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Consider a infinite rod with the temperature at a specific time $u(x,t)$ if the initial temperature is constant in $(-2,2)$ and and zero outside.

I need the boundary and initial conditions for the above mentioned situation.

I know for sure that the boundary condition outside the cross-section must be the limit $\displaystyle\lim_{|x|\to\infty} u(x,t)=0$.

Not sure about the inside, is it $u(x,0) = A$ and $u(x,0) = B$ or $u(-2,0) = A$ and $u(2,0) = B$ (where $A$ and $B$ are constant).

Can someone please exchange their insight and say if I'm on the right trail?

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  • $\begingroup$ Are you looking for more guidance on the solution, or is what I stated all you need? $\endgroup$ – Ron Gordon Jan 30 '14 at 15:02
  • $\begingroup$ So the boundary condition outside was fine what I gave? I will solve the problem using the Fourier transformation method! $\endgroup$ – Sam123 Jan 30 '14 at 15:25
  • $\begingroup$ It was not really fine as given - what I gave below is more like it. Yes, go with Fourier on this one. $\endgroup$ – Ron Gordon Jan 30 '14 at 15:45
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You are given the boundary condition as you specify, and an initial condition as follows:

$$u(x,0) = \begin{cases} A & |x| \lt 2 \\ 0 & |x| \gt 2\end{cases}$$

That is sufficient to solve the equation and get a unique solution.

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