Flow equation for a differential operator - Revision history

Vipul: 12 revisions

2008-05-18T19:40:41Z

12 revisions

← Older revision	Revision as of 19:40, 18 May 2008
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Vipul: /* Definition */

2007-04-08T05:09:02Z

Definition

← Older revision		Revision as of 05:09, 8 April 2007
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	==Definition==		==Definition==

			Note that a flow equation can be associated to any differential operator on a manifold (and in particular, to one on <math>\R^n</math>). There is a kind of correspondence between the differential operator and the associated flow equation. Thus, any property over flow equations gives a property over differential operators, and vice versa.

	===Definition for Euclidean space===		===Definition for Euclidean space===

Vipul: /* Definition for a differential manifold */

2007-04-08T03:51:19Z

Definition for a differential manifold

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	* <math>u: \R \times M \to \R</math> is the dependent variable, written as <math>u(t,x)</math>		* <math>u: \R \times M \to \R</math> is the dependent variable, written as <math>u(t,x)</math>
	* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(t,x)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)		* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(t,x)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)

			The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation) are termed '''trajectories'''. By the property of invariance of the differential equation under time-translation, we conclude that the various trajectories through a point are just time-translaets of each other, so there is a unique trajectory curve through each point.

	==Properties of the flow equation==		==Properties of the flow equation==

Vipul: /* Definition for Euclidean space */

2007-04-08T03:50:04Z

Definition for Euclidean space

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	* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(x,t)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)		* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(x,t)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)

	The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation).		The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation) are termed '''trajectories'''. By the property of invariance of the differential equation under time-translation, we conclude that the various trajectories through a point are just time-translaets of each other, so there is a unique trajectory curve through each point.

	===Definition for a differential manifold===		===Definition for a differential manifold===

Vipul: /* Trajectories of the flow */

2007-04-08T03:46:32Z

Trajectories of the flow

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 * There exists a limiting point at <math>\infty</math>, which is hence a solution to the original differential equation
 * {{fillin}}
 ==Supersolutions and subsolutions==

Vipul: /* Properties of the flow equation */

2007-04-08T03:32:14Z

Properties of the flow equation

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	The flow equation is autonomous with respect to the time parameter -- the only place where the time parameter occurs is where we differentiate <math>u</math> with respect to time.		The flow equation is autonomous with respect to the time parameter -- the only place where the time parameter occurs is where we differentiate <math>u</math> with respect to time.

			===Dependence on linear change in the differential operator===

			Suppose <math>F</math> is a differential operator, and <math>k \in \R</math>. Then if <math>u</math> is a solution for <math>kF</math>, then the map <math>(t,x) \mapsto u(kt,x)</math> is a solution for <math>F</math>.

	==Fixed points and trajectories==		==Fixed points and trajectories==

Vipul at 03:09, 8 April 2007

2007-04-08T03:09:18Z

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 * <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(t,x)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)
 ==Fixed points and trajectories==
 ===Fixed points are solutions of the original equation===
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 * There exists a limiting point at <math>\infty</math>, which is hence a solution to the original differential equation
 * {{fillin}}

Vipul: /* Definition */

2007-04-08T02:55:34Z

Definition

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	==Definition==		==Definition==

	===~~Basic definition~~ for Euclidean space===		===Definition for Euclidean space===
	~~Suppose we have a differential equation <math>F(y) = 0</math> where <math>y: \R^n \to \R</math> is the unknown function.~~

	~~Then consider the~~ differential equation:		Suppose <math>F</math> is a [[differential operator]] on <math>\R^n</math> that is, <math>F</math> takes as input a (sufficiently many times differentiable) function from <math>\R^n</math> to <math>\R</math> and outputs another function from <math>\R^n</math> to <math>\R</math>.

			The flow equation for <math>F</math> is described as follows:

	<math>\frac{\partial u}{\partial t} = F(u)</math>		<math>\frac{\partial u}{\partial t} = F(u)</math>
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	* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(x,t)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)		* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(x,t)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time)

	~~This is called the flow equation for <math>F</math>.~~ The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation).		The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation).

	===Definition for a differential manifold===		===Definition for a differential manifold===

	~~Note that~~ <math>~~\R^n~~</math> ~~can be replaced by any open set in <math>\R^n</math>, or even more generally, by any~~ [[differential manifold]] <math>M</math>~~. In the general case~~, <math>y</math> ~~becomes~~ a ~~map~~ from <math>M</math> to <math>\R</math>, and <math>F</math> ~~is a differential operator on the manifold~~.		Suppose <math>F</math> is a [[differential operator]] on a [[differential manifold]] <math>M</math> that is, <math>F</math> takes as input a (sufficiently many times differentiable) function from <math>M</math> to <math>\R</math> and outputs another function from <math>M</math> to <math>\R</math>.

	The flow equation becomes:		The flow equation becomes:

Vipul: Flow for a scalar differential equation moved to Flow equation for a differential operator

2007-04-08T02:51:06Z

Flow for a scalar differential equation moved to Flow equation for a differential operator

← Older revision	Revision as of 02:51, 8 April 2007
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Vipul at 02:40, 8 April 2007

2007-04-08T02:40:26Z

← Older revision		Revision as of 02:40, 8 April 2007
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			''This article defines the notion of flow for a scalar differential equation, viz a differential equation with one real variable. For the more general notion, refer [[flow for a vector differential equation]]''

	==Definition==		==Definition==