Wiener–Hopf method: Difference between revisions

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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]].

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== Wiener–Hopf decomposition ==

The fundamental equation that appears in the Wiener-Hopf method is of the form

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

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: <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}}

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: <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>

(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^{+}\widehat{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)\widehat{g\,}(k)=G^{+}(k)+G^{-}(k). </math>

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

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*[[Riemann–Hilbert problem]]

== Notes ==

{{reflist}}

== References ==

* {{Cite web|title=Category:Wiener-Hopf - WikiWaves|url=https://wikiwaves.org/Category:Wiener-Hopf|website=wikiwaves.org|access-date=2020-05-19}}

* {{SpringerEOM |id=W/w097910|title=Wiener-Hopf method}}

* {{Cite book|last=Fornberg, Bengt|url=https://www.worldcat.org/oclc/1124781689|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}}