Separable partial differential equation

A separable partial differential equation 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 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.

The most common form of separation of variables is simple separation of variables in which a solution is obtained by assuming a solution of the form given by a product of functions of each individual coordinate. There is a special form of separation of variables called [math]\displaystyle{ R }[/math]-separation of variables which is accomplished by writing the solution as a particular fixed function of the coordinates multiplied by a product of functions of each individual coordinate. Laplace's equation on [math]\displaystyle{ {\mathbb R}^n }[/math] is an example of a partial differential equation which admits solutions through [math]\displaystyle{ R }[/math]-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

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

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

[math]\displaystyle{ V(x_1,x_2,x_3) = V_1(x_1) + V_2(x_2) + V_3(x_3), }[/math]

then it turns out that the problem can be separated into three one-dimensional ODEs for functions [math]\displaystyle{ \psi_1(x_1) }[/math], [math]\displaystyle{ \psi_2(x_2) }[/math], and [math]\displaystyle{ \psi_3(x_3) }[/math], and the final solution can be written as [math]\displaystyle{ \psi(\mathbf{x}) = \psi_1(x_1) \cdot \psi_2(x_2) \cdot \psi_3(x_3) }[/math]. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)