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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 513808, 9 pages
http://dx.doi.org/10.1155/2013/513808
Research Article

Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4Department of Mathematics and Computer Sciences, Cankaya University, Eskisehir Yolu 29.km, 06810 Ankara, Turkey
5Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania

Received 24 August 2013; Accepted 18 September 2013

Academic Editor: Soheil Salahshour

Copyright © 2013 A. H. Bhrawy 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. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer, New York, NY, USA, 1988. View at MathSciNet
  2. L. N. Trefethen, Spectral Methods in MATLAB, vol. 10 of Software, Environments, and Tools, SIAM, Philadelphia, Pa, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  3. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931–4943, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. H. Bhrawy and M. Al-shomrani, “A shifted Legendre spectral method for fractional order multi-point boundary value problems,” Advances in Difference Equations, vol. 2012, article 8, 8 pages, 2012. View at Publisher · View at Google Scholar
  5. A. H. Bhrawy, M. A. Alghamdi, and T. M. Taha, “A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line,” Advances in Difference Equations, vol. 2012, article 179, 12 pages, 2012. View at Publisher · View at Google Scholar
  6. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Z. Liu and D. Li, “Properties of analytic solution and numerical solution of multi-pantograph equation,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 853–871, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. G. A. Bocharov and F. A. Rihan, “Numerical modelling in biosciences using delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 183–199, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. A. Jafari and A. Aminataei, “Method of successive approximations for solving the multi-pantograph delay equations,” General Mathematics Notes, vol. 8, pp. 23–28, 2012.
  10. S. Yüzbaşı, “An efficient algorithm for solving multi-pantograph equation systems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 589–603, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. Yüzbaşi, N. Şahin, and M. Sezer, “A Bessel collocation method for numerical solution of generalized pantograph equations,” Numerical Methods for Partial Differential Equations, vol. 28, no. 4, pp. 1105–1123, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. E. Tohidi, A. H. Bhrawy, and K. Erfani, “A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4283–4294, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. J. Zhao, Y. Xu, H. X. Wang, and M. Z. Liu, “Stability of a class of Runge-Kutta methods for a family of pantograph equations of neutral type,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1170–1181, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W.-S. Wang and S.-F. Li, “On the one-leg θ-methods for solving nonlinear neutral functional differential equations,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 285–301, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  15. D. Trif, “Direct operatorial tau method for pantograph-type equations,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 2194–2203, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. A. H. Bhrawy, L. M. Assas, E. Tohidi, and M. A. Alghamdi, “Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays,” Advances in Differential Equations, vol. 2013, article 63, 16 pages, 2013. View at Publisher · View at Google Scholar
  17. Z.-Z. Sun and Z.-B. Zhang, “A linearized compact difference scheme for a class of nonlinear delay partial differential equations,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 742–752, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. L. F. Cordero and R. Escalante, “Segmented tau approximation for test neutral functional differential equations,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 725–740, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Muslim, “Approximation of solutions to history-valued neutral functional differential equations,” Computers & Mathematics with Applications, vol. 51, no. 3-4, pp. 537–550, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. O. R. Işik, Z. Güney, and M. Sezer, “Bernstein series solutions of pantograph equations using polynomial interpolation,” Journal of Difference Equations and Applications, vol. 18, no. 3, pp. 357–374, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. E. K. Ifantis, “An existence theory for functional-differential equations and functional-differential systems,” Journal of Differential Equations, vol. 29, no. 1, pp. 86–104, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Iserles and Y. Liu, “On neutral functional-differential equations with proportional delays,” Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 73–95, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. C. He, X. Lv, and J. Niu, “A new method based on the RKHSM for solving systems of nonlinear IDDEs with proportional delays,” Abstract and Applied Analysis, vol. 2013, Article ID 541935, 13 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. E. Ishiwata, “On the attainable order of collocation methods for the neutral functional-differential equations with proportional delays,” Computing, vol. 64, no. 3, pp. 207–222, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. H. Bhrawy and A. S. Alofi, “A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 62–70, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,” Boundary Value Problems, vol. 2012, article 62, 13 pages, 2012. View at Publisher · View at Google Scholar
  27. M. Maleki, I. Hashim, M. Tavassoli Kajani, and S. Abbasbandy, “An adaptive pseudospectral method for fractional order boundary value problems,” Abstract and Applied Analysis, vol. 2012, Article ID 381708, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. H. Bhrawy, “A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients,” Applied Mathematics and Computation, vol. 222, pp. 255–264, 2013.
  29. W. Wang, Y. Zhang, and S. Li, “Stability of continuous Runge-Kutta-type methods for nonlinear neutral delay-differential equations,” Applied Mathematical Modelling, vol. 33, no. 8, pp. 3319–3329, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. X. Kuang and Y. H. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, China, 2005.
  31. X. Lv and Y. Gao, “The RKHSM for solving neutral functional-differential equations with proportional delays,” Mathematical Methods in the Applied Sciences, vol. 36, no. 6, pp. 642–649, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. X. Chen and L. Wang, “The variational iteration method for solving a neutral functional-differential equation with proportional delays,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2696–2702, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. W. Wang, T. Qin, and S. Li, “Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay,” Applied Mathematics and Computation, vol. 213, no. 1, pp. 177–183, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  34. Z. Jackiewicz and B. Zubik-Kowal, “Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 3-4, pp. 433–443, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet