This book represents an attempt to implement a general approach that in essence views the theory of partial differential equations (PDEs) of mathematical physics as the language of continuous processes, that is, an interdisciplinary science that considers the hierarchy of mathematical phenomena as a reflection of their physical counterparts. A comprehensive, mathematically rigorous account of the classical theory of PDEs in mathematical physics is thus inseparably bound with the features of the corresponding natural continuum objects. We shall therefore endeavor to trace the simultaneous origins of some basic mathematical objects in different natural contexts (continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics). In parallel, we shall trace the interrelation between different types of problems (elliptic, parabolic, and hyperbolic) as mathematical counterparts of their natural prototypes: steady-state and evolutionary processes (dissipative and conservative). This will be done by an asymptotic analysis of the behavior of these processes in time and their dependence on the relevant governing parameters.

In view of the almost complete absence of a physics background in undergraduate and graduate curricula of mathematics and applied mathematics, it seems important, in a course of mathematical physics, to provide an introduction to the basic concepts of different natural sciences and their relation with PDEs in terms of certain typical boundary-value problems that recur in different scientific contexts. Chapters 1 and 2 are therefore addressed primarily to students of mathematics.