Heat equation solution does not apparently satisfy initial condition

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The problem statement is-

A rod has its $x=0$ end at $0^\circ C$ and the $x=l$ end at $100^\circ C$. Now at $t>0$, we keep $x=0=l$ at $0^\circ C$. The time dependent heat equation solution is $$u=\frac{200}{\pi}\sum\frac{(-1)^{n-1}}{n}e^{-(n\pi\alpha/l)^2t}\sin\frac{n\pi x}{l}$$

Clearly, putting $t=0$ and $x=l$ should give $100^\circ C$ as given but putting $x=l$ gives $\sin(n\pi)=0$, hence the whole thing goes to $0$ instead of $100$.

What am I missing here?

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A partial differential equation (PDE) is only well defined if the initial conditions (IC) and boundary conditions (BC) are compatible to each other on the whole spacetime domain $\Omega = [0,t]\times[x_a,x_b]$.

So in your case the problem (or your thinking) is not well defined.

As a hint answer the following question: Which condition has priority at $t=0$?

  • The initial condition or
  • the boundary condition?

By the way, you have not even stated the initial condition.

Regards