Applied Mathematical Sciences, Vol. 1, 2007, no. 25, 1217 - 1229 Approximate Eigenvalues of Periodic Sturm-Liouville Problems Using Differential Quadrature Method U˘ gur Y¨ ucel Department of Mathematics, Faculty of Science and Art, Pamukkale University, 20020 Denizli, Turkey uyucel@pau.edu.tr Abstract Computation of eigenvalues of regular Sturm-Liouville problems with periodic boundary conditions is considered. It is shown through nu- merical illustrations that both polynomial-based differential quadra- ture (PDQ) and Fourier expansion-based differential quadrature (FDQ) methods can be successfully used to accurately predict the first kth (k =1, 2, 3, ··· ) eigenvalues of the problem using (at least) 2k mesh points in the computational domain. The errors in computed solutions are compared with other published results in the literature for the nu- merical illustrations considered in this work. Mathematics Subject Classification: 34L16, 65L10, 65L15 Keywords: Eigenvalues, Differential quadrature method, Periodic bound- ary conditions, Schr¨ odinger equation, Sturm-Liouville problem 1 Introduction Problems in the fields of elasticity and vibration, including applications of the wave equations of modern physics, fall into a special class of boundary-value problems known as characteristic-value problems. Certain problems in statics also reduce to such problems. In this work, we consider a special type of boundary-value problem called Sturm-Liouville problem. A general Sturm-Liouville problem consists of the linear homogeneous second-order differential equation of the form d dx  p(x) dy dx  +  r(x)λ - q 1 (x)  y =0. (1)