinear Integral Equations, Methods and Applications, Springer, Berlin, Germany, 2011.
  23. I. L. El-Kalla, “Convergence of the Adomian method applied to a class of nonlinear integral equations,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 21, no. 4, pp. 372–376, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. G. Gumah, K. Moaddy, M. Al-Smadi, and I. Hashim, “Solutions to uncertain Volterra integral equations by fitted reproducing kernel Hilbert space method,” Journal of Function Spaces, vol. 2016, Article ID 2920463, 11 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. V. Sizikov and D. Sidorov, “Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography,” Applied Numerical Mathematics. An IMACS Journal, vol. 106, pp. 69–78, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, “New direct method to solve nonlinear Volterra-Fredholm integral and integro-differential equations using operational matrix with block-pulse functions,” in Progress in Electromagnetics Research B, vol. 8, pp. 59–76, EMW Publishing, Cambridge, Mass, USA, 2008. View at Google Scholar
  27. D. Alpay, Ed., Reproducing kernel spaces and applications, vol. 143 of Operator Theory: Advances and Applications, Birkhäuser Basel, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  28. S. Saitoh, D. Alpay, J. A. Ball, and T. Ohsawa, Reproducing Kernels and Their Applications, vol. 11, Springer Science & Business Media, Berlin, Germany, 2013.
  29. S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications, vol. 369 of Pitman Research Notes in Mathematics Series, Longman, Harlow, UK, 1997. View at MathSciNet
  30. E. Babolian and A. Shahsavaran, “Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 87–95, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. R. Ketabchi, R. Mokhtari, and E. Babolian, “Some error estimates for solving Volterra integral equations by using the reproducing kernel method,” Journal of Computational and Applied Mathematics, vol. 273, pp. 245–250, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  32. A. Alvandi, M. Paripour, and Z. Roshani, “Reproducing kernel method for solving a class of Fredholm integro-differential equations,” in Proceedings of the 46th Annual Iranian Mathematics Conference (AIMC 46), p. 505, Yazd University, 2015.
  33. I. Komashynska and M. Al-Smadi, “Iterative reproducing kernel method for solving second-order integrodifferential equations of fredholm type,” Journal of Applied Mathematics, vol. 2014, Article ID 459509, 11 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  34. E. Babolian, S. Javadi, and E. Moradi, “Error analysis of reproducing kernel Hilbert space method for solving functional integral equations,” Journal of Computational and Applied Mathematics, vol. 300, pp. 300–311, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus