Introduction

Abstract

In this first chapter we begin in Sect. 1.1 by introducing the partial differential equations and associated initial and boundary value problems that we shall study in the following chapters. The equations are classified into elliptic, parabolic, and hyperbolic equations, and we indicate the corresponding type of problems in physics that they model. We discuss briefly the concept of a well posed boundary value problem, and the various techniques used in our subsequent presentation. In Sect. 1.2 we introduce some notation and concepts that will be used throughout the text, and in Sect. 1.3 we include a detailed derivation of the heat equation from physical principles explaining the meaning of all terms that occur in the equation and the boundary conditions. In the problem section, Sect. 1.4, we add some further illustrative material.