A Differential Quadrature proposed here can be used to solve boundary-value and initial-value differential equations with a linear or nonlinear nature. Unlike the classic Differential Quadrature Method (DQM), the newly proposed Differential Quadrature chooses the function values and some derivatives wherever necessary as independent variables. Therefore, the d-type grid arrangement used in the classic DQM is exempt while applying the boundary conditions exactly. Most importantly, the explicit weighting coef®cients can be obtained using the proposed procedures. The present method is used to solve two types of differential equations which are the singlespan Bernoulli±Euler beam's buckling equation and the one-degree-of-freedom solid dynamic equation. Excellent results were obtained.

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.

The meshless local Petrov±Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants. It is, however, computationally expensive for some problems. A coupled MLPG/®nite element (FE) method and a coupled MLPG/boundary element (BE) method are proposed in this paper to improve the solution ef®ciency. A procedure is developed for the coupled MLPG/FE method and the coupled MLPG/BE method so that the continuity and compatibility are preserved on the interface of the two domains where the MLPG and FE or BE methods are applied. The validity and ef®ciency of the MLPG/FE and MLPG/BE methods are demonstrated through a number of examples.

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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.