View source for Fourier method

''method of separation of variables''

A method for finding particular solutions of differential equations

$$  \begin{equation}  (Lu)(x,y)=Mu-Nu=0, \quad x \in \mathbf{R}^n, \quad y \in \mathbf{R}^m \tag{1} \end{equation} $$

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410702.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410703.png" />) are linear differential expressions containing only derivatives with respect to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410704.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410705.png" />) with coefficients also depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410706.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410707.png" />). A function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410708.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is a solution of (1) if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f0410709.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

For example, for the equation of a vibrating string

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107012.png" />, and the solution (2) takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107016.png" />, are arbitrary constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107017.png" />. The solutions (5) satisfy the boundary conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107021.png" />. The so-called Fourier series composed of these functions,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

gives the solution of the initial-boundary value problem (4), (6) and initial condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107025.png" /> are the Fourier coefficients

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107027.png" /></td> </tr></table>

of the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107029.png" /> which are assumed to be sufficiently smooth. In a similar way solutions of initial-boundary value problems can be obtained for more general classes of equations (1), where the role of the theory of Fourier series associated with the expansion (7) is played by the spectral theory of linear operators.

The Fourier method is closely connected with the special functions solving equations (3) for particular cases of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107032.png" />. Many of these functions originally arose in this way. For example, when applying this method to the Helmholtz equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107033.png" /></td> </tr></table>

one obtains after transition to polar coordinates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041070/f04107034.png" /></td> </tr></table>

and the first equation in (3) becomes the Bessel differential equation.

Generally speaking, a differential equation has a whole family of coordinate systems in which it admits separation of variables, that is, reduces to the form (1). The problem of finding such coordinate systems is closely connected with group properties of differential equations. Applying methods from the theory of Lie groups enables one to describe all the separable solutions of many classical equations in mathematical physics (those of Laplace, Helmholtz and Schrödinger, the wave equation, etc.). In this way also a whole series of relations is obtained in the theory of special functions.

The method of separation of variables was suggested by J. d'Alembert (1749) for solving the wave equation, the method was developed fairly thoroughly at the beginning of the 19th century by J. Fourier and was formulated in complete generality by M.V. Ostrogradski in 1828.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  U. Miller,   "Symmetry and separation of variables" , Addison-Wesley  (1977)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Bluman,   G.D. Cole,   "Similarity methods for differential equations" , Springer  (1974)</TD></TR></table>