The Physical Origins of Partial Differential Equations
Many important ideas in mathematics are developed within the framework of physical science, and mathematical equations, especially partial differential equations, provides the language to formulate these ideas. In reverse, advances in mathematics provides the stimulus for new advancements in science. Over the years mathematicians and scientists extended these methodologies to include nearly all areas of science and technology, and a paradigm emerged called mathematical modeling. A mathematical model is an equation, or set of equations, whose solution describes the physical behavior of the related physical system. In this context we say, for example, that Maxwell’s equations form a model for electromagnetic phenomena. Like most mathematical models, Maxwell’s equations are based on physical observations. But the model is so accurate, we regard the model itself as describing an actual physical law. Other models, for example a model of how a disease spreads in a population, are more conceptual. Such models often explain observations, but only in a highly limited sense. In general, a mathematical model is a simplified description, or caricature, of reality expressed in mathematical terms. Mathematical modeling involves observation, selection of relevant physical variables, formulation of the equations, analysis of the equations and simulation, and, finally, validation of the model to ascertain whether indeed it is predictive. The subject of partial differential equations encompasses all types of models, from physical laws like Maxwell’s equations in electrodynamics, to conceptual laws that describe the spread of an plant invasive species on a savanna.