Separation of variables

Separation of variables is a classical technique to construct special solutions u to an ODE by using the ansatz $u(t,x)=f(x)g(t)$ for functions f and g of space and time separately. Using the fact that a function of space and a function of time can only be equal if both are constant, one can then decouple the PDE into an ODE in time and a PDE in space, which are presumably easier to solve.

In the case of linear time-translation-invariant equations, one can often express a general solution (under suitable regularity conditions) as a superposition of solutions constructed via separation of variables, for instance by using spectral theory. However this phenomenon is generally unavailable in non-linear or time-dependent contexts.

Separation of variables can be used to construct stationary soliton solutions, and is also used in the ODE method for constructing blowup solutions.