The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.
| Published in | Applied and Computational Mathematics (Volume 3, Issue 6) |
| DOI | 10.11648/j.acm.20140306.15 |
| Page(s) | 315-322 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2014. Published by Science Publishing Group |
Rational Chebyshev Functions, Higher-Order Ordinary Differential Equations, Rational Chebyshev Collocation Method
| [1] | G. Yuksel, M. Gulsu and M. Sezer,” A Chebyshev Polynomial Approach for Higher – Order Linear Fredholm- Volterra Integro-Differential Equations, Gazi University Journal of Science,25(2), 393 – 401 , 2012. |
| [2] | S. Abbasbandy, H. Ghehsareh and I. Hashim, “An approximate solution of the MHD flow over a non-linear stretching sheet by rational Chebyshev collocation method “, U.P.B. Series A, Vol.74, Issue. 4, 2012. |
| [3] | K. Parand and M. Razzaghi, Rational Chebyshev tau method for solving Volterra population model, Applied Mathematics and Computation 149 (2004) 893–900. |
| [4] | D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite function, Math. Comp. 57, 597-619, 1990. |
| [5] | B.Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equation, Math. Comp. 68, 1067-1078, 1999. |
| [6] | B.Y. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math. 86, 635-654, 2000. |
| [7] | J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal. 38, 1113-1133, 2000. |
| [8] | B.Y. Guo, Jacobi spectral approximation and its applications to differential equations on half line, J. Comput. Math. 18, 95-112, 2000. |
| [9] | J.P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys. 70, 63-88, 1987. |
| [10] | J.P. Boyd, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys. 69, 112-142, 1987. |
| [11] | H.I. Siyyam, Laguerre tau methods for solving higher-order ordinary differential equations, J. Comput. Anal. Appl. 3, 173-182, 2001. |
| [12] | K. Parand and M. Razzaghi, Rational Chebyshev tau method for solving higher- order ordinary differential equations, Inter. J. Comput. Math. 81, 73-80, 2004. |
| [13] | M. Sezer and M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol. 27(4), 607-618, 1996. |
| [14] | Salih Yalçınbaş, Nesrin Özsoy and Mehmet Sezer, Approximate solution of higher order linear differential equations by means of a new rational Chebyshev collocation method, Mathematical and Computational Applications, Vol. 15, No. 1, pp. 45-56, 2010. |
| [15] | L. Fox and I.B. Parker, Chebyshev polynomials in Numerical Analysis, Oxford University press, Ely House, London, 1968. |
| [16] | Daniel Zwillinger, Handbook of Di®erential Equations, Academic Press, Boston, 1997. |
| [17] | M. Sezer, M. Gulsu and Bekir Tanay, Rational Chebyshev collocation method for solving higher-order linear ordinary differential equations, Wiley Online Library ,DOI 10.1002/num.20573, 2010. |
APA Style
Mohamed A. Ramadan, Kamal R. Raslan, Mahmoud A. Nassar. (2014). An Approximate Analytical Solution of Higher-Order Linear Differential Equations with Variable Coefficients Using Improved Rational Chebyshev Collocation Method. Applied and Computational Mathematics, 3(6), 315-322. https://doi.org/10.11648/j.acm.20140306.15
ACS Style
Mohamed A. Ramadan; Kamal R. Raslan; Mahmoud A. Nassar. An Approximate Analytical Solution of Higher-Order Linear Differential Equations with Variable Coefficients Using Improved Rational Chebyshev Collocation Method. Appl. Comput. Math. 2014, 3(6), 315-322. doi: 10.11648/j.acm.20140306.15
AMA Style
Mohamed A. Ramadan, Kamal R. Raslan, Mahmoud A. Nassar. An Approximate Analytical Solution of Higher-Order Linear Differential Equations with Variable Coefficients Using Improved Rational Chebyshev Collocation Method. Appl Comput Math. 2014;3(6):315-322. doi: 10.11648/j.acm.20140306.15
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.acm.20140306.15,
author = {Mohamed A. Ramadan and Kamal R. Raslan and Mahmoud A. Nassar},
title = {An Approximate Analytical Solution of Higher-Order Linear Differential Equations with Variable Coefficients Using Improved Rational Chebyshev Collocation Method},
journal = {Applied and Computational Mathematics},
volume = {3},
number = {6},
pages = {315-322},
doi = {10.11648/j.acm.20140306.15},
url = {https://doi.org/10.11648/j.acm.20140306.15},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140306.15},
abstract = {The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.},
year = {2014}
}
TY - JOUR T1 - An Approximate Analytical Solution of Higher-Order Linear Differential Equations with Variable Coefficients Using Improved Rational Chebyshev Collocation Method AU - Mohamed A. Ramadan AU - Kamal R. Raslan AU - Mahmoud A. Nassar Y1 - 2014/12/27 PY - 2014 N1 - https://doi.org/10.11648/j.acm.20140306.15 DO - 10.11648/j.acm.20140306.15 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 315 EP - 322 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20140306.15 AB - The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy. VL - 3 IS - 6 ER -
Email: