The Pseudospectral Method for Solving Differential Eigenvalue Problems

A MATLAB differentiation matrix suite

“…The computation of spectral collocation differentiation matrices for derivatives of arbitrary order has been considered by Huang and Sloan [1994] (constant weights) and Welfert [1997] (arbitrary ␣͑x͒). The algorithm implemented in poldif.m and chebdif.m follows these references closely.…”

“…An accurate solution to the Orr-Sommerfeld equation was first obtained in 1971 by S. Orszag, who used a Chebyshev tau method [Orszag 1971]. Here we use the differentiation matrix approach considered in Huang and Sloan [1994]. Let D ͑4͒ be the fourth-derivative Chebyshev matrix that implements the clamped boundary conditions (60), and let D ͑2͒ be the second-derivative Chebyshev matrix with boundary conditions y͑Ϯ1͒ ϭ 0.…”

Fourier series expansion type of spectral collocation method for vibration analysis of cylindrical shells

“…It is known that physically spurious eigenvalues occur when solving fourth-order differential equations with spectral methods. A strategy to avoid the phenomenon is to use polynomials of different degrees [2,12]. For this reason, a displacement w n is approximated by an Hermite interpolating polynomial for its fourth-order derivative, and a Lagrange interpolating polynomial, whose degree is two less than that of the Hermite interpolating polynomial, for its second-order derivative.…”

“…Huang has proposed a form of an interpolating polynomial whose values and derivatives are zeros at boundary nodes so that clamped boundary conditions can be imposed [2]. It also has been introduced in Trefethen's literature [1].…”

Numerical Computation of Fractional Second-Order Sturm-Liouville Problems

“…Recently P. Antunes and R. Ferreira [16] constructed numerical schemes using radial basis functions while B. Jin and et [17] used Galerkin finite element method to solve the problem. In this paper, we develop a numerical technique for approximating the eigenvalues of the following nonsingular fractional Sturm-Liouville problem of the form [ ( ) ′ 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 [18], [19], [21], [26], [29], [31]. Note that Equation (1.1) is often referred to as the circular ring structure with constraints which has rectangular cross-sections of constant width and parabolic variable thickness; see [27] and [32].…”