KdV Equation Derivation - Revision history

Adi Kurniawan: /* Derivation */

2010-11-06T09:56:31Z

Derivation

← Older revision		Revision as of 09:56, 6 November 2010
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	where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.		where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.

	In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})ds</math> and introduce further transformation to remove <font size='3'><math>\epsilon</math></font> from the equations,		In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})\mathrm{d}s</math> and introduce further transformation to remove <font size='3'><math>\epsilon</math></font> from the equations,

	<center><math>		<center><math>

Meylan at 10:21, 6 September 2010

2010-09-06T10:21:21Z

← Older revision		Revision as of 10:21, 6 September 2010
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	For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]		For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]

	[[Category:Nonlinear Water ~~Waves~~]]		[[Category:Nonlinear Water-Wave Theory]]

Meylan at 10:18, 6 September 2010

2010-09-06T10:18:29Z

← Older revision		Revision as of 10:18, 6 September 2010
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	{{~~incomplete~~ pages}}		{{complete pages}}


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	For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]		For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]

	[[Category:~~789~~]]		[[Category:Nonlinear Water Waves]]

Jcline1: corrected exponents alpha of subleading terms of eqs. (5) and (6)

2010-08-14T02:35:14Z

corrected exponents alpha of subleading terms of eqs. (5) and (6)

← Older revision		Revision as of 02:35, 14 August 2010
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	<center><math>		<center><math>
	\begin{matrix}		\begin{matrix}
	&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^2) &(5)\\ \\		&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^3) &(5)\\ \\
	&H &= &H_0 + \alpha H_1 + o(\alpha) &(6)		&H &= &H_0 + \alpha H_1 + o(\alpha^2) &(6)
	\end{matrix}		\end{matrix}
	</math></center>		</math></center>

Meylan at 09:29, 20 October 2009

2009-10-20T09:29:52Z

← Older revision		Revision as of 09:29, 20 October 2009
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			{{incomplete pages}}


	We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].		We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

Mala058 at 15:24, 14 October 2008

2008-10-14T15:24:09Z

← Older revision		Revision as of 15:24, 14 October 2008
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	We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].		We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

	== Introduction ==		==Introduction==

	In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math>~~, are~~ involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The <b>Korteweg-de Vries Equation</b> verifies the relation between dispersion and nonlinearity properties.		In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The <b>Korteweg-de Vries Equation</b> verifies the relation between dispersion and nonlinearity properties.

	== Derivation ==		==Derivation==

	We begin with the equations for waves on water,		We begin with the equations for waves on water,
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	which is named <b>Korteweg-de Vries (KdV)</b> equation.		which is named <b>Korteweg-de Vries (KdV)</b> equation.

	== Interpretation ==		==Interpretation==

	KdV equation includes dispersive effects through the term <font size='3'><math>H_{0,zzz}</math></font> and nonlinear effects through the term <font size='3'><math>H_0H_{0,z}</math></font> and governs the behavior of the small amplitude waves, with <font size='3'><math>\alpha<<1</math></font>. It is reasonable to ask when and where the independent variables, <font size='3'><math>z</math></font> and <font size='3'><math>\tau</math></font>, are of <font size='3'><math>O(1)</math></font> in order to determine more precisely the region in physical space where the KdV equation is valid as an approximation of the actual flow. According to the definition of <font size='3'><math>z</math></font> and <font size='3'><math>\tau</math></font>, if <font size='3'><math>\alpha=O(\epsilon^2)</math></font>, then <math>\bar{t}>>1</math> and <math>\bar{x}=\bar{t}+O(1)</math>. This leads us to interpret any waveform that arises as a solution of the KdV equation as the large time limit of an initial value problem.		KdV equation includes dispersive effects through the term <font size='3'><math>H_{0,zzz}</math></font> and nonlinear effects through the term <font size='3'><math>H_0H_{0,z}</math></font> and governs the behavior of the small amplitude waves, with <font size='3'><math>\alpha<<1</math></font>. It is reasonable to ask when and where the independent variables, <font size='3'><math>z</math></font> and <font size='3'><math>\tau</math></font>, are of <font size='3'><math>O(1)</math></font> in order to determine more precisely the region in physical space where the KdV equation is valid as an approximation of the actual flow. According to the definition of <font size='3'><math>z</math></font> and <font size='3'><math>\tau</math></font>, if <font size='3'><math>\alpha=O(\epsilon^2)</math></font>, then <math>\bar{t}>>1</math> and <math>\bar{x}=\bar{t}+O(1)</math>. This leads us to interpret any waveform that arises as a solution of the KdV equation as the large time limit of an initial value problem.

Mala058 at 15:22, 14 October 2008

2008-10-14T15:22:42Z

@@ Line 98: / Line 98: @@
 </math></center>
-== Summery ==
+Differentiating (b) and eliminating <math>H_1</math> and <math>B_1</math> from (c) allow us to write,
 [[Category:789]]

Mala058 at 14:56, 14 October 2008

2008-10-14T14:56:58Z

← Older revision		Revision as of 14:56, 14 October 2008
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	</math></center>		</math></center>

	Now we use asymptotic expansions of the form,		The boundary condition (4) expresses <math>\Psi</math> at the flat bed, <math>\bar{y}=0</math>. The boundary condition (3) is <i> Bernoulli equation</i> and (2) is <i>kinematic </i> boundary condition. Now we use asymptotic expansions of the form,

	<center><math>		<center><math>
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	</math></center>		</math></center>

	to derive an equation for each <font size='3'><math>H_i</math></font>.		to derive an equation for each <font size='3'><math>H_i</math></font> according to the boundary conditions (2) to (4).


	<b>* Derivation of <font size='3'><math>~~H_1~~</math></font>:</b>		<b>* Derivation of <font size='3'><math>H_i</math></font>'s:</b>

	Substituting (5) and (6), (1) must be true for all powers of <font size='3'><math>\alpha</math></font>. Therefore,		Substituting (5) and (6), (1) must be true for all powers of <font size='3'><math>\alpha</math></font>. Therefore,
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	</math></center>		</math></center>

			Now at leading order the Bernoulli and kinematic equations, (3) and (2), gives,

			<center><math>
			\begin{matrix}
			&H_0(z,\tau) = \Psi_{0,z} = B_{0,z} &(a) \\ \\
			&H_1-B_{1,z}+\frac{1}{2}B_{0,zzz}+B_{0,\tau}+\frac{1}{2}B^2_{0,z} = 0 &(b) \\ \\
			&-H_0B_{0,zz}+\frac{1}{6}B_{0,zzzz}-B_{1,zz} = -H_{1,z}+H_{0,\tau}+B_{0,z}H_{0,z} &(c)
			\end{matrix}
			</math></center>

	== Summery ==		== Summery ==
	[[Category:789]]		[[Category:789]]

Mala058 at 14:32, 14 October 2008

2008-10-14T14:32:14Z

← Older revision		Revision as of 14:32, 14 October 2008
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	<b>* Derivation of <font size='3'><math>H_1</math></font>:</b>		<b>* Derivation of <font size='3'><math>H_1</math></font>:</b>

	Substituting (5) and (6), (1) must be true for all powers of <font size='3'><math>\alpha</math></font> ~~therefore~~,		Substituting (5) and (6), (1) must be true for all powers of <font size='3'><math>\alpha</math></font>. Therefore,

	<center><math>		<center><math>
	\Psi_{0, \bar{y}\bar{y}} = 0 ~~\quad~~ \rArr \Psi_0 = B_0(z, \tau)		\begin{matrix}
			&O(\alpha^0) &: &\Psi_{0, \bar{y}\bar{y}} = 0 &\rArr &\Psi_0 = B_0(z, \tau) \\ \\
			&O(\alpha) &: &\Psi_{1, \bar{y}\bar{y}} = -\Psi_{0, zz} &\rArr &\Psi_1 = -\frac{1}{2}{\bar{y}}^2 B_{0, zz}+B_1(z, \tau) \\ \\
			&O(\alpha^2) &: &\Psi_{2, \bar{y}\bar{y}} = -\Psi_{1, zz} &\rArr &\Psi_2 = \frac{1}{24}{\bar{y}}^4B_{0,zzzz}-\frac{1}{2}{\bar{y}}^2 B_{1,zz}+ B_2(z, \tau)
			\end{matrix}
	</math></center>		</math></center>

Mala058 at 12:07, 14 October 2008

2008-10-14T12:07:17Z

← Older revision		Revision as of 12:07, 14 October 2008
Line 1:		Line 1:
	We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].		We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

	===Introduction===		== Introduction ==

	In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math>, are involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The <b>Korteweg-de Vries Equation</b> verifies the relation between dispersion and nonlinearity properties.		In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math>, are involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The <b>Korteweg-de Vries Equation</b> verifies the relation between dispersion and nonlinearity properties.

	===Derivation===		== Derivation ==

	We begin with the equations for waves on water,		We begin with the equations for waves on water,
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	where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.		where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.

	~~----~~

	In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})ds</math> and introduce further transformation to remove <font size='3'><math>\epsilon</math></font> from the equations,		In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})ds</math> and introduce further transformation to remove <font size='3'><math>\epsilon</math></font> from the equations,
Line 70:		Line 68:
	<center><math>		<center><math>
	\begin{matrix}		\begin{matrix}
	&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^2) \\ \\		&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^2) &(5)\\ \\
	&H &= &H_0 + \alpha H_1 + o(\alpha)		&H &= &H_0 + \alpha H_1 + o(\alpha) &(6)
	\end{matrix}		\end{matrix}
	</math></center>		</math></center>

	to derive an equation for each <math>H_i</math>.		to derive an equation for each <font size='3'><math>H_i</math></font>.


			<b>* Derivation of <font size='3'><math>H_1</math></font>:</b>

			Substituting (5) and (6), (1) must be true for all powers of <font size='3'><math>\alpha</math></font> therefore,

			<center><math>
			\Psi_{0, \bar{y}\bar{y}} = 0 \quad \rArr \Psi_0 = B_0(z, \tau)
			</math></center>

	~~----~~

	===Summery===		== Summery ==
	[[Category:789]]		[[Category:789]]