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Consider a metal rod, with $K=1$ and longitude $\pi$ .

On its extremes, there is heat transmission with the ambient, according to the differential equation: $u_t(x,t)=u_{xx}(x,t)-hu(x,t),$ with $h>0$.

The extremes of the rod have fixed temperatures of $0^\circ C$ on the left, and $1^\circ C$ on the right. The initial temperature of the rod is the function: $f(x)=x(\pi-x).$

Find the temperature distribution $u(x,t)$ on the rod.

Could someone please explain me how to start? I've never seen something like that (with an $hu(x,t)$ added or taken from the equation). I'm really stuck.

Thank you beforehand, and sorry for my english, it's not my natural language.

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    Do you mean to have primes on some $u$ terms but not all? If so, I don't know what it means.2012-12-16

1 Answers 1

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It sounds like you know how to solve the "usual" heat equation, $u_t=u_{xx}$ by separation of variables. Is that correct? If so, proceed in exactly the same way as you did before, i.e. letting $u(x,t)=X(x)T(t)$.

The eigenvalue problem will be just like before, but the temporal problem will look like ${T'\over T}+h=-\lambda$.

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    No, I just mistyped. Fixed now.2012-12-16