Separable partial differential equation explained

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 -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 that 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

[ - \nabla ^{2} + V( x )]\psi( x ) = E\psi( x )

for the function

\psi( x ) (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function V( x ) in three dimensions is of the formV(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

\psi _{1(x} _{1)} , \psi _{2(x} _{2)} , and \psi _{3(x} _{3)} , and the final solution can be written as \psi( x ) = \psi _{1(x} _{1)} ⋅ \psi _{2(x} _{2)} ⋅ \psi _{3(x} _{3)} . (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.^[1])

References

1. Eisenhart . L. P. . Enumeration of Potentials for Which One-Particle Schroedinger Equations Are Separable . Physical Review . American Physical Society (APS) . 74 . 1 . 1948-07-01 . 0031-899X . 10.1103/physrev.74.87 . 87–89.