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.)

Example

For example, consider the time-independent Schrödinger equation

:$[-\; abla^2\; +\; V(mathbf\{x\})]\; psi(mathbf\{x\})\; =\; Epsi(mathbf\{x\})$

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

:$V(x\_1,x\_2,x\_3)\; =\; V\_1(x\_1)\; +\; V\_2(x\_2)\; +\; V\_3(x\_3),$

then it turns out that the problem can be separated in to three one-dimensional ODEs for functions $psi\_1(x\_1)$, $psi\_2(x\_2)$, and $psi\_3(x\_3)$, and the final solution can be written as $psi(mathbf\{x\})\; =\; psi\_1(x\_1)\; cdot\; psi\_2(x\_2)\; cdot\; psi\_3(x\_3)$. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948. [L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," "Phys. Rev." 74, 87-89 (1948).] )

References