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If we solve the heat equation $u_t=u_{xx}$ by separation of variables, we assume that $u(x,t)=f(x)g(t)$, and solving 2 ordinary differential equations we can derive that $u(x,t)=e^{\omega^2t}(b\cdot \sin(\omega x)+ a\cdot \cos(\omega x))$ for some $a,b$, is a solution.

Now, if we assume the boundary conditions $u(0,t)=u(1,t)=0$, we know that $a=0$, and $\omega=k\pi, k\in \mathbb N$. This gives us: $u(x,t)= be^{k^2\pi^2t}\sin(k\pi x)$

We then know that, since the heat equation is linear, any $u(x,t)=\sum_{k=0}^nb_ke^{k^2\pi^2t}\sin(k\pi x)$ Is also a solution. Under some assumptions we can let $n$ go to infinity.

My question is: How do we know that the set of such infinite sums of sines will contain all solutions to the heat equation (with those boundary conditions)?

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  • $\begingroup$ By proving the uniqueness of solution under some restrictions $\endgroup$ – Ivon Mar 30 '17 at 14:10

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