Mathematical Methods in Interdisciplinary Sciences

14Generalized Differential and Integral Quadrature: Theory and Applications

Francesco Tornabeneand Rossana Dimitri

Department of Innovation Engineering, University of Salento, Lecce, Italy

14.1 Introduction

In the past decades, the simplicity and versatility of the differential quadrature (DQ) method, as proposed by Bellman at the beginning of the 1970s, has gradually increased its application in many branches of engineering and science. Remarkably, the DQ approach represents an efficient numerical tool to solve complex differential equations, and it yields accurate results even with a limited number of collocation points discretizing the domain. This limits many common computational difficulties, especially in a context where nonlinear algorithms and high computational effort are required among complex engineering applications. More in detail, this procedure approximates derivatives of a function at a certain point through a linear sum of all function values in the definition domain, whereby a key aspect is related to the evaluation of weighting coefficients.

In some pioneering works by Bellman and coworkers, two approaches were proposed to compute these weighting coefficients for the first‐order derivatives of functions. The first method was based on an algebraic system of equations featuring ill‐conditioning drawbacks, whereas the second method was a simple algebraic formulation that assumes the roots of shifted Legendre polynomials as grid points. In the preliminary DQ‐based ...