Mathematical Fundamentals of Differential Quadrature Method: Linear Vector Space Analysis and Function Approximation

Abstract

Most engineering problems are governed by a set of partial differential equations (PDEs) with proper boundary conditions. For example, Newtonian fluid flows are modeled by the Navier-Stokes equations; the vibration of thin plates is governed by a fourth order partial differential equation; whereas acoustic waves and microwaves can be simulated by the Helmholtz equation. In general, it is very difficult for us to obtain the closed-form solution of these equations. On the other hand, the solution of these PDEs is always demanded due to practical interests. For example, when we design an aircraft, we need to know the curve of c_l (lift coefficient) versus c_d (drag coefficient) for a given airfoil shape. The values of c_l and c_d can be obtained from the solution of Navier-Stokes equations. Therefore, it is important for us to develop some approximate solutions to the given PDEs.