Solving differential equations by numerical methods: Differential quadrature

Abstract

This thesis presents numerical schemes to solve differential equations using the method of polynomial differential quadrature. Differential quadrature (DQ) is a discrete derivative approximation method which provides highly accurate numerical solutions for differential equations. Determination of the weighting coefficients is investigated in detail, which is essential for the implementation of the present method. Simplified formulations of these coefficients are also stated by using zeros of the Chebyshev polynomials. The implementation of DQ to linear or nonlinear ordinary differential equations is established by producing the set of algebraic equations. The application of DQ to partial differential equations results in systems of time-dependent differential equations. These time dependent differential equations systems are integrated by the Runge-Kutta method. The DQ method is then applied to solve five different time-dependent partial differential equations. Each of the five equation is solved numerically by giving illustrative test problems with the stability analysis. Linear and nonlinear diffusion equations, reaction-diffusion equation in non-homogeneous media, the Kuramoto-Sivashinsky equation, the inviscid Burgers' equation and the Ginzburg-Landau equation are studied by using the proposed method. Solutions accomplished by the present scheme are compared to the analytical or well-known numerical methods.