1.1: 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.

Recall (see the appendix on differential equations) that an \(n\)-th order ordinary differential equation is an equation for an unknown function \(y(x)\) that expresses a relationship between the unknown function and its first \(n\) derivatives. One could write this generally as

\[\label{eq:1}F(y^{(n)}(x),y^{(n-1)}(x),\ldots ,y'(x),y(x),x)=0.\]

Here \(y^{(n)} (x)\) represents the \(n\)th derivative of \(y(x)\). Furthermore, and initial value problem consists of the differential equation plus the values of the first \(n − 1\) derivatives at a particular value of the independent variable, say \(x_0\):

\[\label{eq:2}y^{(n-1)}(x_0)=y_{n-1},\quad y^{(n-2)}(x_0)=y_{n-2},\quad\ldots ,\quad y(x_0)=y_0.\]

If conditions are instead provided at more than one value of the independent variable, then we have a boundary value problem.

If the unknown function is a function of several variables, then the derivatives are partial derivatives and the resulting equation is a partial differential equation. Thus, if \(u = u(x, y,\ldots )\), a general partial differential equation might take the form

\[\label{eq:3}F\left(x,y,\ldots ,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\ldots ,\frac{\partial^2u}{\partial x^2},\ldots\right)=0.\]

Since the notation can get cumbersome, there are different ways to write the partial derivatives. First order derivatives could be written as

\[\frac{\partial u}{\partial x},u_x,\partial_xu,D_xu.\nonumber\]

\[\frac{\partial^2u}{\partial x^2},u_{xx},\partial_{xx}u,D_x^2u.\nonumber\]

\[\frac{\partial^2u}{\partial x\partial y}=\frac{\partial^2u}{\partial y\partial x},u_{xy},\partial_{xy}u,D_yD_xu.\nonumber\]

Note, we are assuming that \(u(x, y,\ldots )\) has continuous partial derivatives. Then, according to Clairaut’s Theorem (Alexis Claude Clairaut, 1713-1765) , mixed partial derivatives are the same.

Examples of some of the partial differential equation treated in this book are shown in Table 2.1.1. However, being that the highest order derivatives in these equation are of second order, these are second order partial differential equations. In this chapter we will focus on first order partial differential equations. Examples are given by

\[\begin{aligned}u_t+u_x&=0. \\ u_t+uu_x&=0. \\ u_t+uu_x&=u. \\ 3u_x-2u_y+u&=x.\end{aligned}\]

For function of two variables, which the above are examples, a general first order partial differential equation for \(u = u(x, y)\) is given as

\[\label{eq:4}F(x,y,u,u_x,u_y)=0,\quad (x,y)∈ D\subset R^2.\]

This equation is too general. So, restrictions can be placed on the form, leading to a classification of first order equations. A linear first order partial differential equation is of the form

\[\label{eq:5}a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y).\]

Note that all of the coefficients are independent of \(u\) and its derivatives and each term in linear in \(u,\: u_x,\) or \(u_y\).

We can relax the conditions on the coefficients a bit. Namely, we could assume that the equation is linear only in \(u_x\) and \(u_y\). This gives the quasilinear first order partial differential equation in the form

\[\label{eq:6}a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u).\]

Note that the \(u\)-term was absorbed by \(f(x, y, u)\).

In between these two forms we have the semilinear first order partial differential equation in the form

\[\label{eq:7}a(x,y)u_x+b(x,y)u_y=f(x,y,u).\]

Here the left side of the equation is linear in \(u\), \(u_x\) and \(u_y\). However, the right hand side can be nonlinear in \(u\).

For the most part, we will introduce the Method of Characteristics for solving quasilinear equations. But, let us first consider the simpler case of linear first order constant coefficient partial differential equations.