The Chebyshev collocation-path following method for solving sixth-order Sturm-Liouville problems

In this paper, we implement a Chebyshev collocation method to approximate the eigenvalues 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.

[1]  M. Syam,et al.  An efficient technique for finding the eigenvalues of fourth-order Sturm-Liouville problems , 2009 .

[2]  Guirong Liu,et al.  The generalized differential quadrature rule for fourth‐order differential equations , 2001 .

[3]  Ibrahim Çelik,et al.  Approximate calculation of eigenvalues with the method of weighted residuals-collocation method , 2005, Appl. Math. Comput..

[4]  Quan Yuan,et al.  An improvement for Chebyshev collocation method in solving certain Sturm-Liouville problems , 2008, Appl. Math. Comput..

[5]  Ibrahim Çelik,et al.  Approximate solution of periodic Sturm-Liouville problems with Chebyshev collocation method , 2005, Appl. Math. Comput..

[6]  D. Lesnic,et al.  An Efficient Method for Sixth-order Sturm-Liouville Problems , 2007 .

[7]  Edward H. Twizell,et al.  Numerical methods for special nonlinear boundary-value problems of order 2 m , 1993 .

[8]  Wesley H. Huang,et al.  The pseudospectral method for solving di8erential eigenvalue problems , 1994 .

[9]  C. P. Gupta,et al.  Existence and uniqueness results for the bending of an elastic beam equation at resonance , 1988 .

[10]  M. Scott.  An initial value method for the eigenvalue problem for systems of ordinary differential equations , 1973 .

[11]  Donal O'Regan,et al.  Solvability of some fourth (and higher) order singular boundary value problems , 1991 .

[12]  Edward H. Twizell,et al.  Finite-difference methods for twelfth-order boundary-value problems , 1991 .

[13]  M. Sezer,et al.  Chebyshev polynomial solutions of linear differential equations , 1996 .

[14]  Ibrahim Çelik,et al.  Approximate computation of eigenvalues with Chebyshev collocation method , 2005, Appl. Math. Comput..

[15]  Marco Marletta,et al.  Algorithm 775: the code SLEUTH for solving fourth-order Sturm-Liouville problems , 1997, TOMS.

[16]  Guo-Wei Wei,et al.  A note on the numerical solution of high-order differential equations , 2003 .

[17]  M. Marletta,et al.  Oscillation Theory and Numerical Solution of Sixth Order Sturm--Liouville Problems , 1998 .

[18]  P.A.A. Laura,et al.  Vibrations of non-uniform rings studied by means of the differential quadrature method , 1995 .

[19]  Marco Marletta,et al.  Oscillation theory and numerical solution of fourth-order Sturm—Liouville problems , 1995 .

[20]  M. Syam.  Finding all real zeros of polynomial systems using multi-resultant , 2004 .

[21]  Leon Greenberg,et al.  An oscillation method for fourth-order, self-adjoint, two-point boundary value problems with nonlinear eigenvalues , 1991 .