en>Tawharanui: removed CS from the nominator

2013-06-10T04:10:20Z

removed CS from the nominator

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The '''Wiener–Hopf method''' is a mathematical technique widely used in [[applied mathematics]]. It was initially developed by [[Norbert Wiener]] and [[Eberhard Hopf]] as a method to solve systems of [[integral equation]]s, but has found wider use in solving two-dimensional [[partial differential equation]]s with mixed [[boundary conditions]] on the same boundary. In general, the method works by exploiting the [[Complex analysis|complex-analytical]] properties of transformed functions. Typically, the standard [[Fourier transform]] is used, but examples exist using other transforms, such as the [[Mellin transform]].

In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively [[analytic function|analytic]] in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the [[complex plane]], typically, a thin strip containing the [[real line]]. [[Analytic continuation]] guarantees that these two functions define a single function analytic in the entire complex plane, and [[Liouville's theorem (complex analysis)|Liouville's theorem]] implies that this function is an unknown [[polynomial]], which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial.

== Wiener–Hopf decomposition ==
The key step in many Wiener–Hopf problems is to decompose an arbitrary function <math>\Phi</math> into two functions <math>\Phi_{\pm}</math> with the desired properties outlined above. In general, this can be done by writing

: <math>\Phi_+(\alpha) = \frac{1}{2\pi i} \int_{C_1} \Phi(z) \frac{dz}{z-\alpha}</math>

and

: <math>\Phi_-(\alpha) = - \frac{1}{2\pi i} \int_{C_2} \Phi(z) \frac{dz}{z-\alpha},</math>

where the contours <math>C_1</math> and <math>C_2</math> are parallel to the real line, but pass above and below the point <math>z=\alpha</math>, respectively.

Similarly, arbitrary scalar functions may be decomposed into a product of +/− functions, i.e. <math>K(\alpha) = K_+(\alpha)K_-(\alpha)</math>, by first taking the logarithm, and then performing a sum decomposition. Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.

==Example==
Let us consider the linear [[partial differential equation]]

:<math>\boldsymbol{L}_{xy}f(x,y)=0,</math>

where <math>\boldsymbol{L}_{xy}</math> is a linear operator which contains
derivatives with respect to <math>x</math> and <math>y</math>,
subject to the mixed conditions on <math>y=0</math>, for some prescribed
function <math>g(x)</math>,

:<math>f=g(x)\text{ for }x\leq 0, \quad f_{y}=0\text{ when }x>0</math>

and decay at infinity i.e. <math>f\rightarrow 0</math> as <math>\boldsymbol{x}\rightarrow \infty</math>. Taking a [[Fourier transform]] with respect to <math>x</math> results in the following [[ordinary differential equation]]

: <math>\boldsymbol{L}_{y}\hat{f}(k,y)-P(k,y)\hat{f}(k,y)=0,</math>

where <math>\boldsymbol{L}_{y}</math> is a linear operator containing <math>y</math> derivatives only, <math>P(k,y)</math> is a known function of <math>y</math> and <math>k</math> and

: <math> \hat{f}(k,y)=\int_{-\infty}^{\infty} f(x,y)e^{-ikx}\textrm{d}x. </math>

If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity is denoted <math>F(k,y)</math>, a general solution can be written as

: <math> \hat{f}(k,y)=C(k)F(k,y), </math>

where <math>C(k)</math> is an unknown function to be determined by the boundary conditions on <math>y=0</math>.

The key idea is to split <math>\hat{f}</math> into two separate functions, <math>\hat{f}_{+}</math> and <math>\hat{f}_{-}</math> which are analytic in the lower- and upper-halves of the complex plane, respectively

: <math> \hat{f}_{+}(k,y)=\int_{0}^{\infty} f(x,y)e^{-ikx}\textrm{d}x, </math>

: <math> \hat{f}_{-}(k,y)=\int_{-\infty}^{0} f(x,y)e^{-ikx}\textrm{d}x. </math>

The boundary conditions then give

: <math> \hat{g}(k)+\hat{f}_{+}(k,0) = \hat{f}_{-}(k,0)+\hat{f}_{+}(k,0) = \hat{f}(k,0) = C(k)F(k,0) </math>

and, on taking derivatives with respect to <math>y</math>,

: <math> \hat{f}'_{-}(k,0) = \hat{f}'_{-}(k,0)+\hat{f}'_{+}(k,0) = \hat{f}'(k,0) = C(k)F'(k,0). </math>

Eliminating <math>C(k)</math> yields

: <math> \hat{g}(k)+\hat{f}_{+}(k,0) = \hat{f}'_{-}(k,0)/K(k), </math>

where

: <math> K(k)=\frac{F'(k,0)}{F(k,0)}. </math>

Now <math>K(k)</math> can be decomposed into the product of functions <math>K^{-}</math> and <math>K^{+}</math> which are analytic in the upper and lower half-planes respectively. To be precise, <math> K(k)=K^{+}(k)K^{-}(k), </math> where

: <math> \hbox{log}K^{-} = \frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{\hbox{log}(K(z))}{z-k} \textrm{d}z, \quad \hbox{Im}k>0, </math>

: <math> \hbox{log}K^{+} = -\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{\hbox{log}(K(z))}{z-k} \textrm{d}z, \quad \hbox{Im}k<0. </math>

(Note that this sometimes involves scaling <math>K</math> so that it tends to <math>1</math> as <math>k\rightarrow\infty</math>.) We also decompose <math>K^{+}\hat{g}</math> into the sum of two functions <math>G^{+}</math> and <math>G^{-}</math> which are analytic in the lower and upper half-planes respectively – i.e.,
:: <math> K^{+}(k)\hat{g}(k)=G^{+}(k)+G^{-}(k). </math>

This can be done in the same way that we factorised <math> K(k). </math>
Consequently,

: <math> G^{+}(k) + K_{+}(k)\hat{f}_{+}(k,0) = \hat{f}'_{-}(k,0)/K_{-}(k) - G^{-}(k). </math>

Now, as the left-hand side of the above equation is analytic in the lower half-plane, whilst the right-hand side is analytic in the upper half-plane, analytic continuation guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large <math>k</math>, an application of [[Liouville's theorem (complex analysis)|Liouville's theorem]] shows that this entire function is identically zero, therefore

:<math> \hat{f}_{+}(k,0) = -\frac{G^{+}(k)}{K^{+}(k)}, </math>

and so

: <math> C(k) = \frac{K^{+}(k)\hat{g}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}. </math>

== See also ==

* [[Wiener filter]]

==External links==
* {{SpringerEOM |id=W/w097910}}
* [http://www.wikiwaves.org/index.php/Category:Wiener-Hopf Wiener–Hopf method] at Wikiwaves

{{DEFAULTSORT:Wiener-Hopf method}}
[[Category:Partial differential equations]]

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en>Tawharanui: removed CS from the nominator