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Mathematical Problems in Engineering
Volume 2015, Article ID 217348, 10 pages
http://dx.doi.org/10.1155/2015/217348
Research Article

Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems

1Department of Management Information Systems, Bartin University, 74100 Bartin, Turkey
2Department of Mathematical Engineering, Yildiz Technical University, 34220 Istanbul, Turkey

Received 10 June 2014; Accepted 13 July 2014

Academic Editor: Abdon Atangana

Copyright © 2015 Sertan Alkan and Aydin Secer. 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. S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters A, vol. 370, no. 5-6, pp. 379–387, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. Y. Zhang, “A finite difference method for fractional partial differential equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 524–529, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. Q. Wang, “Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1048–1055, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. V. Daftardar-Gejji and S. Bhalekar, “Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 113–120, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Q. Wang, “Homotopy perturbation method for fractional KdV equation,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1795–1802, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. Q. Wang, “Homotopy perturbation method for fractional KdV-Burgers equation,” Chaos, Solitons and Fractals, vol. 35, no. 5, pp. 843–850, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. O. Abdulaziz, I. Hashim, and E. S. Ismail, “Approximate analytical solution to fractional modified KdV equations,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 136–145, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. M. U. Rehman and R. A. Khan, “Numerical solutions to initial and boundary value problems for linear fractional partial differential equations,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 5233–5244, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. A. Akinlar, A. Secer, and M. Bayram, “Numerical solution of fractional Benney equation,” Applied Mathematics and Information Sciences, vol. 8, no. 4, pp. 1633–1637, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. Secer, M. A. Akinlar, and A. Cevikel, “Efficient solutions of systems of fractional PDEs by the differential transform method,” Advances in Difference Equations, vol. 2012, article 188, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. M. Kurulay and M. Bayram, “Approximate analytical solution for the fractional modified KdV by differential transform method,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1777–1782, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. M. Kurulay, M. A. Akinlar, and R. Ibragimov, “Computational solution of a fractional integro-differential equation,” Abstract and Applied Analysis, vol. 2013, Article ID 865952, 4 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. A. Akinlar, A. Secer, and M. Bayram, “Stability, synchronization control and numerical solution of fractional Shimizu-Morioka dynamical system,” Applied Mathematics & Information Sciences, vol. 8, no. 4, pp. 1699–1705, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. Atangana and A. Secer, “A note on fractional order derivatives and table of fractional derivatives of some special functions,” Abstract and Applied Analysis, vol. 2013, Article ID 279681, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-Vries equations,” Abstract and Applied Analysis, vol. 2013, Article ID 947986, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. F. Stenger, “Approximations via Whittaker's cardinal function,” Journal of Approximation Theory, vol. 17, no. 3, pp. 222–240, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. F. Stenger, “A sinc-Galerkin method of solution of boundary value problems,” Mathematics of Computation, vol. 33, no. 145, pp. 85–109, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  19. E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proceedings of the Royal Society of Edinburgh, vol. 35, pp. 181–194, 1915. View at Google Scholar
  20. J. M. Whittaker, Interpolation Function Theory, vol. 33 of Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, London, UK, 1935.
  21. M. El-Gamel and A. I. Zayed, “Sinc-Galerkin method for solving nonlinear boundary-value problems,” Computers & Mathematics with Applications, vol. 48, no. 9, pp. 1285–1298, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M. El-Gamel, S. H. Behiry, and H. Hashish, “Numerical method for the solution of special nonlinear fourth-order boundary value problems,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 717–734, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. M. Zarebnia and M. Sajjadian, “The sinc-Galerkin method for solving Troesch's problem,” Mathematical and Computer Modelling, vol. 56, no. 9-10, pp. 218–228, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. A. Seçer, S. Alkan, and M. Bayram, “Sinc-galerkin method for approximate solutions of space fractional partial differential equations,” in Proceedings of the International Conference and Workshop on Mathematical Analysis (ICWOMA '14), p. 48, 2014.
  25. A. Mohsen and M. El-Gamel, “On the Galerkin and collocation methods for two-point boundary value problems using sinc bases,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 930–941, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. El-Gamel and A. I. Zayed, “A comparison between the wavelet-GALerkin and the sinc-GALerkin methods in solving nonhomogeneous heat equations,” in Inverse Problem, Image Analysis, and Medical Imaging, vol. 313, pp. 97–116, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  27. A. Secer and M. Kurulay, “The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems,” Boundary Value Problems, vol. 2012, article 126, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  28. A. Secer, S. Alkan, M. A. Akinlar, and M. Bayram, “Sinc-Galerkin method for approximate solutions of fractional order boundary value problems,” Boundary Value Problems, vol. 2013, article 281, 2013. View at Publisher · View at Google Scholar
  29. R. Almeida and D. F. M. Torres, “Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1490–1500, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  31. M. El-Gamel, “A numerical scheme for solving nonhomogeneous time-dependent problems,” Zeitschrift für Angewandte Mathematik und Physik, vol. 57, no. 3, pp. 369–383, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. M. ur Rehman and R. A. Khan, “Numerical solutions to initial and boundary value problems for linear fractional partial differential equations,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 5233–5244, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus