Fourier Expansion-based Differential Quadrature (FDQ)

Abstract

In Chapter 1, we have shown that the solution of a PDE can be approximated by a polynomial of high degree or by the Fourier series expansion, depending on the feature of the problem. It is noted that the expressions for polynomial approximation and Fourier series expansion are quite different. Thus, when the DQ approximation is applied to these two cases, the formulations to compute the weighting coefficients in the DQ approximation may be different. In Chapter 2, the computation of the weighting coefficients for polynomial-based differential quadrature (PDQ) was described in detail. In PDQ, the solution of a PDE is approximated by a high degree polynomial. The polynomial approximation is suitable for most engineering problems. However, for some problems, especially for those with periodic behaviors such as the Helmholtz problems, polynomial approximation may not be the best choice for the true solution. In contrast, Fourier series expansion could be the best approximation. In this chapter, we will demonstrate that using Fourier series expansion and linear vector space analysis, the weighting coefficients in the Fourier expansion-based differential quadrature (FDQ) can also be calculated by explicit formulations.