Separable partial differential equation
A separable partial differential equation is one that can be broken into a set of separate equations of lower dimensionality 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 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.Example
For example, consider the time-independent Schrödinger equation
for the function . 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.
