Introduction to Modeling and Simulation

12Partial Differential Equation Models

12.1 Introduction

In this chapter we discuss systems modeled by partial differential equations (PDEs), also called distributed‐parameter systems. Whereas ordinary differential equations (ODEs) describe the time evolution of a single vector function , PDEs involve functions of multiple variables , that may include both time and spatial variables. PDEs arise in applications involving sound, heat transfer, vibrating strings and plates, fluid flow, electromagnetics, transmission lines, gravity and other physical phenomena, including quantum mechanics.

12.1.1 Existence and Uniqueness of Solutions

The questions of existence and uniqueness of solutions of PDEs are more difficult than for ODEs. A general mathematical discussion of existence and uniqueness of solutions of PDEs is beyond the scope of this text. We will instead describe a few of the most important classical linear PDE systems, their applications, and their solutions.

In particular, we will discuss the wave equation, the heat equation, and Laplace's equation as important examples of linear second‐order PDEs of the form

(12.1)

where the unknown function is a function of two variables ...