`Journal of Applied MathematicsVolume 2012 (2012), Article ID 803503, 11 pageshttp://dx.doi.org/10.1155/2012/803503`
Research Article

## Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

1Department of Mathematics, Islamic Azad University, Khorasgan Branch, 81515-158 Isfahan, Iran
2Department of Mathematics, Khansar Faculty of Computer and Mathematics, University of Isfahan, Isfahan 81746-73441, Iran
3Department of Physics, Faculty of Science, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia
4Department of Mathematics, Islamic Azad University, Mobarakeh Branch, 84819-14411 Isfahan, Iran

Received 22 September 2012; Accepted 8 December 2012

Copyright © 2012 M. Tavassoli Kajani 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.

1. J. Abdul Jerri, Introduction to Integral Equations with Applications, John Wiley & Sons, New York, NY, USA, 1999.
2. J. P. Boyd, “Evaluating of Dawson's integral by solving its differential equation using orthogonal rational Chebyshev functions,” Applied Mathematics and Computation, vol. 204, no. 2, pp. 914–919, 2008.
3. P. Linz, Analytical and Numerical Methods for Volterra Equations, vol. 7, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1985.
4. 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.
5. Y. Ren, B. Zhang, and H. Qiao, “A simple Taylor-series expansion method for a class of second kind integral equations,” Journal of Computational and Applied Mathematics, vol. 110, no. 1, pp. 15–24, 1999.
6. 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.
7. 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.
8. 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.
9. S. Abbasbandy and A. Taati, “Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 106–113, 2009.
10. G. Ebadi, M. Y. Rahimi, and S. Shahmorad, “Numerical solution of the system of nonlinear Fredholm integro-differential equations by the operational tau method with an error estimation,” Scientia Iranica, vol. 14, no. 6, pp. 546–554, 2007.
11. A. Khani, M. Mohseni Moghadam, and S. Shahmorad, “Numerical solution of special class of systems of non-linear Volterra integro-differential equations by a simple high accuracy method,” Bulletin of the Iranian Mathematical Society, vol. 34, no. 2, pp. 141–152, 2008.
12. K. Maleknejad and M. Tavassoli Kajani, “Solving linear integro-differential equation system by Galerkin methods with hydrid functions,” Applied Mathematics and Computation, vol. 159, no. 3, pp. 603–612, 2004.
13. K. Maleknejad, F. Mirzaee, and S. Abbasbandy, “Solving linear integro-differential equations system by using rationalized Haar functions method,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 317–328, 2004.
14. J. Biazar, H. Ghazvini, and M. Eslami, “He's homotopy perturbation method for systems of integro-differential equations,” Chaos, Solitons & Fractals, vol. 39, no. 3, pp. 1253–1258, 2009.
15. E. Yusufoğlu, “An efficient algorithm for solving integro-differential equations system,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 51–55, 2007.
16. 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.
17. B.-Y. Guo and J. Shen, “Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval,” Numerische Mathematik, vol. 86, no. 4, pp. 635–654, 2000.
18. Y. Maday, B. Pernaud-Thomas, and H. Vandeven, “Shock-Fitting techniques for solving hyperbolic problems with spectral methods,” Recherche Aerospatiale, vol. 6, pp. 1–9, 1985.
19. H. I. Siyyam, “Laguerre tau methods for solving higher-order ordinary differential equations,” Journal of Computational Analysis and Applications, vol. 3, no. 2, pp. 173–182, 2001.
20. J. Shen, “Stable and efficient spectral methods in unbounded domains using Laguerre functions,” SIAM Journal on Numerical Analysis, vol. 38, no. 4, pp. 1113–1133, 2000.
21. J. P. Boyd, Chebyshev and Fourier Spectral Methods, Springer, Berlin, Germany, 2nd edition, 2000.
22. B.-Y. Guo, “Jacobi spectral approximations to differential equations on the half line,” Journal of Computational Mathematics, vol. 18, no. 1, pp. 95–112, 2000.
23. J. P. Boyd, “Orthogonal rational functions on a semi-infinite interval,” Journal of Computational Physics, vol. 70, no. 1, pp. 63–88, 1987.
24. B.-Y. Guo, J. Shen, and Z.-Q. Wang, “A rational approximation and its applications to differential equations on the half line,” Journal of Scientific Computing, vol. 15, no. 2, pp. 117–147, 2000.
25. J. P. Boyd, C. Rangan, and P. H. Bucksbaum, “Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions,” Journal of Computational Physics, vol. 188, no. 1, pp. 56–74, 2003.
26. K. Parand and M. Razzaghi, “Rational Chebyshev tau method for solving higher-order ordinary differential equations,” International Journal of Computer Mathematics, vol. 81, no. 1, pp. 73–80, 2004.
27. H. I. Siyyam, “Laguerre Tau methods for solving higher-order ordinary differential equations,” Journal of Computational Analysis and Applications, vol. 3, no. 2, pp. 173–182, 2001.
28. K. Parand and M. Razzaghi, “Rational Chebyshev tau method for solving Volterra's population model,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 893–900, 2004.
29. K. Parand and M. Razzaghi, “Rational legendre approximation for solving some physical problems on semi-infinite intervals,” Physica Scripta, vol. 69, no. 5, pp. 353–357, 2004.
30. M. Zarebnia and M. G. Ali Abadi, “Numerical solution of system of nonlinear second-order integro-differential equations,” Computers & Mathematics with Applications, vol. 60, no. 3, pp. 591–601, 2010.
31. M. Tavassoli Kajani and F. Ghasemi Tabatabaei, “Rational Chebyshev approximations for solving Lane-Emde equation of index m,” in Proceeding of the International Conference on Computational and Applied Mathematics, pp. 840–844, Bangkok, Thailand, March 2011.
32. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 10th edition, 1972.
33. M. Dadkhah Tirani, F. Ghasemi Tabatabaei, and M. Tavassoli Kajani, “Rational second (third) kind Chebyshev approximations for solving Volterras population model,” in Proceeding of the International Conference on Computational and Applied Mathematics, pp. 835–839, Bangkok, Thailand, March 2011.