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The last six lectures of this course are dedicated to finding solutions to partial
differential equations. An ordinary differential equation is an equation which has only
one independent variable and therefore it is reasonably easy to work with. A partial
differential equation, however, equations partial derivatives with multiple variables
and therefore there it becomes much more difficult to solve such equations or to even
determine if a solution exists.

With ordinary differential equations, we began with 1st-order ODEs followed by 2nd-order ODEs
we saw that the tools we used for 2nd-order linear ODEs with constant coefficients could be used
for similar ODEs with an arbitrary order; however, for partial differential equations, it is
more interesting to look at very specific 2nd-order PDEs; specifically, we will be looking at solutions
to

- Laplace's equation,
- The heat-conduction/diffusion equation, and
- The wave equation.

Each of these are 2nd-order PDEs, but to begin, we must begin by defining the
Laplacian operator.

**Note:** to this point, we have used the function to
represent the unit step function. By common convention and by the convention in the
text book, we will use to represent the unknown solution to
our differential equation.

When we consider other forms of energy, again, in the absence of a forcing function,
there is a natural response can at most affect the acceleration of a particle and
therefore we are, again, required to consider the total acceleration. When we are
no longer restricting ourselves to a single dimension, we must consider the sums of
the second derivatives in all directions, or

.

Rather than always having to write out the number of required partial derivatives, we
abbreviate note that the gradient operator is

and therefore

.

We abbreviate the inner product removing
the arrow to emphasize that the result is a scalar value. This
operator is called the Laplacian operator.

# Laplace's Equation

The simplest form of Laplacian operator is to have a steady-state solution where the
sum of the forces in all directions is zero:

.

Note that this is not the equation of a moving particle, but rather, it is the description of a scalar field: something, perhaps a temperature or an
electric field, satisfies this equation at each point. For example, consider two infinitely long parallel plates set a unit distance apart where
one is maintained at 10°C and the other is maintained at 30°C as shown in Figure 2.

Figure 2. Two *infinitely large* plates with different temperatures.

Assuming the temperature has settled (or maintained a *steady state*) we would expect a uniform transition of temperature from 10°C to 30°C across this
boundary and we would expect the transition to be the same with any cross section of the plate.
Consequently, we may consider such a situation to be one-dimensional (in which
case, ), in a steady state,
the temperature obeys Laplace's equation, or in this case,
which must satisfy the boundary conditions
and
. The obvious solution is the expected solution .

Unfortunately, things become much more complex in higher dimensions. Suppose you have two concentric pipes, one
within the other, and the inner pipe is kept at 100°C while outer pipe is kept at 0°C as shown in Figure 3.

Figure 2. Two *infinitely long* pipes with different temperatures.

It is no longer reasonable to assume that the temperature will drop linearly as we move from away from the centre
of the pipe, and yet physics indicates that the temperature must satisfy Laplace's equation: the concavity in one
direction must perfectly match the concavity in the other.

Other more complicated situations, for example, consider a buried pipe where the walls of the pipe are
kept at 5°C and through this pipe is flowing water heated at 20°C. What is the air temperature at
any point in the pipe? (Assume that there is no turbulence in the water or the air flow.)

In general, Laplace's equation is satisfied by a scalar property within a region and the boundary of that
region has the property constrained to specific values. In one dimension, this is nothing more than the
boundary-value problems seen previously. In two dimensions, the boundary (curves) defines a enclosed area in the
plane, and in three dimensions, the boundary (surfaces) defines an enclosed volume.

As you may have guessed by now, it is impossible in general to find solutions to partial differential
equations and there are many numerical tools to assist you in solving such problems, but we will look at
some specific examples where we can find solutions so that we can examine and understand the behaviours.

# The Heat-conduction/Diffusion Equation

Consider the example of two plates shown in Figure 2. Suppose now that we add a time-varying
element. For example, suppose at time , we fill the volume between the
two plates with air that is 0°C and then we wait. Initially, the air near the plates will begin
to heat up, but after some time, the temperature will approach the steady-state scenario.

The equation which describes such a phenomena is the *heat-conduction* equation or
also known as the *diffusion* equation:

.

What this says is that the rate of change of *u* is proportional to how
far the system is from stability as defined by Laplace's equation. If the system already
satisfies Laplace's equation, that is,
, then
and therefore there
*u* will be constant with respect to time.

James derives this equation in the one-dimensional case in Section 9.2.2 around page 666.

# The Wave Equation

Finally, a third equation which appears often is the *wave* equation which describes the
motion of a wave through a medium. This includes:

- The plucking of a string (one dimensional),
- Dropping a stone in the water (two dimensional), and
- Making a sound wave in air (three dimensional).

In each of these cases, a medium at rest satisfies Laplace's equation: a string is straight,
the pond is smooth, and air is uniform and motionless. The introduction of a disturbance
causes the medium to begin to move and the propagation of that disturbance is, under appropriate
circumstances, called a wave and they usually satisfy the equation

.

Again, what this equation says is that the acceleration of *u* is proportional to how far
the system is away from stability as defined by Laplace's equation
.