Application of generalized differential quadrature rule to sixth-order differential equations

Abstract

A generalized di erential quadrature rule (GDQR) has been proposed as a general numerical method to solve high-order di erential equations. Applications are given to sixth-order di erential equations that govern the free vibration analysis of ring structures. The DQM uses the function values at all grid points as independent variables in the establishment of algebraic equations. This leads to di culties in implementing the boundary conditions at a point. The GDQR regards the function values at all grid points and their derivatives at grid points wherever necessary as independent variables. The given multiple conditions at any point can therefore be imposed in the GDQR in a direct manner. A procedure is also proposed for calculating GDQR's explicit weighting coe cients for an e ective establishment of algebraic equations.

Solving high order ordinary differential equations with radial basis function networks

“…The vibration problem governed by the sixth order differential equation was studied by Gutierrez and Laura [6] using the differential quadrature method (DQM) and the optimized Rayleigh-Ritz method and by Wu and Liu [7] using the generalized differential quadrature rule (GDQR).…”

“…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),…”

“…Firstly, the initial value problem governed by (55) with initial values y(0) = y (0) = y (0) = y (0) = y [4] (0) = y [5] (0) = y [6] (0) = y [7] (0) = 1,…”

“…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.…”