In my class we began to study the Fourier transform, applying it to differential equations. But, I do not fully understand the issue of initial conditions and border, using the Fourier sine transform.
\begin{equation} \frac{\partial^{2} Y(X,t)}{\partial t^{2}} = 4\frac{\partial^{2} Y(X,t)}{\partial X^{2}}\\ With: \\ Y(0,t) = 0\\ Y(4,t) = 0\\ Y(X,0) = a_{0}X\\ Y_{t}(X,0)=b_{0}X(d_{0}-X)\\ 0 < X < 4; 0 < t \end{equation} Where $Y_{t}(X,0)$ It's the derivative with respect to t
Also, if I have \begin{equation} \frac{\partial U}{\partial t} = \frac{\partial^{2} U}{\partial X^{2}}\\ U(0,t) = 1\\ U(\pi,t) = 3\\ U(X,0)=2\\ 0 < X < \pi; 0<t \end{equation}
It would be a similar process as the first one? Or it changes due to what the system asks
Thanks you