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Dirichlet Problems

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1. Consider the Dirichlet Problem where the temperature within a rectangular plate R is steady-state and does not change with respect to time. Find the temperature u(x,y) within the plate for the boundary conditions below and where (see attached).

2. Solve the Dirichlet problem for steady-state (constant with respect to time) temperature within the wedge shown below. Find the temperature (see attached) within the wedge. The temperature along the arc is (see attached).

Hint: For the solution in ρ, which is the radius, the coefficient for one of the two independent solutions must be zero, since the solution must be finite (can't go to infinity) when ρ=0.

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Solution Preview

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1.
The two-dimensional heat equation is:
(1.1)
Under steady state conditions and the problem turns into Laplace equation:
(1.2)
In the case of the rectangle the boundary conditions are:
(1.3)
And:
(1.4)
Note that (1.4) are periodic boundary conditions that tells us w should find the eigenfunctions in the form of .
We start with separation of variables. We write the temperature function as a product of two single-variable independent functions:
(1.5)
Therefore:
(1.6)

Plugging it back into the equation we obtain:

(1.7)
So each side of equation (1.7) is completely independent of the other, and since it is true for any both sides must equal the same constant:
(1.8)
The conditions for are given in (1.4):
(1.9)
Now we need to solve equation (1.8) for :
(1.10)

Case 1:
The equation becomes:
(1.11)
Its solution is:
(1.12)
Applying boundary ...

Solution Summary

The solution finds the temperature within the plate for the boundary conditions. Dirichlet problems are analyzed.

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