I need to know if there is a mistake in these notes:
In the second page we have a representation of a function $f(x)$ as a $\sin$ series. Dont we need to have $f(0)=0=f'(l)$ for such a representation to be true ? Again in that second page we have an example with $f(x)=5$. I am totally confussed !
On
You're correct that the below series does not converge to $f$ such that $f$ is identically $5$ on $\mathbb{R}$: $$u(x,0)=\sum_{n=1}^\infty\frac{10}{(2n-1)\pi}\sin\left(\frac{(2n-1)\pi x}{2l}\right)$$If you read the paper closely, it specifically states that, instead, this series converges to something that matches $f$ on $(0,\infty)$, with an odd extension; it converges to: $$\tilde f(x)=\left\{\begin{array}{l}5\quad\text{ where } x> 0\\0\quad\text{ where }x=0\\-5\ \text{ where }x<0\end{array}\right.$$
This is 'good enough' here, however, since $\lim\limits_{t\to0^+} u(x,t)=5$.
No. At this level, for this type of problem, the function's behavior at individual points doesn't really matter. This is because you are now translating problems into linear algebra: You search for a basis of functions that fits the problem at hand, you express the problem in terms of that basis, and use it to solve the problem.
In this case, you are trying to solve the heat equation on $[0,10]$ with Dirichlet boundary conditions. Refer to page 1: the collection of sine waves of different frequencies satisfying the boundary values is a set of eigenfunctions of the second derivative operator. If possible, it would be nice to express your initial condition $f(x) = u(x,0)$ as a linear combination of these eigenfunctions in order to use 46.3 to arrive at a solution $u(x,t)$.
It turns out that these sines are orthogonal according to the inner product $\langle u,v\rangle = \int uv\ dx$, so after normalizing them, you can (hopefully) write $$ f(x) = \sum \langle f,X_n\rangle X_n. $$ Once you've done this, you apply 46.3 and win.
Now here's the critical point pertaining to your question. Integration doesn't "see" what happens at individual points, or indeed, on sets of measure zero. This sum converges to $f$ almost everywhere. It turns out, and this is the point of this whole theoretical machine, that convergence almost everywhere is enough to talk about differential equations and their solutions.
One of the big breakthroughs in PDEs was the reinterpretation --- I believe in the middle of the twentieth century --- of a general linear partial differential equation as a linear algebra problem. This is the generalization of the Fourier analysis you're doing now.
Studying function spaces from the perspective of linear algebra, and introducing tools like the $L^2$ inner product, is immensely productive in the study of PDEs. For a class of operators called "elliptic," which includes the second-derivative operator of the heat equation, one can find so-called "weak" solutions: solutions which work almost everywhere.