Difference between revisions of "Category:Wiener-Hopf"

Introduction

A method for solving equations in which properties of the function in the complex plane are used to find the solution. It was developed initially for integral equations over a semi-infinite interval and it is used in water waves for linear problems in which there is a semi-infinite region. The theory is described in Noble 1958 and Carrier, Krook and Pearson 1966 and is mentioned in Linton and McIver 2001.

The Wiener-Hopf method is extremely powerful and it furnishes what is virtually an explicit solution to very complicated problems which at first sight seem tractable only by numerical methods. It is far from simply to apply and it relies very much on a simple situation so that the method often does not generalise or is not applicable to even slightly more complicated problems.

It has been applied to several problems in linear water wave theory such as scattering by an infinitely thin semi-infinite breakwater or a semi-infinite Floating Elastic Plate (Wiener-Hopf Elastic Plate Solution).

Basic problem

The Basic Wiener-Hopf problem is to find a pair of unknown functions [math]\displaystyle{ \Psi^\pm(\alpha) }[/math] given an equation the form

[math]\displaystyle{ K(\alpha)\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)\,\! }[/math]

which is satisfied on a line in the complex plane (usually the real line, but not necessarily), and where the functions [math]\displaystyle{ K }[/math] and [math]\displaystyle{ F }[/math] are known. The function [math]\displaystyle{ \Psi^+(\alpha) }[/math] is analytic on one side of the line (we shall call this side the "left") and [math]\displaystyle{ \Psi^-(\alpha) }[/math] is analytic on the other side (the "right"), and all other functions with [math]\displaystyle{ \pm }[/math] superscripts will have be analytic in the same regions as the [math]\displaystyle{ \Psi^\pm }[/math] functions (respectively).

The first step in the solution is to write [math]\displaystyle{ K(\alpha)=K^+(\alpha)/K^-(\alpha) }[/math], making the original equation become

[math]\displaystyle{ \,\!K^+(\alpha)\Psi^+(\alpha)+K^-(\alpha)\Psi^-(\alpha)=K^-(\alpha)F(\alpha); }[/math]

the next step is to write [math]\displaystyle{ K^-(\alpha)F(\alpha)=G^+(\alpha)+G^-(\alpha) }[/math], which implies

[math]\displaystyle{ \,\!K^+(\alpha)\Psi^+(\alpha)-G^+(\alpha)=G^-(\alpha)-K^-(\alpha)\Psi^-(\alpha). }[/math]

Now, the left hand side of the above equation will be analytic to the left of the line that it holds on and the right hand side will be analytic to its left. From the Riemann principle they are consequently the analytic continuations of each other across the line and are both equal to a single entire function [math]\displaystyle{ J(\alpha). }[/math]

Hence

[math]\displaystyle{ \Psi^\pm(\alpha)=\frac{G^\pm(\alpha)\pm J(\alpha)}{K^\pm(\alpha)}. }[/math]

[math]\displaystyle{ J(\alpha) }[/math] is usually limited by Liouville's theorem to being a polynomial whose coefficients are determined by applying some additional conditions on the [math]\displaystyle{ \Psi^\pm. }[/math]

Note that the main difficulty is to factorise the functions [math]\displaystyle{ K^-(\alpha)F(\alpha) }[/math] etc.

Solution of a semi-infinite integral equation

As mentioned above, the Wiener-Hopf method was originally devised to solve semi-infinite integral equations of the form

[math]\displaystyle{ \psi(x)=f(x)+\int_0^\infty k(x-\xi)\psi(\xi)\mathrm{d}\xi, }[/math]

where we shall assume that [math]\displaystyle{ \psi }[/math] is integrable. To demonstrate the method we take the Fourier transform of the above equation to give

[math]\displaystyle{ \,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)+\bar{K}(\alpha)\Psi^+(\alpha), }[/math]

where

[math]\displaystyle{ \Psi^+(\alpha)=\int_0^\infty\psi(x)e^{i\alpha x}\mathrm{d}x, }[/math] [math]\displaystyle{ \Psi^-(\alpha)=\int_{-\infty}^0 \psi(x) e^{i\alpha x}\mathrm{d}x, }[/math] [math]\displaystyle{ F(\alpha)=\int_{\infty}^\infty f(x)e^{i\alpha x}\mathrm{d}x, }[/math]

and

[math]\displaystyle{ \bar{K}(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x, }[/math]

In the above formulae, [math]\displaystyle{ \pm }[/math] superscripts denote functions that are in the upper or lower complex half planes, and the Fourier transform of the integral equation is clearly a Wiener-Hopf equation on the real line, as described in the previous section.