chapter  1
42 Pages

Approximation and Differential Quadrature

Differential Quadrature (DQ) is a numerical method for evaluating derivatives of a sufficiently smooth function, proposed by Bellman and Casti in 1971. The basic idea of DQ comes from Gauss Quadrature, a useful numerical integration method. Gauss quadrature is characterized by approximating a definite integral with a weighted sum of integrand values at a group of so-called Gauss points. Extending it to finding the derivatives of various orders of a sufficiently smooth function gives rise to DQ (Bellman and Casti, 1971; Bellman et al., 1972). In other words, the derivatives of a smooth function are approximated with weighted sums of the function values at a group of so-called nodes. It should be noted that node is also called grid point or mesh point by various authors. Throughout the book all these names are used without distinction. Differential Quadrature can be formulated either through approximation

theory or solving a system of linear equations. In their original paper, Bellman and Casti (1971) used the latter to derive DQ. Throughout the book, however, we will employ the former to formulate DQ and DQ methods for the sake of simplicity. Thus in this chapter, an introduction to function approximation theory is first briefed in sections 1.1 and 1.2, followed by the fundamentals of the direct differential quadrature method in the subsequent of the sections.