Separable partial differential equation - Alchetron, the free social 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 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 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 is an example of a partial differential equation which admits solutions through 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 )

for the function ψ ( x ) (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function V ( 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 into three one-dimensional ODEs for functions ψ 1 ( x 1 ) , ψ 2 ( x 2 ) , and ψ 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 ) . (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)

References

Separable partial differential equation Wikipedia

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