New insights in solving distributed system equations by the quadrature method-I. Analysis

This paper provides new insights in solving distributed system equations by the method of differential quadrature (Bellman et al., 1972). In this study, the method is shown to be essentially equivalent to the general collocation method (Finlayson and Scriven, 1966). Explicit formulae of the quadrature coefficients are derived for arbitrarily-distributed nodes and for nodes located at the zeros of an orthogonal polynomial. Due to their simplicity and flexibility, these formulae allow us to calculate the values of the coefficients accurately and efficiently. Since the accuracy of the equation solutions depends on the locations of the nodes, a systematic procedure for placing the grid points is developed. In Part II of this paper, this proposed approach is demonstrated to be superior than that of the orthogonal collocation. Methods for approximating partial derivatives in a symmetric problem are also proposed to minimize the computational effort. The above techniques are then extended to cases in which the given partial differential equation and associated constraints can not be transformed into an initial-value problem. Instead of the conventional approach of converting the problem into a large set of algebraic equations, a two-point boundary-value problem is formulated to reduce the number of iteration parameters. This strategy is especially suitable for cases with highly nonlinear equations.