- 5.11.1 The physical problem
- 5.11.2 The mathematical problem
- 5.11.3 Step 1: Find the eigenfunctions
- 5.11.4 Step 2: Solve the problem
- 5.11.5 Summary of the solution

5.11 A Problem in Three Independent Variables

This example addresses a much more complex case. It involves three independent variables and eigenfunctions that turn out to be Bessel functions.

5.11.1 The physical problem

Find the unsteady heat conduction in a disk if the perimeter is
insulated. The initial temperature is given.

5.11.2 The mathematical problem

- Finite domain :
- Unknown temperature
- Parabolic partial differential equation:

- One homogeneous Neumann boundary condition at :

- One initial condition at :

We will solve using separation of variables in the form

The eigenfunctions will get rid of the variable in the partial differential equation, and the eigenfunctions will get rid of the variable, leaving ordinary differential equations for the Fourier coefficients .

5.11.3 Step 1: Find the eigenfunctions

Let's start trying to get rid of one variable first. We might try
a solution of the form

where the would be the eigenfunctions and the the corresponding Fourier coefficients. Unfortunately, if we try to substitute a single term of the form into the homogeneous partial differential equation, we are not able to take all terms to the same side of the equation and and terms to the other side. So we do not get a Sturm-Liouville problem for .

Try again, this time

If we substitute into the homogeneous partial differential equation we get:

This, fortunately,

So we have a Sturm-Liouville problem for :

with boundary conditions that are periodic of period . This problem was already fully solved in 7.38. It was the standard Fourier series for a function of period . In particular, the eigenfunctions were , , and , .

Like we did in 7.38, in order to cut down on writing, we will indicate those eigenfunctions compactly as , where and .

So we can concisely write

Now, if you put this into the partial differential equation, you will see that you get rid of the coordinate as usual, but that still leaves you with and . So instead of ordinary differential equations in , you get partial differential equations involving both and derivatives. That is not good enough.

We must go one step further: in addition we need to expand each
Fourier coefficient in a generalized Fourier series in :

Now, if you put a single term of the form
into the homogeneous partial differential equation, you get

Since , this is separable:

So we get a Sturm-Liouville problem for with eigenvalue

with again the same homogeneous boundary conditions as :

We need to find all solutions to this problem.

Unfortunately, the ordinary differential equation above is not a constant coefficient one, so we
cannot write a characteristic equation. However, we have seen the
special case that before, 7.38. It was a Euler equation.
We found in 7.38 that the only solutions that are regular at
were found to be . But over here, the only one of that form
that also satisfies the boundary condition at is the
case . So, for , we only get a single eigenfunction

For the case , the trick is to define a stretched
coordinate as

This equation can be found in any mathematical handbook in the section on Bessel functions. It says there that solutions are the Bessel functions of the first kind and of the second kind :

Now we need to apply the boundary conditions. Now if you look up the graphs for the functions , or their power series around the origin, you will see that they are all singular at . So, regularity at requires .

The boundary condition at the perimeter is

Since is nonzero, nontrivial solutions only occur if

Now if you look up the graphs of the various functions , , , you will see that they are all oscillatory functions, like decaying sines, and have an infinity of maxima and minima where the derivative is zero.

Each of the extremal points gives you a value of , so you will get an infinite of values , , , , , . There is no simple formula for these values, but you can read them off from the graph. Better still, you can find them in tables for low values of and . (Schaum's gives a table containing both the zeros of the Bessel functions and the zeros of their derivatives.)

So the -eigenvalues and eigenfunctions are:

where is the counter over the nonzero stationary points of . To include the special case , we can simply add , to the list above.

In case of negative , the Bessel function of imaginary argument becomes a modified Bessel function of real argument, and looking at the graph of those, you see that there are no solutions.

5.11.4 Step 2: Solve the problem

We again expand all variables in the problem in generalized Fourier
series:

Let's start with the initial condition:

To find the Fourier coefficients , we need orthogonality
for both the and eigenfunctions. Now the ordinary differential equation for the
eigenfunctions was in standard form,

but the one for was not:

The derivative of the first coefficient is , not . To fix it up, we must divide the equation by . And that makes the weight factor that we need to put in the orthogonality relationship equal to .

As a result, our orthogonality relation for the Fourier coefficients
of initial condition
becomes

The integral within the square brackets turns into its -Fourier coefficient and the outer integral turns that coefficient in its generalized -Fourier coefficient . Note that the total numerator is an integral of over the area of the disk against a mode shape .

The -integral in the denominator can be worked out using Schaum's
Mathematical Handbook 24.88/27.88:

(setting the second term to zero for .)

Hence, while akward, there is no fundamental problem in evaluating as many as you want numerically. We will therefor consider them now ``known''.

Next we expand the desired temperature in a generalized Fourier series:

Put into partial differential equation
:

Because of the SL equation satisfied by the :

Because of the SL equation satisfied by the :

Hence the ordinary differential equation for the Fourier coefficients is:

with solution:

At time zero, the series expansion for must be the same as the
one for the given initial condition :

Hence we have found the Fourier coefficients of and solved the problem.

5.11.5 Summary of the solution

Find the set of positive stationary points of the Bessel functions , and add .

Find the generalized Fourier coefficients of the initial condition:

Then:

That was not too bad!