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
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
. This generally relies upon the problem having some special form or
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. In this way, the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
(PDE) can be solved by solving a set of simpler PDEs, or even
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s (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
-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
is an example of a partial differential equation which admits solutions through
-separation of variables; in the three-dimensional case this uses
6-sphere coordinates
In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere x^2+y^2+z^2=1. They are so named because the loci where one coordinate is const ...
.
(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
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s; see
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
.)
Example
For example, consider the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
:
for the function
(in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous
Helmholtz equation.) If the function
in three dimensions is of the form
:
then it turns out that the problem can be separated into three one-dimensional ODEs for functions
,
, and
, and the final solution can be written as
. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.
[{{cite journal , last=Eisenhart , first=L. P. , title=Enumeration of Potentials for Which One-Particle Schroedinger Equations Are Separable , journal=Physical Review , publisher=American Physical Society (APS) , volume=74 , issue=1 , date=1948-07-01 , issn=0031-899X , doi=10.1103/physrev.74.87 , pages=87–89])
References
Differential equations