Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem
Applied and Computational Mathematics
Volume 4, Issue 5, October 2015, Pages: 369-373
Received: Aug. 31, 2015;
Accepted: Sep. 11, 2015;
Published: Sep. 22, 2015
Views 3923 Downloads 127
Dong Yun Shen, Department of Mathematics, Foshan University, Foshan, Guangdong, China
Yong Huang, Department of Mathematics, Foshan University, Foshan, Guangdong, China
This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples of estimating eigenvalues are given. By comparison with the exact results in open literatures, the correctness and effectiveness of the present approach are verified.
Dong Yun Shen,
Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem, Applied and Computational Mathematics.
Vol. 4, No. 5,
2015, pp. 369-373.
MA Al-Gwaiz, Sturm–Liouville Theory and its Applications, Springer-Verlag, London, 2008.
Veerle Ledoux, Study of special algorithms for solving Sturm–Liouville and Schrödinger equations. Ph.D Thesis, Department of Applied Mathematics and Computer Science, Ghent University, 2007.
Alan L Andrew, Correction of finite difference eigenvalues of periodic Sturm–Liouville problems, J. Austral. Math. Soc. Ser. B 30 (1989) 460-469.
Paolo Ghelardoni, Approximations of Sturm--Liouville eigenvalues using boundary value methods, Appl. Numer. Math. 23 (1997) 311--325.
DJ Condon, Corrected finite difference eigenvalues of periodic Sturm–Liouville problems, Appl. Numer. Math. 30 (1999) 393–401.
İbrahim Çelik, Approximate computation of eigenvalues with Chebyshev collocation method, Appl. Math. Comput. 168 (2005) 125–134.
İbrahim Çelik, Guzin Gokmen, Approximate solution of periodic Sturm–Liouville problems with Chebyshev collocation method, Appl. Math. Comput. 170 (2005) 285–295.
Quan Yuan, Zhiqing He, Huinan Leng, An improvement for Chebyshev collocation method in solving certain Sturm–Liouville problems, Appl. Math. Comput. 195 (2008) 440–447.
Lan Chen, He-Ping Ma, Approximate solution of the Sturm–Liouville problems with Legendre–Galerkin–Chebyshev collocation method, Appl. Math. Comput. 206 (2008) 748–754.
Xuecang Zhang, Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm–Liouville problems, Appl. Math. Comput. 217 (2010) 2266-2276.
Saeid Abbasbandy, A new application of the homotopy analysis method: Solving the Sturm–Liouville problems, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 112–126.
Mohamed El-gamel, Mahmoud Abd El-hady, Two very accurate and efficient methods for computing eigenvalues of Sturm–Liouville problems, Appl. Math. Modell. 37 (2013) 5039–5046.
Alan L Andrew, Twenty years of asymptotic correction for eigenvalue computation, ANZIAM J. 42 (2000) C96–C116.