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. 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 {displaystyle 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 {displaystyle {mathbb {R} }^{n}} is an example of a partial differential equation which admits solutions through R {displaystyle 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.) For example, consider the time-independent Schrödinger equation for the function ψ ( x ) {displaystyle psi (mathbf {x} )} (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function V ( x ) {displaystyle V(mathbf {x} )} in three dimensions is of the form then it turns out that the problem can be separated into three one-dimensional ODEs for functions ψ 1 ( x 1 ) {displaystyle psi _{1}(x_{1})} , ψ 2 ( x 2 ) {displaystyle psi _{2}(x_{2})} , and ψ 3 ( x 3 ) {displaystyle psi _{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 ) {displaystyle psi (mathbf {x} )=psi _{1}(x_{1})cdot psi _{2}(x_{2})cdot psi _{3}(x_{3})} . (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.)