Application of the generalized differential quadrature rule to eighth‐order differential equations

Abstract

SUMMARYThis work extends the application of the generalized di erential quadrature rule (GDQR) to an eighthorder boundary-value di erential equation with four boundary conditions at boundaries. The di erential quadrature expression and explicit weighting coe cients for the eighth-order di erential equation are formulated for a ÿrst time to implement the GDQR more accurately. A circular cylindrical single-barrel roof is employed as an example. The numerical results show good accuracy and convergence with only a few sampling points. The application of the GDQR is straightforward and has clear advantages in the implementation of multiple boundary conditions over the existing -point technique. The GDQR has demonstrated itself as a general numerical method to solve high-order di erential equation with multiple boundary conditions.

Solving high order ordinary differential equations with radial basis function networks

“…This problem is taken from reference (Kelkar and Sewell [33]) and was studied by Wu and Liu [9] using the GDQR (Figure 10 …”

“…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.…”

“…Results obtained (V h and its derivatives) are then substituted into (65)-(69) to produce the force and moment resultants. The analytic homogeneous results can be found in (Kelkar and Sewell [33]) and (Wu and Liu [9]). …”

“…To obtain higher accuracy, Wu and Liu [9] recalculated those values. The analytic homogeneous results of Wu and Liu [9] are used here for comparison. Tables 6-8 present the final computed force and moment resultants using 5, 9 and 13 equally spaced points.…”