On different methods of solving differential equations

Question

I've studied the basic concepts of partial differential equations, and one question comes to my mind. What are the propuse of the diferent methods of resolution of differential equations. For example if you start with: $$ \frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2} - \frac{\partial^2 \Psi}{\partial x^2} = 0 $$ and assume a solution of the type: $$ \Psi(x, t) = X(x)T(x) $$ you will arrive a someting like: $$ \Psi_n(x) = \left ( A \sin{(k_nx) + B\cos(k_n x)} \right)e^{ick_nt} $$ where the set of all eigenvectors $\Psi_n(x, t)$, forms a basis of a Hilbert Space. From what I understand, by using this method, you find eigenfunctions (or eigenvectors), that are the stationary solutions to the equation. Solutions which are in equilibrium. (in a similar way, that of finding a matrix in a diagonal basis, which is the simplest form of a linear application you can find)

But now imagine you want dynamical solutions, solutions which are not stationary. You would use something like the Fourier Series (or Fourier-Bessel Series, etc.) (given an initial conditions), or a Fourier Transform. Also, in this case, for the wave equation, performing the change of variables: $$ \eta = x + ct \\ \xi = x - ct $$ also yields a non-stationary solution. So my question is? Is this right? I mean, you always will get stationary solution by separation of variables, and non-stationary ones, with all the other methods? Is this an universal thing? Are there any other methods?

Answer 1

I do not understand what you mean when you say that the solutions you get with separation of variables are stationary, since they depend explicitly on time. Maybe it will help you to realize that your function \begin{equation} \Psi_n(t,x) = \left( A_n \sin(k_n x) + B_n \cos(k_n x) \right) e^{i c k_n t} \end{equation} is actually of the form $f(x+ct) + g(x-ct)$, which is what you seem to be referring to in the last paragraph. Using the relations \begin{align} \sin(k_n x) = \frac{e^{i k_n x} - e^{-i k_n x}}{2i}, && \cos(k_n x) = \frac{e^{i k_n x} + e^{-i k_n x}}{2} \end{align} in your $\Psi_n(t,x)$, you find that it becomes \begin{equation} \Psi_n(t,x) = \left( C_ne^{i k_n x} + \bar{C}_n e^{- ik_n x} \right)e^{ic k_n t} = C_n e^{ik_n(x+ct)} + \bar{C}_n e^{-i k_n(x-ct)}, \end{equation} where \begin{align} C_n = \frac{B_n -iA_n}{2}, && \bar{C}_n = \frac{B_n +iA_n}{2}, \end{align} which is of the form $f(x+ct) + g(x-ct)$. Note that the wave equation is linear, which means that the sum of any number of solutions is also a valid solution. With separation of variables you found an infinite set of solutions (one for each $n$), so the most general one is a linear combination of all of them. Indeed, any two functions $f(x+ct)$ and $g(x-ct)$ defined on some interval of the real line can be decomposed in Fourier modes as \begin{align} f(x+ct) = \sum_{n= - \infty}^{\infty} c_n e^{ik_n(x+ct)} && g(x-ct) = \sum_{n= - \infty}^{\infty} d_n e^{ik_n(x-ct)}, \end{align} which confirms that the two methods you mentioned lead to the same set of solutions for the equation.