Wiener–Hopf method: Difference between revisions

Revision as of 13:14, 19 May 2020

Revision as of 17:38, 10 December 2020

Line 1:

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.

Line 21:

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]]

Line 30:

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,

Line 46:

: <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

Line 72:

* {{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,.|isbn=978-1-61197-597-0|___location=Philadelphia|oclc=1124781689}}

{{DEFAULTSORT:Wiener-Hopf method}}

[[Category:Partial differential equations]]