Question

Find the steady state temperature function on a pi x pi rectangular plate, where the temperature at the top edge is kept at sin(4x)^oC and the remaining 3 sides are kept at 0^oC

Answer 1

Solution:

The initial conditions are shown below:

Now, Starting of the following equation:

Subject to initial conditions:

It is known as the Dirichlet Problem whose solution is:

Where:

So, in this problem:

Therefore, equation (3) can be written as:

or

But, we know that:

Therefore:

Replacing equation (6) into equation (5) we have:

This equation can be written as:

Solving we obtain:

Finally, replacing this result into equation (2) for we obtain: