|Solve Dirichlet's problem for a circular annulus. The domain is the space between two concentric circles, being the innermost circle with radius , and being the outermost circle with radius .|
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First transform the Laplacian operator to polar coordinates, as explained in this problem.
Now the DE is
The BC's are
An auxiliary condition required from the nature of the problem and coordinate system is
Separate variables and plug back into the original equation, separate variables and set each side equal to a negative (for convenience) constant lambda.
The first ODE is
with the auxiliary condition
In order for , the square root of lambda must be an integer value.
The second ODE is
If so that , the solution is:
If , the solution is:
This ODE is in Cauchy-Euler form, so guess the solution and plug it back in.
The solution is
The Fourier coefficients are given with the help of the boundary conditions.
Now you have this system of equations to solve to determine the coefficients: