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.
The most common form of separation of variables is simple separation of variables. 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 R {\displaystyle R} -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 R n {\displaystyle {\mathbb {R} }^{n}} is an example of a partial differential equation that admits solutions through R {\displaystyle R} -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
[ − ∇ 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) {\displaystyle [-\nabla ^{2}+V(\mathbf {x} )]\psi (\mathbf {x} )=E\psi (\mathbf {x} )}for the function ψ ( x ) {\displaystyle \psi (\mathbf {x} )} (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function V ( x ) {\displaystyle 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 ) , {\displaystyle 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 into three one-dimensional ODEs for functions ψ 1 ( x 1 ) {\displaystyle \psi _{1}(x_{1})} , ψ 2 ( x 2 ) {\displaystyle \psi _{2}(x_{2})} , and ψ 3 ( x 3 ) {\displaystyle \psi _{3}(x_{3})} , and the final solution can be written as ψ ( x ) = ψ 1 ( x 1 ) ⋅ ψ 2 ( x 2 ) ⋅ ψ 3 ( x 3 ) {\displaystyle \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 1948.1)
References
Eisenhart, L. P. (1948-07-01). "Enumeration of Potentials for Which One-Particle Schroedinger Equations Are Separable". Physical Review. 74 (1). American Physical Society (APS): 87–89. Bibcode:1948PhRv...74...87E. doi:10.1103/physrev.74.87. ISSN 0031-899X. /wiki/Bibcode_(identifier) ↩