Delay differential equation - Revision history

Vipul at 02:16, 9 July 2012

2012-07-09T02:16:45Z

← Older revision		Revision as of 02:16, 9 July 2012
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	===Note on autonomous case===		===Note on autonomous case===

	The delay differential equations that we study are typically [[autonomous delay differential equation]]s: an equation in the general form above is autonomous if, for any <math>\tau \in \R</math>, the function <math>F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t)</math> is invariant under replacing <math>x(t)</math> by the function <math>t \mapsto x(t - \tau)</math>.		The delay differential equations that we study are typically [[autonomous delay differential equation]]s: an equation in the general form above is autonomous if, for any <math>\tau \in \R</math>, the function <math>F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t)</math> is invariant under replacing <math>x(t)</math> by the function <math>t \mapsto x(t - \tau)</math>. Intuitively, what this means is that <math>t</math> does not appear explicitly in <math>F</math>, and all the behavior at previous points is specified in terms of how much earlier they were than <math>t</math>.

Vipul at 02:13, 9 July 2012

2012-07-09T02:13:12Z

← Older revision		Revision as of 02:13, 9 July 2012
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	==Definition==		==Definition==

	The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations). If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math> (Which we think of as time), the first-order first-degree case is:		The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations).

			===First-order first-degree case===

			If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math> (Which we think of as time), the first-order first-degree case is:

	<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>		<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>

	The delay differential equations that we study are typically [[autonomous delay differential equation]]s: the ~~nature of~~ the ~~expression for~~ <math>d(x(t))~~/dt~~</math> is invariant under ~~translation of~~ <math>t</math> by ~~any constant~~.		===General case===

			The general case of a delay differential equation is of the form:

			<math>F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t) = 0</math>

			===Note on autonomous case===

			The delay differential equations that we study are typically [[autonomous delay differential equation]]s: an equation in the general form above is autonomous if, for any <math>\tau \in \R</math>, the function <math>F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t)</math> is invariant under replacing <math>x(t)</math> by the function <math>t \mapsto x(t - \tau)</math>.

Vipul at 15:50, 8 July 2012

2012-07-08T15:50:52Z

← Older revision		Revision as of 15:50, 8 July 2012
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	<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>		<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>

			The delay differential equations that we study are typically [[autonomous delay differential equation]]s: the nature of the expression for <math>d(x(t))/dt</math> is invariant under translation of <math>t</math> by any constant.

Vipul at 15:49, 8 July 2012

2012-07-08T15:49:32Z

← Older revision		Revision as of 15:49, 8 July 2012
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	==Definition==		==Definition==

	The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations). If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math>, the first-order first-degree case is:		The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations). If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math> (Which we think of as time), the first-order first-degree case is:

	<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>		<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>

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2012-07-08T15:49:10Z

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==Definition==

The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations). If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math>, the first-order first-degree case is:

<math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>