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| 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 equation]]s (ODEs) if the problem can be broken down into one-dimensional equations.
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| 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 <math>R</math>-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 <math>{\mathbb R}^n</math> is an example of a partial differential equation which admits solutions through <math>R</math>-separation of variables.
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| (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 [[integral]]s; see [[separation of variables]].)
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| == Example ==
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| For example, consider the time-independent [[Schrödinger equation]]
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| :<math>[-\nabla^2 + V(\mathbf{x})]\psi(\mathbf{x}) = E\psi(\mathbf{x})</math> | |
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| for the function <math>\psi(\mathbf{x})</math> (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous [[Helmholtz equation]].) If the function <math>V(\mathbf{x})</math> in three dimensions is of the form
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| :<math>V(x_1,x_2,x_3) = V_1(x_1) + V_2(x_2) + V_3(x_3),</math>
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| then it turns out that the problem can be separated into three one-dimensional ODEs for functions <math>\psi_1(x_1)</math>, <math>\psi_2(x_2)</math>, and <math>\psi_3(x_3)</math>, and the final solution can be written as <math>\psi(\mathbf{x}) = \psi_1(x_1) \cdot \psi_2(x_2) \cdot \psi_3(x_3)</math>. (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.<ref>L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," ''Phys. Rev.'' '''74''', 87-89 (1948).</ref>)
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| == References == | |
| <references/>
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| [[Category:Differential equations]]
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Latest revision as of 20:31, 9 December 2014
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