The Chebyshev collocation-path following method for solving sixth-order Sturm–Liouville problems Qasem M. Al-Mdallal ⇑ , Muhammed I. Syam Department of Mathematical Sciences, UAE University, College of Science, P.O. Box 17551, Al-Ain, United Arab Emirates article info Keywords: Sixth-order Sturm–Liouville problem A Chebyshev collocation method Eigenvalues Path following method abstract In this paper, we implement a Chebyshev collocation method to approximate the eigen- values of nonsingular sixth-order Sturm–Liouville problem. This method transforms the Sturm–Liouville problem to a sparse singular linear system which is solved by the path following technique. Numerical results demonstrate the accuracy and efficiency of the present algorithm. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction In this paper, we develop a numerical technique for approximating the eigenvalues of the following non-singular sixth- order Sturm–Liouville problem pðxÞy 000 ðxÞ ½  000 ¼ sðxÞy 00 ðxÞ ½  00  rðxÞy 0 ðxÞ ½  0  kwðxÞ qðxÞ ½ yðxÞ; ð1:1Þ subject to a j y ðjÞ ð1Þþ b j y ðjÞ ð1Þ¼ 0; ðj ¼ 0; ... ; 5Þ; ð1:2Þ where p; s; r; w and q are piecewise continuous functions with pðxÞ > 0; and wðxÞ P 0 for all x 2 ð1; 1Þ. Here a j and b j (for j ¼ 0; ... ; 5) are constants. The numerical solution of eigenvalue problems has received considerable interest in recent years because they have large number of applications in different areas of physics and engineering. A few examples of such applications are pendulums, vibrating and rotating shafts, viscous flow between rotating cylinders, the thermal instability of fluid spheres and spherical shells, earth’s seismic behavior and ring structures; for more details see [1,5,12,10,14,20]. Note that Eq. (1) is often referred to as the circular ring structure with constraints which has rectangular cross-sections of constant width and parabolic variable thickness; see [11,21]. In the literature, problem (1)-(2) had been studied theoretically by [6,7] who showed that its eigenvalues form a countable, increasing sequence k 0 6 k 1 6 k 2 6  with lim n!1 k n ¼1; and each eigenvalue has multiplicity at most 3. However, the numerical treatment of such problems has always been far from trivial which, therefore, attracts several researchers to initiate and/or implement different numerical methods to obtain http://dx.doi.org/10.1016/j.amc.2014.01.083 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: q.almdallal@uaeu.ac.ae (Q.M. Al-Mdallal), m.syam@uaeu.ac.ae (M.I. Syam). Applied Mathematics and Computation 232 (2014) 391–398 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc