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International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 486013, 9 pages
http://dx.doi.org/10.1155/2013/486013
Research Article

Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations

Department of Mathematics, Celal Bayar University, Muradiye, 45140 Manisa, Turkey

Received 8 May 2013; Accepted 27 June 2013

Academic Editor: Irena Lasiecka

Copyright © 2013 Ayşe Kurt 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. K. Kadalbajoo and K. K. Sharma, “Numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type,” Journal of Optimization Theory and Applications, vol. 115, no. 1, pp. 145–163, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. E. Elmer and E. S. Van Vleck, “Traveling wave solutions for bistable differential-difference equations with periodic diffusion,” SIAM Journal on Applied Mathematics, vol. 61, no. 5, pp. 1648–1679, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. D. Bainov, M. B. Dimitrova, and A. B. Dishliev, “Oscillation of the bounded solutions of impulsive differential-difference equations of second order,” Applied Mathematics and Computation, vol. 114, no. 1, pp. 61–68, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. K. Kadalbajoo and K. K. Sharma, “Numerical analysis of singularly perturbed delay differential equations with layer behavior,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 11–28, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. T. Rashed, “Numerical solution of functional differential, integral and integro-differential equations,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 485–492, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. K. Maleknejad and Y. Mahmoudi, “Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 799–806, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. W. Wang and C. Lin, “A new algorithm for integral of trigonometric functions with mechanization,” Applied Mathematics and Computation, vol. 164, no. 1, pp. 71–82, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Wang, “An algorithm for solving the high-order nonlinear Volterra-Fredholm integro-differential equation with mechanization,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 1–23, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. E. Elmer and E. S. Van Vleck, “A variant of Newton's method for the computation of traveling waves of bistable differential-difference equations,” Journal of Dynamics and Differential Equations, vol. 14, no. 3, pp. 493–517, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. P. Rai and K. K. Sharma, “Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 118–132, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. P. Rai and K. K. Sharma, “Parameter uniform numerical method for singularly perturbed differential-difference equations with interior layers,” International Journal of Computer Mathematics, vol. 88, no. 16, pp. 3416–3435, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. V. Kumar and K. K. Sharma, “An optimized B-spline method for solving singularly perturbed differential difference equations with delay as well as advance,” Neural, Parallel & Scientific Computations, vol. 16, no. 3, pp. 371–385, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. K. Kadalbajoo and K. K. Sharma, “An exponentially fitted finite difference scheme for solving boundary-value problems for singularly-perturbed differential-difference equations: small shifts of mixed type with layer behavior,” Journal of Computational Analysis and Applications, vol. 8, no. 2, pp. 151–171, 2006. View at Google Scholar · View at MathSciNet
  14. M. K. Kadalbajoo and K. K. Sharma, “Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations: small shifts of mixed type with rapid oscillations,” Communications in Numerical Methods in Engineering, vol. 20, no. 3, pp. 167–182, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. H. Reihani and Z. Abadi, “Rationalized Haar functions method for solving Fredholm and Volterra integral equations,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 12–20, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. H. AliAbadi and E. L. Ortiz, “Numerical treatment of moving and free boundary value problems with the tau method,” Computers & Mathematics with Applications, vol. 35, no. 8, pp. 53–61, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Pour-Mahmoud, M. Y. Rahimi-Ardabili, and S. Shahmorad, “Numerical solution of the system of Fredholm integro-differential equations by the Tau method,” Applied Mathematics and Computation, vol. 168, no. 1, pp. 465–478, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. M. Hosseini and S. Shahmorad, “Numerical solution of a class of integro-differential equations by the tau method with an error estimation,” Applied Mathematics and Computation, vol. 136, no. 2-3, pp. 559–570, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. K. Maleknejad and F. Mirzaee, “Numerical solution of integro-differential equations by using rationalized Haar functions method,” Kybernetes, vol. 35, no. 10, pp. 1735–1744, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. M. Hosseini and S. Shahmorad, “Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases,” Applied Mathematical Modelling, vol. 27, no. 2, pp. 145–154, 2003. View at Publisher · View at Google Scholar · View at Scopus
  21. R. Farnoosh and M. Ebrahimi, “Monte Carlo method for solving Fredholm integral equations of the second kind,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 309–315, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. B. Asady, M. Tavassoli Kajani, A. Hadi Vencheh, and A. Heydari, “Solving second kind integral equations with hybrid Fourier and block-pulse functions,” Applied Mathematics and Computation, vol. 160, no. 2, pp. 517–522, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Sezer, “Taylor polynomial solutions of Volterra integral equations,” International Journal of Mathematical Education in Science and Technology, vol. 25, no. 5, pp. 625–633, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. S. Yalçınbaş and M. Sezer, “The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials,” Applied Mathematics and Computation, vol. 112, no. 2-3, pp. 291–308, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Sezer and M. Gülsu, “Polynomial solution of the most general linear Fredholm integrodifferential-difference equations by means of Taylor matrix method,” Complex Variables, vol. 50, no. 5, pp. 367–382, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. M. Gülsu and M. Sezer, “Approximations to the solution of linear Fredholm integrodifferential-difference equation of high order,” Journal of the Franklin Institute, vol. 343, no. 7, pp. 720–737, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  27. S. Yalçinbaş, “Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations,” Applied Mathematics and Computation, vol. 127, no. 2-3, pp. 195–206, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. Akyüz-Daşcıoğlu and M. Sezer, “Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations,” Journal of the Franklin Institute, vol. 342, no. 6, pp. 688–701, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. S. Yalçinbaş, M. Sezer, and H. H. Sorkun, “Legendre polynomial solutions of high-order linear Fredholm integro-differential equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 334–349, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. Gülsu and Y. Öztürk, “A new collocation method for solution of mixed linear integro-differential-difference equations,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 2183–2198, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. S. Yüzbaşı, N. Şahin, and M. Sezer, “Bessel collocation method for solving high-order linear Fredholm integro-differential-difference equations,” Journal of Advanced Research in Differential Equations, vol. 3, no. 2, pp. 1–23, 2011. View at Google Scholar · View at MathSciNet
  32. K. Erdem, S. Yalçinbas, and M. Sezer, “A new collocation method for solution of linear Fredholm integro differential-difference equations using Bernoulli polynomials,” in Proceedings of the International Conference on Applied Analysis and Algebra (ICAAA), Yildiz Technical University, Istanbul, Turkey, 1992.
  33. A. Akyüz-Daşcıoğlu and M. Sezer, “A Taylor polynomial approach for solving the most general linear Fredholm integro-differential-difference equations,” Mathematical Methods in the Applied Sciences, vol. 35, no. 7, pp. 839–844, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  34. S. Yalçınbaş, N. Özsoy, and M. Sezer, “Approximate solution of higher order linear differential equations by means of a new rational Chebyshev collocation method,” Mathematical & Computational Applications, vol. 15, no. 1, pp. 45–56, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. Yalçınbaş, M. Aynigül, and T. Akkaya, “Legendre series solutions of Fredholm integral equations,” Mathematical & Computational Applications, vol. 15, no. 3, pp. 371–381, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. A. Kurt, Fibonacci polynomial solutions of linear differential, integral and integro-differential equations [M.S. thesis], Graduate School of Natural and Applied Sciences, Mugla University, Mugla, Turkey, 2012.