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I'm trying to solve this PDE problem:


on $[-1,1]$ with periodic boundary conditions, and taking as initial data the function $u_0=1+\sin^2(\pi x)+\sin^2(2\pi x)$

I want to obtain the analytic solution. I think i can solve it separating variables and applying Fourier after that, ¿but anyone knows a shorter or easier way to get the solution?

Thanks a lot.

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Here is a related problem. – Mhenni Benghorbal Jan 23 '13 at 19:46
How is this question different from the one you asked the other day? Also, it is not clear to me that you know what constitutes "periodic boundary conditions". It would be better if you explicitly state the PDE, BC(s), IC(s) you are considering. – JohnD Jan 23 '13 at 20:02
Hi, John. I know how to solve it as I said, but I was asking for some "trick" method working in these cases, like the one 5PM has proposed, to avoid applying separation of variables and Fourier techniques. – Mark_Hoffman Jan 23 '13 at 20:14
up vote 1 down vote accepted

The basic thing to remember is that the function $u=Ce^{- \alpha \beta^2 t} \cos \beta x$ satisfies $u_t = \alpha u_{xx}$ (direct verification), for any $C$ and $\beta$. In your case $\alpha=1/5$.

Thus, whenever your initial data is given as a sum of trigonometric functions $C_k \cos \beta_k x$, you are in luck: just multiply each by exponential term $e^{- \alpha \beta_k^2 t} $ and you have the solution.

Your initial data is indeed of the above form, thanks to the identity $\sin^2y =\frac{1-\cos 2y}{2}$.

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