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I'm interested in using the Fourier transform to solve the heat equation. I've been poring over this wikipedia article: http://en.wikipedia.org/wiki/Heat_equation#Solving_the_heat_equation_using_Fourier_series trying to understand it but every time I go through the solution I get stuck at the step where they generalize the solution to $u(x, t) = X(x)T(t)$ ** by summing all solutions.

Can someone explain to me why this is necessary? My hunch is that when you solve the PDE via separation of variables with equation **, you are assuming the independence of $x$ and $t$, and so summing all solutions somehow relaxes this constraint. If this is true, what's the mathematical justification for this.

user21154
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The heat equation is linear in $u(x,t)$, so any multiple of a solution is again a solution and and sum of two solutions is again a solution. You can check this for yourself: let $u_1(x,t)$ and $u_2(x,t)$ be two solutions. Then $v(x,t)=au_1(x,t)+bu_2(x,t)$ is again a solution for any real $a,b$. Just plug it into the equation and see that the equation is satisfied.

It is true that when you sum two separated solutions, the sum is no longer separated, but it is a solution. You need to do this to get all solutions. A question not addressed in the article is whether you can get all solutions by summing over separated solutions in this way.

Ross Millikan
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  • Thank-you. Yes, I guess that's part of what I'm asking. How does one know that summing over separated solutions in this way gets you all the solutions. – user21154 Feb 17 '13 at 22:43
  • @user21154: I'm pretty sure you do get all the solutions that way, but I don't remember seeing a proof. – Ross Millikan Feb 17 '13 at 22:48