Separable partial differential equation
Encyclopedia
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 equation
s (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 integral
s; see separation of variables
.)
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 in to 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.)
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(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
Separation of variables
In mathematics, separation of variables 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 on a different side of the equation....
. This generally relies upon the problem having some special form or symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
. In this way, the PDE can be solved by solving a set of simpler PDEs, or even ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s (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 integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s; see separation of variables
Separation of variables
In mathematics, separation of variables 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 on a different side of the equation....
.)
Example
For example, consider the time-independent Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for the function
(in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equationHelmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
.) If the function
in three dimensions is of the formthen it turns out that the problem can be separated in to 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.)