Introduction

This chapter begins with the discussion of the basic notion of partial differential equation (PDE). It provides a definite real physical process, interested to describe it in a deterministic, unique way; hence finding the general solution of the corresponding PDE is not the final goal Existence and uniqueness of the solution, and its continuous dependence on the initial data and the heating rate mean that model is physically sound, or well-posed. The chapter discusses the initial and boundary conditions for that equation, which make the initial-boundary value problem well-posed. Generally, if a PDE contains only derivatives with respect to one variable, it can be solved by the same methods which are used for solving ordinary differential equation (ODE), and the solution looks like a solution of an ODE with constants replaced by functions. Thus, in a contradistinction to general solutions of ODEs that contain arbitrary numbers, general solutions of PDEs contain arbitrary functions.