T-II-1 - Revision history

Rosso: /* The Method of characteristics */

2011-10-06T11:57:30Z

The Method of characteristics

← Older revision		Revision as of 13:57, 6 October 2011
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	* Using the method of characteristics show that the solution of Eq.1 writes:		* Using the method of characteristics show that the solution of Eq.1 writes:
	center> <math>\begin{cases} u (x,t) =u_{\text{init}}(x_0,0)\\ x(t)=x_0+ u_{\text{init}}(x_0,0)\, t \end{cases} </math></center>		<center> <math>\begin{cases} u (x,t) =u_{\text{init}}(x_0,0)\\ x(t)=x_0+ u_{\text{init}}(x_0,0)\, t \end{cases} </math></center>
	* Give an interpretation for <math> x_0</math>		* Give an interpretation for <math> x_0</math>
	* Show that <math> u(x,t)=0</math> is alway a solution for <math> \| x \| > 1 </math>. Denote by <math> u_0(x,t)</math> this solution.		* Show that <math> u(x,t)=0</math> is alway a solution for <math> \| x \| > 1 </math>. Denote by <math> u_0(x,t)</math> this solution.

2011-10-06T11:57:18Z

The Method of characteristics

← Older revision		Revision as of 13:57, 6 October 2011
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	* Using the method of characteristics show that the solution of Eq.1 writes:		* Using the method of characteristics show that the solution of Eq.1 writes:
	center> <math>\begin{cases} u (x,t) =u_{\text{init}}(x_0,0)\\ x(t)=x_0+ u_{\text{init}}(x_0,0)\, t ~~1-x^2~~ \end{cases} </math></center>		center> <math>\begin{cases} u (x,t) =u_{\text{init}}(x_0,0)\\ x(t)=x_0+ u_{\text{init}}(x_0,0)\, t \end{cases} </math></center>
	* Give an interpretation for <math> x_0</math>		* Give an interpretation for <math> x_0</math>
	* Show that <math> u(x,t)=0</math> is alway a solution for <math> \| x \| > 1 </math>. Denote by <math> u_0(x,t)</math> this solution.		* Show that <math> u(x,t)=0</math> is alway a solution for <math> \| x \| > 1 </math>. Denote by <math> u_0(x,t)</math> this solution.

2011-10-06T11:56:58Z

The Method of characteristics

← Older revision		Revision as of 13:56, 6 October 2011
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	== The Method of characteristics==		== The Method of characteristics==

	* ~~Write~~ the solution of Eq.1 ~~with the initial condition of Eq.~~0.		* Using the method of characteristics show that the solution of Eq.1 writes:
			center> <math>\begin{cases} u (x,t) =u_{\text{init}}(x_0,0)\\ x(t)=x_0+ u_{\text{init}}(x_0,0)\, t 1-x^2 \end{cases} </math></center>
			* Give an interpretation for <math> x_0</math>
	* Show that <math> u(x,t)=0</math> is alway a solution for <math> \| x \| > 1 </math>. Denote by <math> u_0(x,t)</math> this solution.		* Show that <math> u(x,t)=0</math> is alway a solution for <math> \| x \| > 1 </math>. Denote by <math> u_0(x,t)</math> this solution.
	* Show that two other solutions exist, namely:		* Show that two other solutions exist, namely:

2011-10-03T12:50:55Z

← Older revision		Revision as of 14:50, 3 October 2011
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	* Show that two other solutions exist, namely:		* Show that two other solutions exist, namely:
	<center><math>		<center><math>
	u_{\pm}(x,t)= \frac{1}{4 t^2} \left( 1 \pm \sqrt{\Delta(x,t)}\right)^2-1		u_{\pm}(x,t)= 1-\frac{1}{4 t^2} \left( 1 \pm \sqrt{\Delta(x,t)}\right)^2
	</math></center>		</math></center>
	where <math> \Delta(x,t)= 4 t^2 -4 x t +1 </math>.		where <math> \Delta(x,t)= 4 t^2 -4 x t +1 </math>.

2011-09-21T09:40:14Z

Model 0 : Deep sea

← Older revision		Revision as of 11:40, 21 September 2011
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	Consider the following equation for the shape of the wave:		Consider the following equation for the shape of the wave:

	<center><math>\partial_t u(x,t) + c \, \partial_x u(x,t) = 0 </math></center>		<center><math>\partial_t u(x,t) + v \, \partial_x u(x,t) = 0 </math></center>

	where <math>u(x,t) </math> is the height of the wave with respect to the unperturbed sea level, and <math> c </math> is constant. We will use the following initial condition <math>u(x,t=0)=u_{\text{init}}(x) </math>		where <math>u(x,t) </math> is the height of the wave with respect to the unperturbed sea level, and <math> v </math> is constant. We will use the following initial condition <math>u(x,t=0)=u_{\text{init}}(x) </math>

2011-09-21T09:39:50Z

The Method of characteristics

← Older revision		Revision as of 11:39, 21 September 2011
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	</math></center>		</math></center>
	where <math> \Delta(x,t)= 4 t^2 -4 x t +1 </math>.		where <math> \Delta(x,t)= 4 t^2 -4 x t +1 </math>.
	* Show that for <math> t<0.5 </math> only <math> u_0(x,t)</math> and <math> u_-(x,t)</math> are ~~physically~~ acceptable.		* Show that for <math> t<0.5 </math> only <math> u_0(x,t)</math> and <math> u_-(x,t)</math> are acceptable.
	* Draw the full solution for <math> t=0.5 </math> and compute its derivative at <math> x=1 </math>		* Draw the full solution for <math> t=0.5 </math> and compute its derivative at <math> x=1 </math>
	* Show that for <math> t>0.5 </math> the solution is a multivalued function <math> u_{\text{mult}}(x,t)</math>		* Show that for <math> t>0.5 </math> the solution is a multivalued function <math> u_{\text{mult}}(x,t)</math>

2011-09-21T08:55:54Z

Model 0 : Deep sea

← Older revision		Revision as of 10:55, 21 September 2011
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	Consider the following equation for the shape of the wave:		Consider the following equation for the shape of the wave:

	<center><math>\partial_t u(x,t) + c \partial_x u(x,t) = 0 </math></center>		<center><math>\partial_t u(x,t) + c \, \partial_x u(x,t) = 0 </math></center>

	where <math>u(x,t) </math> is the height of the wave with respect to the unperturbed sea level, and <math> c </math> is constant. We will use the following initial condition <math>u(x,t=0)=u_{\text{init}}(x) </math>		where <math>u(x,t) </math> is the height of the wave with respect to the unperturbed sea level, and <math> c </math> is constant. We will use the following initial condition <math>u(x,t=0)=u_{\text{init}}(x) </math>

2011-09-21T08:55:33Z

Model 0 : Deep sea

← Older revision		Revision as of 10:55, 21 September 2011
Line 12:		Line 12:
	Consider the following equation for the shape of the wave:		Consider the following equation for the shape of the wave:

	<center><math>\partial_t u(x,t) + v \partial_x u(x,t) = 0 </math></center>		<center><math>\partial_t u(x,t) + c \partial_x u(x,t) = 0 </math></center>

	where <math>u(x,t) </math> is the height of the wave with respect to the unperturbed sea level, and <math> v </math> is constant. We will use the following initial condition <math>u(x,t=0)=u_{\text{init}}(x) </math>		where <math>u(x,t) </math> is the height of the wave with respect to the unperturbed sea level, and <math> c </math> is constant. We will use the following initial condition <math>u(x,t=0)=u_{\text{init}}(x) </math>

2011-09-21T08:30:41Z

← Older revision		Revision as of 10:30, 21 September 2011
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	</center>		</center>

	* Derive, from the Burgers equation in Eq. (3), the equation satisfied by <math> \phi </math>. This can be done in three steps :		* Derive, from the Burgers equation in Eq. (2), the equation satisfied by <math> \phi </math>. This can be done in three steps :
	(i) <math> u=-2\nu \partial_x \psi </math>,		(i) <math> u=-2\nu \partial_x \psi </math>,
	(ii) integrate the equation that you have obtained,		(ii) integrate the equation that you have obtained,

2011-09-21T08:29:03Z

← Older revision		Revision as of 10:29, 21 September 2011
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	== The Cole Hopf Transformation==		== The Cole Hopf Transformation==

	Consider ~~Now~~ Eq.2 with <math> \nu>0 </math>. This equation can be integrated via the Cole Hopf transformation which amounts to perform the following		Consider now Eq. (2) with <math> \nu>0 </math>. This equation can be integrated via the Cole Hopf transformation which amounts to perform the following
	change of variable:		change of variable: