Separable partial differential equation

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Differential equations

Scope

Fields

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

* * *

List of named differential equations

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

Existence and uniqueness

* Picard–Lindelöf theorem
* Peano existence theorem
* Carathéodory's existence theorem
* Cauchy–Kowalevski theorem

General topics

* Initial conditions
* Boundary values
* Dirichlet
* Neumann
* Robin
* Cauchy problem
* Wronskian
* Phase portrait
* Lyapunov / Asymptotic / Exponential stability
* Rate of convergence
* Series / Integral solutions
* Numerical integration
* Dirac delta function

Solution methods

* Inspection
* Method of characteristics

Euler
* Exponential response formula
* Finite difference (Crank–Nicolson)
* Finite element
* Infinite element
* Finite volume
* Galerkin
* Petrov–Galerkin
* Green's function
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

List

* Isaac Newton
* Gottfried Leibniz
* Jacob Bernoulli
* Leonhard Euler
* Joseph-Louis Lagrange
* Józef Maria Hoene-Wroński
* Joseph Fourier
* Augustin-Louis Cauchy
* George Green
* Carl David Tolmé Runge
* Martin Kutta
* Rudolf Lipschitz
* Ernst Lindelöf
* Émile Picard
* Phyllis Nicolson
* John Crank

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* t
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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 {\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 that 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.)

## Example

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For example, consider the time-independent Schrödinger equation

[ − ∇ 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) {\displaystyle [-\nabla ^{2}+V(\mathbf {x} )]\psi (\mathbf {x} )=E\psi (\mathbf {x} )}

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

V ( x 1 , x 2 , x 3 ) = V 1 ( x 1 ) + V 2 ( x 2 ) + V 3 ( x 3 ) , {\displaystyle 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 ) {\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.[1])

## References

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

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