Numerical solution of the Sturm-Liouville problem by using Chebyshev cardinal functions
Subject Areas : StatisticsM. Shahriari 1 , B. Nemati Saray 2 , F. Pashaie 3
1 - Department of Mathematics, Faculty of Science, University of Maragheh, P.O. Box 55181-83111, Maragheh, Iran
2 - Faculty of Mathematics, Zanjan University of Basic Sciences, Zanjan, postcode, 45137-66731 Iran
3 - Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran
Keywords: مسأله اشتورم-لیوویل, چندجملهای چبیشف, تابع کاردینال چبیشف, ماتریس عملیاتی مشتق,
Abstract :
In this manuscript, a numerical technique is presented for finding the eigenvalues of the regular Sturm-Liouville problems. The Chebyshev cardinal functions are used to approximate the eigenvalues of a regular Sturm-Liouville problem with Dirichlet boundary conditions.These functions defined by the Chebyshev function of the first kind. By using the operational matrix of derivative the problem is reduced to a set of algebraic equation. Finally we use some numerical examples to show that this method include to demonstrate the validity and applicability of technique.
1. Bujurke, N., C. Salimath, and S. Shiralashetti, Computation of eigenvalues and solutions of regular Sturm–Liouville problems using Haar wavelets. Journal of Computational and Applied Mathematics, 2008. 219(1): p. 90-101.
2. Hille, E., Lectures on ordinary differential equations. 1969.
3. Freiling, G. and V.A. Yurko, Inverse Sturm-Liouville problems and their applications. 2001: NOVA Science Publishers New York.
4. Condon, D., Corrected finite difference eigenvalues of periodic Sturm–Liouville problems. Applied numerical mathematics, 1999. 30(4): p. 393-401.
5. Khmelnytskaya, K., H. Rosu, and A. González, Periodic Sturm–Liouville problems related to two Riccati equations of constant coefficients. Annals of Physics, 2010. 325(3): p. 596-606.
6. Jokar, M., M. Lakestani, and M. Shahriari, Computation of half-eigenvalues periodic sturm-liouville problem using trigonometric wavelets. International Journal of Nonlinear Science, 2012. 13(4): p. 495-504.
7. Yücel, U., Approximate Eigenvalues of Periodic Sturm-Liouville Problems Using Differential Quadrature Method. Applied Mathematical Sciences, 2007. 1(25): p. 1217-1229.
8. Ghelardoni, P., G. Gheri, and M. Marletta, Spectral corrections for Sturm–Liouville problems. Journal of computational and applied mathematics, 2001. 132(2): p. 443-459.
9. Ledoux, V., M. Van Daele, and G.V. Berghe, Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics. Computer Physics Communications, 2009. 180(2): p. 241-250.
10. El-Daou, M.K. and N.R. Al-Matar, An improved Tau method for a class of Sturm–Liouville problems. Applied Mathematics and Computation, 2010. 216(7): p. 1923-1937.
11. Altıntan, D. and Ö. Uğur, Variational iteration method for Sturm–Liouville differential equations. Computers & Mathematics with Applications, 2009. 58(2): p. 322-328.
12. El-Gamel, M., Numerical comparison of sinc-collocation and Chebychev-collocation methods for determining the eigenvalues of Sturm–Liouville problems with parameter-dependent boundary conditions. SeMA Journal, 2014. 66(1): p. 29-42.
13. Lakestani, M. and B.N. Saray, Numerical solution of telegraph equation using interpolating scaling functions. Computers & Mathematics with Applications, 2010. 60(7): p. 1964-1972.
14. Chen, L. and H.-P. Ma, Approximate solution of the Sturm–Liouville problems with Legendre–Galerkin–Chebyshev collocation method. Applied Mathematics and Computation, 2008. 206(2): p. 748-754.
15. Dehghan, M. and M. Lakestani, The use of Chebyshev cardinal functions for solution of the second‐order one‐dimensional telegraph equation. Numerical Methods for Partial Differential Equations, 2009. 25(4): p. 931-938.
16. Lakestani, M. and M. Dehghan, Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions. Computer Physics Communications, 2010. 181(5): p. 957-966.
17. Trif, D., Matlab package for the Schrödinger equation. Journal of Mathematical Chemistry, 2008. 43(3): p. 1163-1176.
18. Miller, K. R., Introduction to Differential Equations. 1991: Prentice Hall.-
