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zezima1
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Homework Statement
Solve the diffusion equation with the boundary conditions v(0,t)=0 for t > 0 and v(x,0) = c for t=0. The method should be separation of variables.
Homework Equations
The separation of variables method.
The Attempt at a Solution
Attempting a solution of the form XT leads you to an exponential for T and a sinusoidal for X:
X = Asin(kx) + Bcos(kx)
where -k^2 was the constant used for solving the two separated differential equations.
However. My solution manuals writes the constants A and B as a continious function of the parameter k, and I don't understand why. Why do the constants, which are chosen from the boundary conditions have anything to do with k?
And going further the full solution is then written as a Fourier integral from 0 to ∞ of XTB(k)dk
Where on Earth does this come from? Note that A(k)=0 from the boundary conditions.
Can someone try to explain why you most impose a continuous superposition like the above to get the general solution?