ABSTRACT

In this chapter, the author is concerned with differential equations and the general use of vector matrix notation. There are many differential equations encountered in engineering and science whose order is higher than the first order. The author discusses the basic concepts involved in replacing a continuous system with a discrete one. He uses the simplest first-order system to illustrate some of the important properties of these basic concepts. The author analyzes three of the more common methods: the explicit, the fully implicit, and the Crank–Nicolson. Two of the important characteristics of any method are stability and accuracy. Accuracy is measured by comparing the approximation to the series expansion. The analyses of stability and accuracy can be extended to multi-degree-of-freedom systems, converting the system to decoupled first-order systems. The author provides examples to emphasize the differences between stability and accuracy.