Separable partial differential equation
- Separable partial differential equation
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.
(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
:
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. [L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," "Phys. Rev." 74, 87-89 (1948).] )
References
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