Editing Separation of variables (section)

===Partial differential equations===

For many PDEs, such as the wave equation, Helmholtz equation and Schrödinger equation, the applicability of separation of variables is a result of the [[spectral theorem]]. In some cases, separation of variables may not be possible. Separation of variables may be possible in some coordinate systems but not others,<ref name="MathWorld">''John Renze, Eric W. Weisstein'', Separation of variables</ref> and which coordinate systems allow for separation depends on the symmetry properties of the equation.<ref name="Miller1984">Willard Miller(1984) ''Symmetry and Separation of Variables'', Cambridge University Press</ref> Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in individual cases (for instance in the biharmonic equation above).

Consider an initial boundary value problem for a function <math> u(x,t) </math> on <math> D = \{(x,t): x \in [0,l], t \geq 0 \}</math> in two variables:

:<math> (Tu)(x,t) = (Su)(x,t) </math>

where <math>T</math> is a differential operator with respect to <math>x</math> and <math>S</math> is a differential operator with respect to <math>t</math> with boundary data:

:<math>(Tu)(0,t) = (Tu)(l,t) = 0</math> for <math> t \geq 0</math> 
:<math>(Su)(x,0)=h(x) </math> for <math> 0 \leq x \leq l</math>

where <math>h</math> is a known function.

We look for solutions of the form <math>u(x,t) = f(x) g(t)</math>. Dividing the PDE through by <math>f(x)g(t)</math> gives

:<math> \frac{Tf}{f} = \frac{Sg}{g} </math>

The right hand side depends only on <math>x</math> and the left hand side only on <math> t</math> so both must be equal to a constant <math> K </math>, which gives two ordinary differential equations

:<math>Tf = Kf, Sg = Kg</math>

which we can recognize as eigenvalue problems for the operators for <math>T</math> and <math>S</math>. If <math>T</math> is a compact, self-adjoint operator on the space <math>L^2[0,l]</math> along with the relevant boundary conditions, then by the Spectral theorem there exists a basis for <math>L^2[0,l]</math> consisting of eigenfunctions for <math>T</math>. Let the spectrum of <math>T</math> be <math>E</math> and let <math>f_{\lambda}</math> be an eigenfunction with eigenvalue <math>\lambda \in E</math>. Then for any function which at each time <math>t</math> is square-integrable with respect to <math>x</math>, we can write this function as a linear combination of the <math>f_{\lambda}</math>. In particular, we know the solution <math>u</math> can be written as

:<math>u(x,t) = \sum_{\lambda \in E} c_{\lambda}(t)f_{\lambda}(x)</math>

For some functions <math>c_{\lambda}(t)</math>. In the separation of variables, these functions are given by solutions to <math> Sg = Kg </math>

Hence, the spectral theorem ensures that the separation of variables will (when it is possible) find all the solutions.

For many differential operators, such as <math>\frac{d^2}{dx^2}</math>, we can show that they are self-adjoint by integration by parts. While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception of zero).<ref name="Benson2007">David Benson (2007) ''Music: A Mathematical Offering'', Cambridge University Press, Appendix W</ref>