Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2015 (2015), Article ID 616438, 9 pages
http://dx.doi.org/10.1155/2015/616438
Research Article

An Efficient Numerical Algorithm for Solving Fractional Higher-Order Nonlinear Integrodifferential Equations

Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al-Ain, UAE

Received 3 July 2015; Accepted 11 October 2015

Academic Editor: Jozef Banas

Copyright © 2015 Muhammed I. Syam 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. R. P. Agarwal, S. K. Ntouyas, B. Ahmad, and M. S. Alhothuali, “Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions,” Advances in Difference Equations, vol. 2013, article 128, 2013. View at Publisher · View at Google Scholar · View at Scopus
  2. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  3. F. Mainardi, “Fractional calculus: Some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 223–276, Springer, Vienna, Austria, 1997. View at Google Scholar
  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  5. B. Ahmad and B. S. Alghamdi, “Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions,” Computer Physics Communications, vol. 179, no. 6, pp. 409–416, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. Y.-K. Chang and J. J. Nieto, “Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators,” Numerical Functional Analysis and Optimization, vol. 30, no. 3-4, pp. 227–244, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. Z. Luo and J. J. Nieto, “New results for the periodic boundary value problem for impulsive integro-differential equations,” Nonlinear Analysis, Theory, Methods & Applications, vol. 70, no. 6, pp. 2248–2260, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Mesloub, “On a mixed nonlinear one point boundary value problem for an integrodifferential equation,” Boundary Value Problems, vol. 2008, Article ID 814947, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. R. P. Agarwal, B. De Andrade, and G. Siracusa, “On fractional integro-differential equations with state-dependent delay,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1143–1149, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 480–487, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. 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 Scopus
  13. Q. M. Al-Mdallal, “On the numerical solution of fractional Sturm-Liouville problems,” International Journal of Computer Mathematics, vol. 87, no. 12, pp. 2837–2845, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. Q. M. Al-Mdallal and M. I. Syam, “An efficient method for solving non-linear singularly perturbed two points boundary-value problems of fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2299–2308, 2012. View at Publisher · View at Google Scholar · View at Scopus
  15. Q. M. Al-Mdallal, M. I. Syam, and M. N. Anwar, “A collocation-shooting method for solving fractional boundary value problems,” Communications in Non-Linear Science and Numerical Simulation, vol. 15, no. 12, pp. 3814–3822, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. J. Cao, Q. Yang, and Z. Huang, “Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 1, pp. 224–234, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. M. H. M. Rashid and Y. El-Qaderi, “Semilinear fractional integro-differential equations with compact semigroup,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 6276–6282, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. R. C. Mittal and R. Nigam, “Solution of fractional integro-differential equations by Adomian decomposition method,” International Journal of Applied Mathematics and Mechanics, vol. 4, no. 2, pp. 87–94, 2008. View at Google Scholar
  19. S. Momani and M. Aslam Noor, “Numerical methods for fourth-order fractional integro-differential equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 754–760, 2006. View at Publisher · View at Google Scholar · View at Scopus
  20. E. A. Rawashdeh, “Numerical solution of fractional integro-differential equations by collocation method,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 1–6, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. J. Zhao, J. Xiao, and N. J. Ford, “Collocation methods for fractional integro-differential equations with weakly singular kernels,” Numerical Algorithms, vol. 65, no. 4, pp. 723–743, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. Y. Nawaz, “Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations,” Computers and Mathematics with Applications, vol. 61, no. 8, pp. 2330–2341, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. K. Sayevand, “Analytical treatment of Volterra integro-differential equations of fractional order,” Applied Mathematical Modelling, vol. 39, no. 15, pp. 4330–4336, 2015. View at Publisher · View at Google Scholar · View at Scopus
  24. A. Arikoglu and I. Ozkol, “Solution of fractional integro-differential equations by using fractional differential transform method,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 521–529, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. D. Nazari and S. Shahmorad, “Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 883–891, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. H. Saeedi and M. M. Moghadam, “Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1216–1226, 2011. View at Publisher · View at Google Scholar · View at Scopus
  27. P. Mokhtary, “Discrete Galerkin method for fractional integro-differential equations,” Acta Mathematica Scientia, http://arxiv.org/abs/1501.01111.
  28. J. Biazar and H. Ebrahimi, “Chebyshev wavelets approach for nonlinear systems of Volterra integral equations,” Computers and Mathematics with Applications, vol. 63, no. 3, pp. 608–616, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. L. Huang, X.-F. Li, Y. Zhao, and X.-Y. Duan, “Approximate solution of fractional integro-differential equations by Taylor expansion method,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1127–1134, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. L. Blank, “Numerical treatment of differential equations of fractional order,” Numerical Analysis Report 287, Manchester Center for Numerical Computational Mathematics, 1996. View at Google Scholar