ABSTRACT
The advent of high-speed digital computers has made it both feasible and, indeed, easy to perform numerical approximation of solutions. The subject of the numerical solution of differential equations is a highly developed one, and is applied daily to problems in engineering, physics, biology, astronomy, and many other parts of science. Not surprisingly—and like many of the other fundamental ideas related to calculus—the basic techniques for the numerical solution of differential equations go back to Newton and Euler. The chapter introduces the most basic ideas in the subject of numerical analysis of differential equations. The notion of error is central to any numerical technique. Numerical methods only give approximate answers. Any type of approximation argument involves some sort of loss of information; that is to say, there will always be an error term. It is also the case that the numerical approximation techniques can give rise to instability phenomena and other unpredictable behavior.
