Advanced Numerical and Semi-Analytical Methods for Differential Equations

15Differential Quadrature Method

15.1 Introduction

We have already discussed Akbari–Ganji's method (AGM), exp‐function method, Adomian decomposition method (ADM), homotopy perturbation method (HPM), variational iteration method (VIM), and homotopy analysis method (HAM) in Chapters 9, 10, 11, 12, 13, and 14, respectively. This chapter presents another effective numerical method that approximates the solution of the PDEs by functional values at certain discretized points. This method can be applied with considerably less number of grid points, whereas the methods like finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) as given in Chapters 5, 6, and 7, respectively, may need more number of grid points to obtain the solution. However, FDM, FEM, and FVM are versatile methods which may certainly be used to handle regular as well as irregular domains.

Approximating partial derivatives by means of weighted sum of function values is known as Differential Quadrature (DQ). The differential quadrature method (DQM) was proposed by Bellman et al. [1–4]. Determination of weighted coefficients plays an important and crucial role in DQM. As such, an effective procedure for finding weighted coefficients suggested by Bellman is to use a simple algebraic formulation of weighted coefficient with the help of coordinates of grid points. Here, the coordinate of the grid points are the roots of the base functions like Legendre polynomials, Chebyshev polynomials, ...