What is the purpose of representing a (partial) differential equation with linear operators?

Question

One of the first things that is covered in a PDE class (and linear algebra, of course) is the concept of linearity and linear operators, i.e. an operator $L$ such that $L(c_1f_1+c_2f_2)=c_1L(f_1)+c_2L(f_2)$.

It is then said that $\frac{\partial}{\partial x}$,$\frac{\partial^2}{\partial x^2}$, etc are linear operators because they fit these definitions.

What is the importance in this? Does it have something to do with the superposition principle? I haven't solved an equation written as $Lu=0$, but rather we are given $\nabla^2u=0$, and although they are in the same form, I don't see the importance in bringing up the fact that those things are indeed linear operators.

Answer 1

This linearity allows you to add any number of different solutions, which satisfy homogeneous boundary conditions. Any combination of those solutions will also satisfy the homogeneous boundary conditions. These solutions can then be combined, such that it also satisfies the initial conditions, which for example can be found with a Fourier series.