Introduction to Partial Differential Equations (Herman)

Introduction

We begin our study of partial differential equations with first order partial differential equations. Before doing so, we need to define a few terms.

Linear Constant Coefficient Equations

Let’s consider the linear first order constant coefficient partial differential equation

Conservation Laws

There are many applications of quasilinear equations, especially in fluid dynamics. The advection equation is one such example and generalizations of this example to nonlinear equations leads to some interesting problems. These equations fall into a category of equations called conservation laws. We will first discuss one-dimensional (in space) conservations laws and then look at simple examples of nonlinear conservation laws.

Shock Waves

Solutions of nonlinear advection equations can become multivalued due to a gradient catastrophe. Namely, the derivatives and become undefined. We would like to extend solutions past the catastrophe. However, this leads to the possibility of discontinuous solutions. Such solutions which may not be differentiable or continuous in the domain are known as weak solutions. In particular, consider the initial value problem.

Traffic Flow

An interesting application is that of traffic flow. We had already derived the flux function. Let’s investigate examples with varying initial conditions that lead to shock or rarefaction waves. As we had seen earlier in modeling traffic flow, we can consider the flux function

The Charpit equations. These were named after the French mathematician Paul Charpit Villecourt, who was probably the first to present the method in his thesis the year of his death, 1784. His work was further extended in 1797 by Lagrange and given a geometric explanation by Gaspard Monge (1746-1818) in 1808. This method is often called the Lagrange-Charpit method.