The Physical Origins of Partial Differential Equations

Abstract

Many important ideas of mathematics were developed within the framework of physical science. For example, calculus has its origins in efforts to accurately describe the motion of bodies. Mathematical equations have always provided a context in which to formulate concepts in physics- Maxwell’s equations describe electrodynamical phenomena, Newton’s equations describe mechanical systems, Schrödinger’s equation describes aspects of quantum mechanics, and so on. In recent years, however, mathematicians and scientists have extended these types of connections to include nearly all areas of science and technology, and a new field has emerged called mathematical modeling. A mathematical model is an equation, or set of equations, whose solution describes the physical behavior of a related physical system. In this context, we say, for example, that Maxwell’s equations are a model for electrodynamical phenomena. Like most mathematical models, Maxwell’s equations are based on experiment and physical observations. In general, a mathematical model is a simplified description of physical reality expressed in mathematical terms. Mathematical modeling involves physical observation, selection of the relevant physical variables, formulation of the equations, analysis of the equations and simulation, and, finally, validation of the model.