A separable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation of variables. It generally relies upon the problem having some special form or symmetry. In this way, the partial differential equation (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.

-separation of variables; in the three-dimensional case this uses 6-sphere coordinates.

(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.)

Example

For example, consider the time-independent Schrödinger equation

(in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation in three dimensions is of the form

. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)

References