## Approximations of Sturm-Liouville eigenvalues using differential quadrature (DQ) method.(English)Zbl 1092.65067

Summary: The polynomial-based differential quadrature and the Fourier expansion-based differential quadrature methods are applied to compute eigenvalues of the Sturm-Liouville problem. It is demonstrated through some examples that accurate results for the first $$k$$th eigenvalues of the problem, where $$k=1,2,3,\dots$$, can be obtained by using (at least) $$2k$$ mesh points. The results of this work are compared with other published results in the literature.

### MSC:

 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
Full Text:

### References:

 [1] Andrew, A.L., Asymptotic correction of computed eigenvalues of differential equations, Ann. numer. math., 1, 41-51, (1994) · Zbl 0823.65080 [2] Andrew, A.L., Asymptotic correction of Numerov’s eigenvalue estimates with natural boundary conditions, J. comput. appl. math., 125, 359-366, (2000) · Zbl 0970.65086 [3] Andrew, A.L.; Paine, J.W., Correction of Numerov’s eigenvalue estimates, Numer. math., 47, 289-300, (1985) · Zbl 0554.65060 [4] Andrew, A.L.; Paine, J.W., Correction of finite element estimates for sturm – liouville eigenvalues, Numer. math., 50, 205-215, (1986) · Zbl 0588.65062 [5] Bellman, R.; Kashef, B.G.; Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. comput. phys., 10, 40-52, (1972) · Zbl 0247.65061 [6] Bert, C.W.; Malik, M., Differential quadrature method in computational mechanics: a review, Appl. mech. rev., 49, 1-28, (1996) [7] Ghelardoni, P., Approximations of sturm – liouville eigenvalues using boundary value methods, Appl. numer. math., 23, 311-325, (1997) · Zbl 0877.65056 [8] Paine, J.W.; de Hoog, F.R.; Anderssen, R.S., On the correction of finite difference eigenvalue approximations for sturm – liouville problems, Computing, 26, 123-139, (1981) · Zbl 0436.65063 [9] Pryce, J.D., Numerical solution of sturm – liouville problems, (1993), Oxford University Press Oxford · Zbl 0795.65053 [10] C. Shu, Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, Ph.D. Thesis, University of Glasgow, UK, 1991. [11] Shu, C., Differential quadrature and its application in engineering, (2000), Springer Berlin · Zbl 0944.65107 [12] Shu, C.; Khoo, B.C.; Chew, Y.T.; Yeo, K.S., Numerical studies of unsteady boundary layer flows past an impulsively started circular cylinder by GDQ and GIQ approaches, Comput. methods appl. mech. engrg., 135, 229-241, (1996) · Zbl 0894.76066 [13] Shu, C.; Richards, B.E., Application of generalized differential quadrature to solve two-dimensional incompressible navier – stokes equations, Internat. J. numer. methods fluids, 15, 791-798, (1992) · Zbl 0762.76085 [14] Vanden Berghe, G.; De Meyer, H., Accurate computation of higher sturm – liouville eigenvalues, Numer. math., 59, 243-254, (1991) · Zbl 0716.65079 [15] Vanden Berghe, G.; De Meyer, H.; Van Daele, M., A parallel approach to the modified numerov-like eigenvalue determination for sturm – liouville problems, Comput. math. appl., 23, 12, 69-74, (1992) · Zbl 0773.65066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.