(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

The separation of variables method.

3. 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?

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# Homework Help: Diffusion equation

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