Category:Wiener-Hopf - Revision history

Meylan at 23:04, 10 May 2010

2010-05-10T23:04:06Z

← Older revision		Revision as of 23:04, 10 May 2010
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	{{~~incomplete~~ pages}}		{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan: /* Solution of a semi-infinite integral equation */

2010-05-10T23:03:50Z

Solution of a semi-infinite integral equation

← Older revision		Revision as of 23:03, 10 May 2010
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	where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give		where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give
	<center>		<center>
	<math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)+K(\alpha)\Psi^+(\alpha),</math>		<math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)+\bar{K}(\alpha)\Psi^+(\alpha),</math>
	</center>		</center>
	where		where
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	and		and
	<center><math>		<center><math>
	K(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,		\bar{K}(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,
	</math></center>		</math></center>
	In the above formulae, <math>\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.		In the above formulae, <math>\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.

Meylan: /* Basic problem */

2010-05-10T23:02:18Z

Basic problem

← Older revision		Revision as of 23:02, 10 May 2010
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	<center><math>\Psi^\pm(\alpha)=\frac{G^\pm(\alpha)\pm J(\alpha)}{K^\pm(\alpha)}.</math></center>		<center><math>\Psi^\pm(\alpha)=\frac{G^\pm(\alpha)\pm J(\alpha)}{K^\pm(\alpha)}.</math></center>
	<math>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>\Psi^\pm.</math>		<math>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>\Psi^\pm.</math>

			Note that the main difficulty is to factorise the functions <math>K^-(\alpha)F(\alpha)</math> etc.

	== Solution of a semi-infinite integral equation ==		== Solution of a semi-infinite integral equation ==

Meylan: /* Solution of a semi-infinite integral equation */

2010-05-10T23:00:42Z

Solution of a semi-infinite integral equation

← Older revision		Revision as of 23:00, 10 May 2010
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\Psi^-(\alpha)=\int_{-\infty}^0 \psi(x) e^{i\alpha x}\mathrm{d}x,</math>		\Psi^-(\alpha)=\int_{-\infty}^0 \psi(x) e^{i\alpha x}\mathrm{d}x,
	</center>		</math></center>
			<center>
			<math>F(\alpha)=\int_{\infty}^\infty f(x)e^{i\alpha x}\mathrm{d}x,
			</math></center>
	and		and
	<center>		<center><math>
	<math>~~F(\alpha)=\int_{infty}^\infty f(x)e^{i\alpha x}\mathrm{d}x,\quad~~		K(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,
	K(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,</math>		</math></center>
	</center>
	In the above formulae, <math>\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.		In the above formulae, <math>\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.

Meylan: /* Solution of a semi-infinite integral equation */

2010-05-10T22:58:16Z

Solution of a semi-infinite integral equation

← Older revision		Revision as of 22:58, 10 May 2010
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	where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give		where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give
	<center>		<center>
	<math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F^+(\alpha)+K(\alpha)\Psi^+(\alpha),</math>		<math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)+K(\alpha)\Psi^+(\alpha),</math>
	</center>		</center>
	where		where
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\Psi^-(\alpha)=\int_{-\infty}^0\~~left(\int_0^\infty k~~(x~~-\xi)\psi(\xi)d\xi\right~~)e^{i\alpha x}\mathrm{d}x,</math>		\Psi^-(\alpha)=\int_{-\infty}^0 \psi(x) e^{i\alpha x}\mathrm{d}x,</math>
	</center>		</center>
	and		and
	<center>		<center>
	<math>F^+(\alpha)=\~~int_0~~^\infty f(x)e^{i\alpha x}\mathrm{d}x,\quad		<math>F(\alpha)=\int_{infty}^\infty f(x)e^{i\alpha x}\mathrm{d}x,\quad
	K(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,</math>		K(\alpha)=\int_{-\infty}^\infty k(x)e^{i\alpha x}\mathrm{d}x,</math>
	</center>		</center>
	In the above formulae, <math>\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.		In the above formulae, <math>\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.

Meylan: /* A fuller description of the theory */

2010-05-10T22:56:54Z

A fuller description of the theory

Show changes

Meylan at 22:15, 6 September 2009

2009-09-06T22:15:18Z

← Older revision		Revision as of 22:15, 6 September 2009
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			{{incomplete pages}}

	== Introduction ==		== Introduction ==

Adi Kurniawan: /* Solution of a semi-infinite integral equation */

2009-04-28T05:37:51Z

Solution of a semi-infinite integral equation

← Older revision		Revision as of 05:37, 28 April 2009
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	As mentioned above, the Wiener-Hopf method was originally devised to solve semi-infinite integral equations of the form		As mentioned above, the Wiener-Hopf method was originally devised to solve semi-infinite integral equations of the form
	<center>		<center>
	<math>\psi(x)=f(x)+\int_0^\infty k(x-\xi)\psi(\xi)d\xi,</math>		<math>\psi(x)=f(x)+\int_0^\infty k(x-\xi)\psi(\xi)\mathrm{d}\xi,</math>
	</center>		</center>
	where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give		where we shall assume that <math>\psi</math> is integrable. To demonstrate the method we take the Fourier transform of the above equation to give
	<center>		<center>
	<math>\Psi^+(\alpha)+\Psi^-(\alpha)=F^+(\alpha)+K(\alpha)\Psi^+(\alpha),</math>		<math>\,\!\Psi^+(\alpha)+\Psi^-(\alpha)=F^+(\alpha)+K(\alpha)\Psi^+(\alpha),</math>
	</center>		</center>
	where		where

Adi Kurniawan: /* Basic problem */

2009-04-28T05:37:20Z

Basic problem

← Older revision		Revision as of 05:37, 28 April 2009
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	<math>\Psi^\pm(\alpha)</math> given an equation the form		<math>\Psi^\pm(\alpha)</math> given an equation the form
	<center>		<center>
	<math>K(\alpha)\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)</math>		<math>K(\alpha)\Psi^+(\alpha)+\Psi^-(\alpha)=F(\alpha)\,\!</math>
	</center>		</center>

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	The first step in the solution is to write <math>K(\alpha)=K^+(\alpha)/K^-(\alpha)</math>, making the original equation become		The first step in the solution is to write <math>K(\alpha)=K^+(\alpha)/K^-(\alpha)</math>, making the original equation become
	<center>		<center>
	<math>K^+(\alpha)\Psi^+(\alpha)+K^-(\alpha)\Psi^-(\alpha)=K^-(\alpha)F(\alpha);		<math>\,\!K^+(\alpha)\Psi^+(\alpha)+K^-(\alpha)\Psi^-(\alpha)=K^-(\alpha)F(\alpha);
	</math></center>		</math></center>
	the next step is to write <math>K^-(\alpha)F(\alpha)=G^+(\alpha)+G^-(\alpha)</math>,		the next step is to write <math>K^-(\alpha)F(\alpha)=G^+(\alpha)+G^-(\alpha)</math>,
	which implies		which implies
	<center>		<center>
	<math>K^+(\alpha)\Psi^+(\alpha)-G^+(\alpha)=G^-(\alpha)-K^-(\alpha)\Psi^-(\alpha).		<math>\,\!K^+(\alpha)\Psi^+(\alpha)-G^+(\alpha)=G^-(\alpha)-K^-(\alpha)\Psi^-(\alpha).
	</math></center>		</math></center>
	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>J(\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>J(\alpha).</math>

Adi Kurniawan at 05:36, 28 April 2009

2009-04-28T05:36:33Z

← Older revision		Revision as of 05:36, 28 April 2009
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	= Introduction =		== Introduction ==

	A method for solving equations in which properties of the function in the complex plane are		A method for solving equations in which properties of the function in the complex plane are
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	[[:Category:Floating Elastic Plate\|Floating Elastic Plate]] ([[Wiener-Hopf Elastic Plate Solution]]).		[[:Category:Floating Elastic Plate\|Floating Elastic Plate]] ([[Wiener-Hopf Elastic Plate Solution]]).

	= ~~The~~ Basic ~~Problem~~ =		== Basic problem ==

	The Basic Wiener-Hopf problem is to find a pair of unknown functions		The Basic Wiener-Hopf problem is to find a pair of unknown functions
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	<math>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>\Psi^\pm.</math>		<math>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>\Psi^\pm.</math>

	= ~~The~~ Solution of a ~~Semi~~-~~Infinite Integral Equation~~ =		== 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		As mentioned above, the Wiener-Hopf method was originally devised to solve semi-infinite integral equations of the form
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	In the above formulae, <math>\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.		In the above formulae, <math>\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.

	= A ~~Fuller Description~~ of the ~~Theory~~ =		== A fuller description of the theory ==

	The Wiener-Hopf technique is an extension of the [http://en.wikipedia.org/wiki/Fourier_transform Fourier transform] method to semi-infinite domains of simple geometry, such as those with a straight or circular boundary. In the Wiener-Hopf technique the Fourier transform variable <math>\alpha </math> is extended into the complex plane. The transform <math>\hat{\phi }\left( \alpha,z\right) </math> may then have singularities on the complex plane depending on the integrability of <math>\phi\left( x,z\right) </math>.		The Wiener-Hopf technique is an extension of the [http://en.wikipedia.org/wiki/Fourier_transform Fourier transform] method to semi-infinite domains of simple geometry, such as those with a straight or circular boundary. In the Wiener-Hopf technique the Fourier transform variable <math>\alpha </math> is extended into the complex plane. The transform <math>\hat{\phi }\left( \alpha,z\right) </math> may then have singularities on the complex plane depending on the integrability of <math>\phi\left( x,z\right) </math>.