Application of the generalized differential quadrature rule to initial–boundary-value problems

Partial differential equations (PDEs) for the forced vibration of structural beams are solved in this paper using the recently proposed generalized differential quadrature rule (GDQR). The GDQR techniques are first applied to both spatial and time dimensions simultaneously as a whole. No other classical methods are needed in the time dimension. The objective of this paper is to formularize the GDQR expressions and corresponding explicit weighting coefficients, while the derivation of explicit weighting coefficients is one of the most important aspects in the differential quadrature methods. It should be emphasized that the GDQR expressions and weighting coefficients for two-dimensional problems are not a direct application of those for one-dimensional problems, and they are distinctly different for PDEs of different orders. An Euler beam and a Timoshenko beam are employed as examples. Accurate results are obtained. The proposed procedures can be applied to problems in other disciplines of sciences and technology, where the problems may be governed by other PDEs with different orders in the time or spatial dimension.