Subsections

### 5.8 An alternate procedure

This example tries to be clever about handling inhomogeneous boundary equations for the Laplace equations. It does run into a few problems. But the students will of course explain and fix up the problem.

#### 5.8.1 The physical problem

Find the steady temperature distribution in the square plate/cross section below if the heat fluxes out of the sides are known.

#### 5.8.2 The mathematical problem

• Finite domain : , .
• Unknown temperature
• Elliptic
• Four Neumann boundary conditions
• Integral constraint due to all Neumann boundary conditions:

Try separation of variables:

#### 5.8.3 Step 0: Fix the boundary conditions

Standard approach:

All boundary conditions are inhomogeneous. Our standard approach would be to set where

and then set

This would work without any problems. A quadratic in would be fine. Of course, this choice for is quite arbitrary.

Alternative approach:

Instead, we will follow a more elegant procedure that does not require us to arbitrarily choose a . Unfortunately, this alternative procedure will get us into some trouble.

The idea is that the given problem can be seen as the sum of two problems, each with homogeneous boundary conditions in one direction.

If we add the solutions to the two problems together, we should get the solution to the original problem.

The instructor will solve the left hand problem. The students will solve the right hand problem, identify the difficulty, and fix it.

Some people split up the problem into 4, one for each side. That makes the difficulty even worse.

#### 5.8.4 Step 1: Find the eigenfunctions

Substitute into the homogeneous partial differential equation :

Since the instructor's -boundary conditions are homogeneous, he has a Sturm-Liouville problem for :

This was already solved in problem 7.19. Looking back there, substituting ,

#### 5.8.5 Step 2: Solve the problem

Expand all variables in the problem for in a Fourier series:

Remember that the expression you find for the integrals in the bottom, , does not work for , in which case it turns out to be .

Fourier-expand the partial differential equation :

Because of the Sturm-Liouville equation in the previous section

giving the ordinary differential equation

or substituting in the eigenvalue

Fourier-expand the boundary condition :

Fourier-expand the boundary condition :

Solve the above ordinary differential equation and boundary conditions for . It is a constant coefficient one, with a characteristic equation

Caution! Note that both roots are the same when . So we need to do the case separately.

For the solution is

The boundary conditions above give two linear equations for and :

that are best solved using Gaussian elimination. Rewriting the various exponentials in terms of sinh and cosh, the solution for the Fourier coefficients of except is:

For the solution of the ordinary differential equation is

Put in the boundary conditions to get equations for the integration constants and :

Oops! We can only solve this if

Looking above for the definition of those Fourier coefficients, we see we only have a solution if

Unfortunately, these two integrals will normally not be equal! Also, remains unknown.

#### 5.8.6 Summary of the solution

First compute the Fourier coefficients of the given boundary conditions:

Then the solution is equal to:

But this only satisfies the boundary condition on the top of the plate if

No problem! Students will explain and fix the problem.