Diffusion Equation

Colin R. Mitchell

Imagine a strip of ocean beginning at a beach and extending outward, where the harvesting of lobsters is prohibited. If we imagine a line perpendicular to this strip, lying within the strip, we can use it to model the populations of lobsters using a diffusion equation. Suppose the lobsters each lay so many eggs, and that so many of the eggs hatch to become lobsters. This is thus proportional to the population of lobsters at some certain point on the line. We can model this using the following partial differential equation:

We can solve this using the method of separation of variables. Doing so, we seek a solution of the form

with

Substituting these into the original equation, we get

Since the left side is only dependent on t, and the right side is dependent only on x, then both sides must be constant. So we then have

From this, we get

The other equation becomes

If we let , then this becomes . The solution for this is

.

Using the boundary conditions, we get

Since we do not want the trivial solution , we need

We can choose because we want u to always be positive, since we can’t have negative populations of fish. We will then get a model of the form

So we get a general solution of the form

,

which satisfies the boundary conditions. We want to find a lower limit for L that makes the overall population grow. If we take our solution for j and insert it into the ordinary differential equation in j , we can solve for m . Since we have

we also have that

So an L satisfying this will yield a population which grows over time. Finally, using the initial condition , we can find the particular solution

When , the integral on the right side becomes . The rest of the integrals are 0. Since we only wanted the solution for , and we get the final solution of

With L satisfying the proper condition, our solution for u will then grow, as illustrated below.