Wiener–Hopf method: Difference between revisions

Revision as of 13:29, 3 December 2012

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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

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: <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 continutioncontinuation 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>