The development of the high-speed digital computer has had, and continues to have, a revolutionary effect on modern applied science. Immediate evidence is available in the form of a large number of computer-generated numerical solutions of fundamental, unsolved systems of mathematical equations. The diversity of fields being affected includes lunar and planetary astrodynamics, wave diffraction, shock waves, laminar flow of liquids, free-surface fluid flow, weather prediction, thermodynamics, elasticity, electrostatic and gravitational potential, optimal control, n-body problems, vibration theory, molecular interaction, quantum theory, and relativistic collapse. Less obviously, there have been natural, qualitative changes in related mathematical models and theories.

This book attempts to develop a broad spectrum of applications that can be formulated as problems in differential equations in the real domain. Existing analytical theories and techniques will be summarized appropriately so that the reader will understand when he should not use a computer. For those problems which cannot be solved analytically, we will develop finite difference, computeroriented-numerical methods for approximating solutions. Indeed, if a computer algorithm is defined as a finite sequence of computer operations designed to yield an approximate solution of a given mathematical problem, then this book is concerned primarily with the development of computer algorithms. In this connection,it must be understood that the immense power of the modern digital computer lies in its ability to perform arithmetic operations and to store and retrieve numbers with exceptional speed.