# An Introduction to Partial Differential Equations in the
Undergraduate Curriculum

**1.5. The Transport Equation**

One of the driving motivations for studying PDE’s is to describe the

physical world around us. We can use a** flux argument **to derive

equations describing the evolution of a** density**, which is just a fancy

word describing the concentration of something (mass in a region, heat

in a metal bar, traffic on a highway) per unit volume.

Consider a one-dimensional freeway and let be
the density

of cars per unit length on the freeway.

Figure 1.1: Flux argument for cars on a freeway.

( draw your own figure).

Then the mass of cars in the region a < x < b is given by

Now suppose we are measuring the flux, Q, of cars** into**
this region

measured in mass/unit time. It can written in terms of the number of

cars crossing into the region at x = a, called q(a), minus the number

of cars that flow out of the region at x = b, called q(b),

Now, by **conservation of mass**, the rate of change of
the mass between

a and b is given by the flux into the region,

We can rewrite the flux by a clever application of the
fundamental

theorem of calculus:

We can now rewrite the conservation of mass equation as

or, rearranging

Since this is true for **every** interval a < x < b,
the integrand must

vanish identically. So

Equations of this form are called** transport equations
or conservation**

laws – they are a very active area of study in PDE’s.

We can propose a simple model for the flux function q(x, t) – suppose

we assume the cars are all moving at a constant speed C. Then

we can argue that the flux is just equal to the product of the number

of cars time the speed they are moving at,

Substituting into the transport equation yields

which is just the convection equation. If we specify the
initial distribution

of cars,

we can show fairly easily that the solution to the
convection equation

with this initial condition is just

corresponding to cars moving uniformly to the right.

Physically, we just see the distribution of cars
translating to the

right with a speed of C.

Figure 1.2: Solution to the convection equation.

( draw your own figure).

To verify this solution let , and look for a
solution .

Then, by the chain rule

Substituting into the
convection equation (1.17), we find

Moreover, when t = 0, we find
so that the initial condition

is satisfied also.

**1.6. Challenge Problems for Lecture 1**

**Problem 1.** Classify the follow differential equations as ODE’s or

PDE’s, linear or nonlinear , and determine their order. For the linear

equations, determine whether or not they are homogeneous.

(a) The **diffusion equation** for h(x, t):

(b) The **wave equation **for w(x, t):

(c) The** thin film equation **for h(x, t):

(d) The** forced harmonic oscillator** for y(t):

(e) The **Poisson Equation** for the electric potential
:

where is a known
charge density.

(f) **Burger’s equation **for h(x, t):

**Problem 2. **Suppose when deriving the convection
equation, we assumed

the speed of the cars was given by βx for x > 0.

(a) Explain why the flux function now is given by

and the associated transport equation is given by

(b) Explain why

correspond to a** boundary condition** of no flux of
cars in from

the origin and an** initial condition** specifying the distribution

of cars at t = 0.

(c) Verify that

is a solution to both the transport equation given in (a)
and

the initial and boundary conditions given in (b).

**Problem 3.** Show that the helicoid

satisfies the minimal surface equation,

MAPLE may be helpful with the algebra .

**Problem 4.** Show that the soliton

satisfies the the Korteweg-deVries equation,

MAPLE may be helpful with the algebra , in particular if
you don’t

remember your hyperbolic trigonometric identities .