The generalized differential quadrature rule for fourth‐order differential equations

Abstract

SUMMARYThe generalized di erential quadrature rule (GDQR) proposed here is aimed at solving high-order di erential equations. The improved approach is completely exempted from the use of the existing -point technique by applying multiple conditions in a rigorous manner. The GDQR is used here to static and dynamic analyses of Bernoulli-Euler beams and classical rectangular plates. Numerical error analysis caused by the method itself is carried out in the beam analysis. Independent variables for the plate are ÿrst deÿned. The explicit weighting coe cients are derived for a fourth-order di erential equation with two conditions at two di erent points. It is quite evident that the GDQR expressions and weighting coe cients for two-dimensional problems are not a direct application of those for one-dimensional problems. The GDQR are implemented through a number of examples. Good results are obtained in this work.

Solving high order ordinary differential equations with radial basis function networks

“…The unsymmetric IRBF collocation method is further verified here in the solution of eighth order ODE y [8] + y [7] + y [6] + y [5] + y [4] + y + y + y + y = 9 exp(x),…”

“…Sallam and El-Hawary [4]; Esmail et al [5]; Gutierrez and Laura [6]; Wu and Liu [7,8,9]). For example, the Cauchy problems governed by second order and fourth order equations were solved successfully using deficient spline function approximations by Sallam and El-Hawary [4] and Esmail et al [5] respectively.…”

“…For example, the Cauchy problems governed by second order and fourth order equations were solved successfully using deficient spline function approximations by Sallam and El-Hawary [4] and Esmail et al [5] respectively. Recently, the differential quadrature (DQ) method (Bellman and Casti [10]) was developed to solve fourth, sixth and eighth order ODEs without using the δ-point technique for the treatment of multiple boundary conditions in the papers of Wu and Liu [8,7,9] respectively.…”

Numerical Computation of Fractional Second-Order Sturm-Liouville Problems

“…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]. Historically, problem (11.)…”

Free Vibration Analysis of Rotating Functionally Graded Annular Disc of Variable Thickness Using Generalized Differential Quadrature Method

“…Compared to the other numerical methods, the DQ method can lead to almost accurate results using a considerably smaller number of grid points and, hence, requiring relatively little computational e ort [22,23]. The GDQ method is an improvement of the DQ method, especially for solving higher order di erential equations, which is more computationally e cient and accurate.…”