Separable partial differential equation

A separable partial differential equation (PDE) is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some special form or symmetry. In this way, the PDE can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations.

(This should not be confused with the case of a separable ODE, which refers to a somewhat different class of problems that can be broken into a pair of integrals; see separation of variables.)

For example, consider the time-independent Schrödinger equation

$[-\nabla ^{2}+V(\mathbf {x} )]\psi (\mathbf {x} )=E\psi (\mathbf {x} )$

for the function $\psi (\mathbf {x} )$ (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function $V(\mathbf {x} )$ in three dimensions is of the form

$V(x,y,z)=V_{1}(x)+V_{2}(y)+V_{3}(z),$

then it turns out that the problem can be separated in to three one-dimensional ODEs for functions $\psi _{1}(x)$, $\psi _{2}(x)$, and $\psi _{3}(x)$, and the final solution can be written as $\psi (\mathbf {x} )=\psi _{1}(x)\cdot \psi _{3}(x)\cdot \psi _{3}(x)$. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)