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RayonG
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Homework Statement
Find a formal solution of the heat equation u_t=u_xx subject to the following:
u(0,t)=0
u_x(∏,t)=0
u(x,0)=f(x)
for 0≤x≤∏ and t≥0
Homework Equations
u(x,t)=X(x)T(t)
The Attempt at a Solution
I first began with a separation of variables.
T'(t)=λT(t)
T(t) = Ce^(λt).X''(x)=λX(x)
The boundary conditions show that X(0)=0 and X'(∏)=0.
There are three general cases for the solution X(x).
Case 1: When λ=0, I get X(x)=Ax+B, and using the boundary conditions get A=B=0.
So the solution is a trivial solution (not what I need).
Case 2: When λ=μ^2 >0, then X(x)=Acosh(μx)+Bsinh(μx) and X'(x)=Aμsinh(μx)+Bμcosh(μx).
Again, using the boundary conditions lead to C=D=0, leading to the trivial solution X(x)=0.
Case 3: When λ=-μ^2 <0, then X(x)=Acos(μx)+Bsin(μx). X'(x)=-Aμsin(μx)+Bμcos(μx).
This is where I'm stuck. After solving I get A=0 and B≠0, as cos(μ∏)=0. How do I continue?
The solution is along the lines of f(x)=u(x,0)= Summation of n=1 to infinity of (Bn*sin(nx)), but clearly that is wrong. Help?
EDIT: You don't have to show that the Fourier series obtained converges to the solution.
After looking at it some more, I think that X(x)=sin(μx). λ_n=-(μ_n)^2=-n^2/2 (where n is a positive integer). Would that be right?
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