Higher Order Compact Finite Difference Method For Fourth-Order Euler-Bernoulli Beam Partial Differential Equation

Abstract

In this thesis, we develop a higher order compact finite difference method to solve numerically a fourth-order partial differential equation that governs the transverse dis- placement of the Euler–Bernoulli beam equations. We develop an eight-order compact finite difference approximation for both the spatial derivative and the time derivative. An advantage of this method is that it completely satisfies the boundary conditions, eliminating the need for further approximations at the boundaries and it is easy to im- plement. Moreover, the proposed method is eight-order accurate and is based upon a single compact stencil. The discretizing procedure transforms the initial boundary value problem of the partial differential equation into a linear system of algebraic equations that can be solved by matrix inversion method. The stability, consistency and conver- gence were analyzed. Error analysis of compact finite difference method is based on Taylor’s theorem. The test example confirms the eight-order compact finite difference method has small maximum absolute error when compared with those already avail- able in the literature. The results indicate that the eight-order compact finite difference method is an applicable technique and approximates the exact solution very well.