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]].

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 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>

Line 12 ⟶ 17:

: <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==

Consider the linear [[partial differential equation]]

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 {{mvar|x}} and {{mvar|y}}, subject to the mixed conditions on {{mvar|y}}&nbsp;=&nbsp;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}}&nbsp;→&nbsp;0 as <math>\boldsymbol{x}\rightarrow \infty</math>.

Taking a [[Fourier transform]] with respect to {{mvar|x}} results in the following [[ordinary differential equation]]

where <math>\boldsymbol{L}_{xy}</math> is a linear operator which contains

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

derivatives with respect to <math>x</math> and <math>y</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

subject to the mixed conditions on <math>y=0</math>, for some prescribed

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

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

: <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,

and decay at infinity i.e. <math>f\rightarrow 0</math>

: <math> \widehat{f}_{+}(k,y)=\int_0^\infty f(x,y)e^{-ikx}\,\textrm{d}x, </math>

as <math>\boldsymbol{x}\rightarrow \infty</math>. Taking a [[Fourier transform]] with respect to x results in the following [[ordinary differential equation]]

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

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

solution can be written as

: <math>

\hat{f}=C(k)\hat{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> \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>

: <math>

\hat{f}_{-}(k,0)+\hat{f}_{+}(k,0)

=

\hat{g}(k)+\hat{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>

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

: <math>

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

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

To be precise, <math> K(k)=K^{+}(k)K^{-}(k), </math> where

Now <math>K(k)</math> can be decomposed into the product of functions <math>K^{-}</math> and <math>K^{+}</math> which analytic in the upper-half plane or lower-half plane, respectively

: <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)=\log K^{+}(k)K^ = -\frac{1}{2\pi i}\int_{-\infty}^\infty \frac{\log(kK(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>

: <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>

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>

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

\hat{f}'_{-}(k,0)/K_{-}(k) - K_{+}(k)\hat{g}_{-}(k),

</math>

where it has been assumed that <math>g</math> can be broken down into functions analytic in the lower-half plane

<math>g_{+}</math> and upper-half plane <math>g_{-}</math>, respectively.

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

continution 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) = -\hat{g}_{+}(k),

</math>

and so

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

: <math>

C(k) = \frac{\hat{g}(k)-\hat{g}_{+}(k)}{F(k,0)}.

</math>

== See also ==

* [[Wiener filter]]

*[[Riemann–Hilbert problem]]

==External linksNotes ==

{{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}}

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

* {{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]]