Characteristics of Linear and Quasi-Linear First-Order PDEs

We now compare the characteristics of linear PDEs and quasi-linear PDEs. This comparison is useful because the characteristic curves behave differently in these two cases. In a linear equation, the characteristic curves depend only on the independent variables. In a quasi-linear equation, they may also depend on the unknown function

u

. Consider the first-order quasi-linear equation

\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0. \quad \text{(1)} \end{equation*}

The characteristic equations are

\begin{equation*} \frac{dx}{dt}=a(x,y,u), \quad \frac{dy}{dt}=b(x,y,u), \quad \frac{du}{dt}=c(x,y,u). \quad \text{(2)} \end{equation*}

The first two equations of (2) give the direction of the characteristic curve in the

(x,y)

-plane. If

a(x,y,u) \neq 0

, then the slope is

\begin{equation*} \frac{dy}{dx}=\frac{b(x,y,u)}{a(x,y,u)}. \quad \text{(3)} \end{equation*}

The characteristic slope is the slope

\frac{dy}{dx}

of the projected characteristic curve in the

(x,y)

-plane.

u

. Consider the first-order quasi-linear equation

\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0. \quad \text{(1)} \end{equation*}

The characteristic equations are

\begin{equation*} \frac{dx}{dt}=a(x,y,u), \quad \frac{dy}{dt}=b(x,y,u), \quad \frac{du}{dt}=c(x,y,u). \quad \text{(2)} \end{equation*}

The first two equations of (2) give the direction of the characteristic curve in the

(x,y)

-plane. If

a(x,y,u) \neq 0

, then the slope is

\begin{equation*} \frac{dy}{dx}=\frac{b(x,y,u)}{a(x,y,u)}. \quad \text{(3)} \end{equation*}

The characteristic slope is the slope

\frac{dy}{dx}

of the projected characteristic curve in the

(x,y)

-plane.

The characteristic slope

\frac{dy}{dx}=\frac{b}{a}

gives the direction of the projected characteristic curve in the

(x,y)

-plane. Choose a slope rule and click Draw Slope Field to see small line segments showing the local direction. Constant slopes give parallel directions, while variable slopes change from point to point. This helps you read the ratio

b/a

as a geometric direction field.

For the quasi-linear equation (1), the characteristic slope may depend on

u

. Therefore, the characteristic curve in the

(x,y)

-plane may depend on the solution itself.

For a quasi-linear first-order PDE, the characteristic direction in the

(x,y)

-plane is generally controlled by

\begin{equation*} \frac{dy}{dx}=\frac{b(x,y,u)}{a(x,y,u)}. \end{equation*}

Since

u

may appear in this expression, the projected characteristic curves may change with the solution surface.

In a quasi-linear equation, the projected slope may contain

u

. Choose a fixed value of

u

and click Update Field to see the direction field for the sample rule

\frac{dy}{dx}=u

. Changing

u

changes the slope field even at the same

(x,y)

points. This shows why quasi-linear characteristic curves may depend on the solution surface itself.

Now consider the linear case for characteristic curves. Here the coefficients of

u_x

and

u_y

do not depend on

u

. Thus we write these coefficients as

a(x,y)

and

b(x,y)

. In this case, the characteristic slope becomes

\begin{equation*} \frac{dy}{dx}=\frac{b(x,y)}{a(x,y)}. \quad \text{(4)} \end{equation*}

For a linear first-order PDE, the characteristic curves in the

(x,y)

-plane are determined by the equation

\begin{equation*} \frac{dy}{dx}=\frac{b(x,y)}{a(x,y)}. \end{equation*}

These curves depend only on

x

and

y

The main difference is whether the projected slope depends on

u

. Choose a case and click Compare Slopes to see what information is needed to draw the characteristic curves. In the linear case, the slope depends only on

x

and

y

, so it can be studied before solving for

u

. In the quasi-linear case, the slope may contain

u

, so the projected curves may change with the solution.

This is an important simplification. In the linear case, we may find the characteristic curves before knowing the unknown solution

u

Suppose

a

and

b

are constants, with

a \neq 0

. Then

\begin{equation*} \frac{dy}{dx}=\frac{b}{a}. \quad \text{(5)} \end{equation*}

Since the slope is constant, the characteristics are straight lines in the

(x,y)

-plane.

When

a

and

b

are constants, the characteristic slope is the constant ratio

b/a

. Enter constant values of

a

and

b

, then click Compute Lines to find the slope and the characteristic family. The result has the form

y=mx+C

, where

m=b/a

. This calculator helps you practise the straight-line case before working with variable coefficients.

Consider the equation

\begin{equation*} 2u_x+3u_y-c(x,y,u)=0. \end{equation*}

Here

a=2

and

b=3

. Hence the characteristic slope is

\begin{equation*} \frac{dy}{dx}=\frac{3}{2}. \end{equation*}

Integrating, we get

\begin{equation*} y=\frac{3}{2}x+C. \end{equation*}

Thus the characteristics are straight lines.

The solved problem gives the family

y=\frac{3}{2}x+C

. Choose a slope and click Plot Lines to display several members of the family for different values of

C

. Parallel lines appear because all members have the same slope. This makes the phrase family of characteristics visible in the

(x,y)

-plane.

To compare linear and quasi-linear cases, inspect the coefficient functions

a

and

b

. Choose a sample pair and click Check Slope Type to see whether

b/a

is constant, depends only on

x

and

y

, or depends on

u

. This classification tells you whether the projected characteristics can be drawn independently of the unknown solution. It also explains why constant coefficients lead to straight lines.

This topic may also be called characteristic slopes or straight-line characteristics. We use Characteristics of Linear and Quasi-Linear First-Order PDEs because it clearly compares the two cases.

BMLabs Mathematics

BMLabs Mathematics

Characteristics of Linear and Quasi-Linear First-Order PDEs

Characteristics of Linear and Quasi-Linear First-Order PDEs

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