Dynamical system (definition) - Revision history

2001:630:53:B76:156:B45B:42EE:8D22: /* General definition */

2014-10-30T12:43:28Z

General definition

← Older revision		Revision as of 14:43, 30 October 2014
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	A '~~''parabolic partial differential equation''' is a type of second-order~~ [~~[partial differential equation]] (PDE) of the form~~		I'm Luther but I by no means truly liked that name. The preferred pastime for him and his [http://autos.jdpower.com/content/buying-tip/xa345A7/pros-and-cons-of-buying-an-extended-warranty-for-a-car.htm children] is to drive and now he is trying to earn cash with it. Her family members life in Delaware but she requirements to move because of her family members. After becoming out of my occupation for many years I became a manufacturing [http://www.shipzula.com/blogs/post/26656 extended auto warranty] auto warranty auto warranty and distribution officer but I plan on [http://www.spa-it.al/ActivityFeed/MyProfile/tabid/60/userId/11049/Default.aspx extended car warranty] car [http://Www.Nbcnews.com/id/23147777/ns/business-consumer_news/t/auto-warranty-firms-launch-sleazy-scam/ warranty altering] it.<br><br>Also visit my web-site ... car warranty ([http://www.Friscowebsolutions.com/UserProfile/tabid/379/userId/203/Default.aspx sources tell me])
	: ~~<math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,<~~/~~math>~~
	~~that satisfies the condition~~
	~~:<math>B^2~~ - ~~AC = 0.\ <~~/~~math>~~

	~~This definition is analogous to the definition~~ of a ~~planar [[Parabola#Analytic_geometry_equations\|parabola~~]].

	~~This form of partial differential equation~~ is ~~used~~ to ~~describe a wide~~ family of problems in science including [[heat diffusion]], [[underwater acoustics\|ocean acoustic propagation]], physical or mathematical systems with a time variable, and processes that behave essentially like heat diffusing through a solid.

	~~A simple example~~ of a ~~parabolic PDE is the one-dimensional~~ [~~[heat equation]],~~
	:~~<math>u_t = k u_{xx},\ <~~/~~math>~~
	~~where <math>u(t,x)<~~/~~math> is the temperature at time <math>t<~~/~~math> and at position <math>x<~~/~~math>, and <math>k<~~/~~math> is a constant. The symbol <math>u_t</math> signifies the [[partial derivative~~]~~] with respect to the time variable <math>t</math>,~~ and ~~similarly <math>u_{xx}<~~/~~math> is the second partial derivative with respect to <math>x<~~/~~math>~~.

	This equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. ~~The quantity <math>u_{xx}</math> measures how far off the temperature is from satisfying the mean value property of [[harmonic functions]].~~

	~~A generalization of the heat equation is~~

	~~:<math>u_t = -Lu,\ <~~/~~math>~~

	~~where <math>L<~~/~~math> is a [[second order]] [[elliptic operator]] (implying <math>L<~~/~~math> must be [[Positive operator\|positive]] also; a case where <math>L<~~/~~math> is non-positive is described below). Such a system can be hidden in an equation of the form~~

	~~:<math>\nabla \cdot (a(x) \nabla u(x)) + b(x)^T \nabla u(x) + cu(x) = f(x)<~~/~~math>~~

	~~if the matrix-valued function <math>a(x)<~~/~~math> has a [[Kernel (algebra)\|kernel]] of dimension 1.~~

	~~==Solution==~~
	~~Under broad assumptions, parabolic PDEs as given above have solutions for all ''x,y'' and ''t''>0. An equation of the form <math>u_t = -L(u)<~~/~~math> is considered parabolic if ''L'' is a (possibly nonlinear) function of ''u'' and its first and second derivatives, with some further conditions on ''L''~~. ~~With such a nonlinear parabolic differential equation, solutions exist for a short time—but may explode in a [[Mathematical singularity\|singularity]~~] ~~in a finite amount of time. Hence, the difficulty is in determining solutions for all time, or more generally studying the singularities that arise. This is in general quite difficult, as in the~~ [~~[solution of the Poincaré conjecture]] via [[Ricci flow]].~~

	~~==Backward parabolic equation==~~
	~~One may occasionally wish to consider PDEs of the form <math>u_t = Lu,\ <~~/~~math> where <math>L<~~/~~math> is a positive [[elliptic operator]]~~. While these problems are no longer necessarily [[well-posed]] (solutions may grow unbounded in finite time, or not even exist), they occur when studying the reflection of singularities of solutions to various other PDEs.<ref>{{citation \| first=M. E. \| last=Taylor \| authorlink = Michael E. Taylor \| title=Reflection of singularities of solutions to systems of differential equations \| journal=Comm. Pure Appl. Math. \| volume=28 \| issue=4 \| year=1975 \| pages=457–478 \| doi=10.1002/~~cpa.3160280403 }}<~~/~~ref>~~

	~~This class of equations is closely related to standard hyperbolic equations, which can be seen easily by considering the so~~-~~called 'backwards heat equation':~~

	~~:<math>\begin{cases} u_{~~t~~} = \Delta u & \textrm{on} \ \ \Omega \times (0,T), \\ u=0 & \textrm{on} \ \ \partial\Omega \times (0,T), \\ u = f & \textrm{on} \ \ \Omega \times \left \{ T \right \}. \end{cases} <~~/~~math>~~

	~~This is essentially the same as the backward hyperbolic equation:~~

	~~:<math>\begin{cases} u_{t} =~~ -~~\Delta u & \textrm{on} \ \ \Omega \times (0,T), \\ u=0 & \textrm{on} \ \ \partial\Omega \times (0,T), \\ u = f & \textrm{on} \ \ \Omega \times \left \{ 0 \right \}. \end{cases} </math>~~

	~~== Examples ==~~
	* [[Heat equation]]
	* [[Mean curvature flow]]
	* [[Ricci flow]]

	~~== See also == <!~~-- ~~{{Div col}}~~ -->
	*[[Hyperbolic partial differential equation]]
	*[[Elliptic partial differential equation]]
	*[[Autowave]]
	<~~!-- {{Div col end}} --~~>

	~~== Notes ==~~
	<~~references/~~>

	~~== References ==~~
	*{{Citation \| last1=Evans \| first1=Lawrence C. ~~\| title=Partial differential equations \| origyear=1998 \| url=~~http://www.~~ams~~.~~org~~/~~bull~~/~~2000-37-03~~/~~...~~/S0273-0979-00-00868-5.pdf \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| edition=2nd \| series=Graduate Studies in Mathematics \| isbn=978-0-8218-4974-3 \| id={{MathSciNet \| id = 2597943}} \| year=2010 \| volume=19}}
	* {{springer\|title=Parabolic partial differential equation\|id=p/~~p071210}}~~
	* {{springer\|title=Parabolic partial differential equation, numerical methods\|id=p/~~p071220}}~~

	~~{{DEFAULTSORT:Parabolic Partial Differential Equation}}~~
	~~[[Category:Parabolic partial differential equations\|*]~~]

134.93.142.234: corrected a typo: replaced lower case phi by upper case phi

2013-12-05T12:54:36Z

corrected a typo: replaced lower case phi by upper case phi

← Older revision		Revision as of 14:54, 5 December 2013
Line 1:		Line 1:
	~~Wilber Berryhill~~ is ~~what his wife loves~~ to ~~contact him~~ and ~~he completely loves this name~~. ~~Mississippi~~ is ~~exactly~~ where ~~her home~~ is ~~but her husband desires them~~ to ~~transfer~~. ~~Invoicing~~ is ~~my profession~~. ~~To perform domino~~ is ~~something I truly appreciate performing~~.<br><br>~~my web page :: online reader;~~ [~~http:~~//~~galab~~-~~work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl~~=~~1489804~~ http://~~galab~~-~~work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl=1489804~~],		A '''parabolic partial differential equation''' is a type of second-order [[partial differential equation]] (PDE) of the form
			: <math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,</math>
			that satisfies the condition
			:<math>B^2 - AC = 0.\ </math>

			This definition is analogous to the definition of a planar [[Parabola#Analytic_geometry_equations\|parabola]].

			This form of partial differential equation is used to describe a wide family of problems in science including [[heat diffusion]], [[underwater acoustics\|ocean acoustic propagation]], physical or mathematical systems with a time variable, and processes that behave essentially like heat diffusing through a solid.

			A simple example of a parabolic PDE is the one-dimensional [[heat equation]],
			:<math>u_t = k u_{xx},\ </math>
			where <math>u(t,x)</math> is the temperature at time <math>t</math> and at position <math>x</math>, and <math>k</math> is a constant. The symbol <math>u_t</math> signifies the [[partial derivative]] with respect to the time variable <math>t</math>, and similarly <math>u_{xx}</math> is the second partial derivative with respect to <math>x</math>.

			This equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity <math>u_{xx}</math> measures how far off the temperature is from satisfying the mean value property of [[harmonic functions]].

			A generalization of the heat equation is

			:<math>u_t = -Lu,\ </math>

			where <math>L</math> is a [[second order]] [[elliptic operator]] (implying <math>L</math> must be [[Positive operator\|positive]] also; a case where <math>L</math> is non-positive is described below). Such a system can be hidden in an equation of the form

			:<math>\nabla \cdot (a(x) \nabla u(x)) + b(x)^T \nabla u(x) + cu(x) = f(x)</math>

			if the matrix-valued function <math>a(x)</math> has a [[Kernel (algebra)\|kernel]] of dimension 1.

			==Solution==
			Under broad assumptions, parabolic PDEs as given above have solutions for all ''x,y'' and ''t''>0. An equation of the form <math>u_t = -L(u)</math> is considered parabolic if ''L'' is a (possibly nonlinear) function of ''u'' and its first and second derivatives, with some further conditions on ''L''. With such a nonlinear parabolic differential equation, solutions exist for a short time—but may explode in a [[Mathematical singularity\|singularity]] in a finite amount of time. Hence, the difficulty is in determining solutions for all time, or more generally studying the singularities that arise. This is in general quite difficult, as in the [[solution of the Poincaré conjecture]] via [[Ricci flow]].

			==Backward parabolic equation==
			One may occasionally wish to consider PDEs of the form <math>u_t = Lu,\ </math> where <math>L</math> is a positive [[elliptic operator]]. While these problems are no longer necessarily [[well-posed]] (solutions may grow unbounded in finite time, or not even exist), they occur when studying the reflection of singularities of solutions to various other PDEs.<ref>{{citation \| first=M. E. \| last=Taylor \| authorlink = Michael E. Taylor \| title=Reflection of singularities of solutions to systems of differential equations \| journal=Comm. Pure Appl. Math. \| volume=28 \| issue=4 \| year=1975 \| pages=457–478 \| doi=10.1002/cpa.3160280403 }}</ref>

			This class of equations is closely related to standard hyperbolic equations, which can be seen easily by considering the so-called 'backwards heat equation':

			:<math>\begin{cases} u_{t} = \Delta u & \textrm{on} \ \ \Omega \times (0,T), \\ u=0 & \textrm{on} \ \ \partial\Omega \times (0,T), \\ u = f & \textrm{on} \ \ \Omega \times \left \{ T \right \}. \end{cases} </math>

			This is essentially the same as the backward hyperbolic equation:

			:<math>\begin{cases} u_{t} = -\Delta u & \textrm{on} \ \ \Omega \times (0,T), \\ u=0 & \textrm{on} \ \ \partial\Omega \times (0,T), \\ u = f & \textrm{on} \ \ \Omega \times \left \{ 0 \right \}. \end{cases} </math>

			== Examples ==
			* [[Heat equation]]
			* [[Mean curvature flow]]
			* [[Ricci flow]]

			== See also == <!-- {{Div col}} -->
			*[[Hyperbolic partial differential equation]]
			*[[Elliptic partial differential equation]]
			*[[Autowave]]
			<!-- {{Div col end}} -->

			== Notes ==
			<references/>

			== References ==
			*{{Citation \| last1=Evans \| first1=Lawrence C. \| title=Partial differential equations \| origyear=1998 \| url=http://www.ams.org/bull/2000-37-03/.../S0273-0979-00-00868-5.pdf \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| edition=2nd \| series=Graduate Studies in Mathematics \| isbn=978-0-8218-4974-3 \| id={{MathSciNet \| id = 2597943}} \| year=2010 \| volume=19}}
			* {{springer\|title=Parabolic partial differential equation\|id=p/p071210}}
			* {{springer\|title=Parabolic partial differential equation, numerical methods\|id=p/p071220}}

			{{DEFAULTSORT:Parabolic Partial Differential Equation}}
			[[Category:Parabolic partial differential equations\|*]]

en>CarrieVS: /* Construction of dynamical systems */ Repairing links to disambiguation pages - You can help!

2012-06-09T14:25:36Z

Construction of dynamical systems: Repairing links to disambiguation pages - You can help!

New page

Wilber Berryhill is what his wife loves to contact him and he completely loves this name. Mississippi is exactly where her home is but her husband desires them to transfer. Invoicing is my profession. To perform domino is something I truly appreciate performing.<br><br>my web page :: online reader; [http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804],