TABLE OF CONTENTS Various phenomena treated in science and engineering are often described in terms of differential equations formulated by using their continuum mechanics models. Solving differential equations under various conditions such as boundary or initial conditions leads to the understanding of the phenomena and can predict the future of the phenomena (determinism). Exact solutions for differential equations, however, are generally difficult to obtain. Numerical methods are adopted to obtain approximate solutions for differential equations. Among these numerical methods, those which approximate continua with infinite degree of freedom by a discrete body with finite degree of freedom are called "discrete analysis." Popular discrete analyses are the finite difference method, the method of weighted residuals, and the Rayleigh-Ritz method. Via these methods of discrete analysis, differential equations are reduced to simultaneous linear algebraic equations and thus can be solved numerically.
This chapter will explain first the method of weighted residuals and the Rayleigh-Ritz method which furnish a basis for the finite-element method (FEM) by taking examples of one-dimensional boundary-value problems, and then will compare the results with those by the one-dimensional FEM in order to acquire a deeper understanding of the basis for the FEM.
Differential equations are generally formulated so as to be satisfied at any points which belong to regions of interest. The method of weighted residuals determines the approximate solution ? to a differential equation such that the integral of the weighted error of the differential equation of the approximate function ? over the region...
