Mathematical Models and Their Formulation

Abstract

With a few exceptions, the chapters of this handbook are concerned with mathematical methods useful in the quantitative analysis of problems in science and engineering.† An important and challenging aspect of any quantitative study of a real-life phenomenon is the formulation of mathematical problems which are relevant to a better understanding of the phenomenon and to which these mathematical methods can be applied. Real-life phenomena are usually too complex to be analyzed quantitatively without idealization and simplification. It is just not feasible or practical to follow the individual motions of trillions of molecules in a cubic centimeter of air or the evolution of billions of stars in a typical galaxy. For many practical purposes, however, information about a body of matter (or a galaxy) can be obtained by treating the collection of molecules (or stars) in that body as a “continuous medium” having properties, such as density, velocity, etc., that vary smoothly throughout the body. In this continuum model, the equilibrium or motion of the body under external forces and torques, for example, may be taken as a consequence of Euler’s laws of mechanics for continuous media. The mathematical methods described in this handbook may now be used to deduce from Euler’s law an initial/boundary-value problem for differential equations that governs the mechanical behavior of the continuum. Section 4.7 of this handbook gives a sample derivation of some relevant differential equations of this mathematical model, widely known as continuum mechanics, for the study of the mechanics of deformable bodies of matter.