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{{Short description|Mathematical method for integrodifferential equations}}
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 '
==
The fundamental equation that appears in the Wiener-Hopf method is of the form
:<math>A(\alpha)\Xi_+(\alpha) + B(\alpha)\Psi_-(\alpha) + C(\alpha) =0, </math>
where <math>A</math>, <math>B</math>, <math>C</math> are known [[holomorphic function]]s, the functions <math>\Xi_+(\alpha)</math>, <math>\Psi_-(\alpha)</math> are unknown and the equation holds in a strip <math>\tau_- < \mathfrak{Im}(\alpha) < \tau_+</math> in the [[Complex_plane|complex <math>\alpha</math> plane]]. Finding <math>\Xi_+(\alpha)</math>, <math>\Psi_-(\alpha)</math> is what's called the '''Wiener-Hopf problem'''.{{sfn | Noble | 1958 | loc=§4.2}}
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_
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.{{sfn | Noble | 1958 | loc=Chapter 1}}
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.{{citation needed|date=May 2020}}
==Example==
:<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 {{mvar|x}} and {{mvar|y}}, subject to the mixed conditions on {{mvar|y}} = 0, for some prescribed function {{math|''g''(''x'')}},
:<math>f=g(x)\text{ for }x\leq 0, \quad f_y=0\text{ when }x>0</math>
and decay at infinity i.e. {{mvar|f}} → 0 as <math>\boldsymbol{x}\rightarrow \infty</math>.
Taking a [[Fourier transform]] with respect to {{mvar|x}} results in the following [[ordinary differential equation]]
: <math>\boldsymbol{L}_y \widehat{f}(k,y)-P(k,y)\widehat{f}(k,y)=0,</math>
where <math>\boldsymbol{L}_{y}</math> is a linear operator containing {{mvar|y}} derivatives only, {{math|''P''(''k,y'')}} is a known function of {{mvar|y}} and {{mvar|k}} and
: <math> \widehat{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'')}}, a general solution can be written as
: <math> \widehat{f}(k,y)=C(k)F(k,y), </math>
where {{math|''C''(''k'')}} is an unknown function to be determined by the boundary conditions on {{mvar|y}}=0.
The key idea is to split <math>\widehat{f}</math> into two separate functions, <math>\widehat{f}_{+}</math> and <math>\widehat{f}_{-}</math> which are analytic in the lower- and upper-halves of the complex plane, respectively,
: <math> \widehat{f}_{+}(k,y)=\int_0^\infty f(x,y)e^{-ikx}\,\textrm{d}x, </math>
: <math> \widehat{f}_{-}(k,y)=\int_{-\infty}^0 f(x,y)e^{-ikx}\,\textrm{d}x. </math>
The boundary conditions then give
: <math> \widehat{g\,}(k)+\widehat{f}_{+}(k,0) = \widehat{f}_{-}(k,0)+\widehat{f}_{+}(k,0) = \widehat{f}(k,0) = C(k)F(k,0) </math>
and, on taking derivatives with respect to <math>y</math>,
: <math> \widehat{f}'_{-}(k,0) = \widehat{f}'_{-}(k,0)+\widehat{f}'_{+}(k,0) = \widehat{f}'(k,0) = C(k)F'(k,0). </math>
Eliminating <math>C(k)</math> yields
: <math> \widehat{g\,}(k)+\widehat{f}_{+}(k,0) = \widehat{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> \log K^{-} = \frac{1}{2\pi i}\int_{-\infty}^\infty \frac{\log(K(z))}{z-k} \,\textrm{d}z, \quad \operatorname{Im}k>0, </math>
: <math> \log K^{+} = -\frac{1}{2\pi i}\int_{-\infty}^\infty \frac{\log(K(z))}{z-k} \,\textrm{d}z, \quad \operatorname{Im}k<0. </math>
: <math> K^{+}(k)\widehat{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)\widehat{f}_{+}(k,0) = \widehat{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 {{mvar|k}}, an application of [[Liouville's theorem (complex analysis)|Liouville's theorem]] shows that this entire function is identically zero, therefore
:<math> \widehat{f}_{+}(k,0) = -\frac{G^{+}(k)}{K^{+}(k)}, </math>
and so
: <math> C(k) = \frac{K^{+}(k)\widehat{g\,}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}. </math>
== See also ==
* [[Wiener filter]]
*[[Riemann–Hilbert problem]]
==
{{reflist}}
== References ==
*
* {{SpringerEOM |id=W/w097910|title=Wiener-Hopf method}}
* {{Cite book|last=Fornberg, Bengt|title=Complex variables and analytic functions : an illustrated introduction|others=Piret, Cécile.|date=2020 |isbn=978-1-61197-597-0|___location=Philadelphia|oclc=1124781689}}
* {{cite book | last=Noble | first=Ben | title=Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations | publisher=Taylor & Francis US | publication-place=New York, N.Y | date=1958 | isbn=978-0-8284-0332-0}}
{{DEFAULTSORT:Wiener-Hopf method}}
[[Category:Partial differential equations]]
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