I have the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ Boundary conditions are $u=0$ at $x=0$ and $x=1$
The initial condition is $u=\sin(\pi x)$
I know that $$L \left(\frac{\partial u}{\partial t}\right)=sL(u)-\sin(\pi x)$$
Then the next step in my notes is $$L\left( \frac{\partial^2 u}{\partial x^2}\right)=\frac{\partial^2}{\partial x^2}(L(u))$$
Where does this step come from and what does it mean?
To see what the step $$L\left(\frac{\partial^2 u}{\partial x^2}\right) = \frac{\partial^2}{\partial x^2}(L(u))$$ means, write both sides out using the definition of the (one-sided) Laplace transform: $$ \underbrace{\int_0^\infty e^{-st}\left(\frac{\partial^2}{\partial x^2}u(x,t)\right)\,dt}_{L\left(\frac{\partial^2 u}{\partial x^2}\right)} = \underbrace{\frac{\partial^2}{\partial x^2}\int_0^\infty e^{-st}u(x,t)\,dt}_{\frac{\partial^2}{\partial x^2}(L(u))}. $$ Essentially, it says we're assuming $u(x,t)$ is regular enough in $x$ and $t$ to allow differentiation under the integral sign.