# Examples from Section 18.2

Math 421:03

Date: 6 November 2002

Example 1   Solve the boundary-value problem

Proof. [Solution]We will find the solution as , where each is harmonic (just a fancy word for ). We will force to be zero on the entire boundary except the North'' edge, where for all . Likewise, will be zero all around the boundary except the South'' edge, where as well.

The solution for is then

where

As for ,

where

So we have

See Figure 1 for the graph of .

# Application to the Heat Equation

Let's see how we can use this to solve the heat equation with inhomogenous boundary conditions. We will solve the boundary value problem

So our rectangular region has zeros all around it except at the top. We know that we can solve the PDE part by writing , where , . The boundary conditions can be satisfied if

 (1)

and

That is, all the nonzero boundary stuff is given to . We can satisfy the initial conditions if we require

 (2)

Now is easy; it's just from Example 1.

 (3)

where the are just the Fourier sine series of :

As for , we know that the general solution has to be

where to fit the inital conditions (2),

Because of the series expression for , we can simplify:

The integral should be familar; it is nonzero if and only if , in which case it is . So

So we are interested in the Fourier sine coefficients on of the function . They are

Thus

Putting this back into the general solution, we have

Putting it all together,

Example 2   Solve the heat equation with inhomogenous boundary conditions as those given in Example 1.

Proof. [Solution]We will have to , and compute each separately as in the previous development. If we let

we have

As for , the general solution is now of the form

where this time

So putting it all together we have

See Figure 2 for some graphs.

Matthew Leingang 2002-11-08