Ergodicity - Revision history

128.187.97.23: /* Examples in electronics */

2014-12-10T19:16:40Z

Examples in electronics

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	In mathematics, a '''collocation method''' is a method for the [[numerical analysis\|numerical]] solution of [[ordinary differential equation]]s, [[partial differential equation]]s and [[integral equation]]s. The idea is to ~~choose a finite-dimensional space of candidate solutions (usually,~~ [~~[polynomial]]s up to a certain degree) and a number of points in the domain (called ''collocation points''), and to select that solution which satisfies the given equation at the collocation points.~~		Alyson is what my husband loves to contact me but [http://modenpeople.co.kr/modn/qna/292291 free psychic] I don't like when individuals use my complete name. One of the very best issues in the globe for him is doing ballet and he'll be starting something else alongside with it. My wife and [http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna psychic readers] I reside in Mississippi and I adore every working day living right here. Since I was eighteen I've been working as a bookkeeper but quickly my spouse and I will begin our personal company.<br><br>Here is my blog post ... [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 tarot card readings]

	~~== Ordinary differential equations ==~~

	~~Suppose that the [[ordinary differential equation]]~~
	:~~<math> y'(t) = f(t,y(t)), \quad y(t_0)=y_0, <~~/~~math>~~
	~~is to be solved over the interval [''t''<sub>0</sub>, ''t''<sub>0<~~/~~sub> + ''h'']~~. ~~Choose 0 ≤ ''c''<sub>1</sub>< ''c''<sub>2</sub>< … < ''c''<sub>''n''</sub> ≤ 1~~.

	~~The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition ''p''(''t''<sub>0<~~/~~sub>) = ''y''<sub>0<~~/~~sub>, and the differential equation ''p''<nowiki>'<~~/~~nowiki>('~~'t'') = ''f''(''t'',''p''(''t'')) at all points, called the '''collocation points,''' ''t'' = ''t''<sub>0</sub> + ''c''<sub>''k''</sub>''h'' where ''k'' = 1, …, ''n''. ~~This gives ''n'' + 1 conditions, which matches~~ the ~~''n'' + 1 parameters needed to specify a polynomial of degree ''n''.~~

	~~All these collocation methods are in fact implicit [[Runge–Kutta methods]]. The coefficient ''c''<sub>''k''</sub>~~ in the ~~Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.~~
	~~<ref>{{harvnb\|Ascher\|Petzold\|1998}}; {{harvnb\|Iserles\|1996\|pp=43–44}}</ref>~~

	~~=== Example: The trapezoidal rule ===~~

	~~Pick, as an example, the two collocation points ''c''<sub>1</sub> = 0~~ and '~~'c''<sub>2</sub> = 1 (so ''n'' = 2)~~. ~~The collocation conditions are~~

	:~~<math> p(t_0) = y_0, \, <~~/~~math>~~
	~~:<math> p'(t_0) = f(t_0, p(t_0)), \, <~~/~~math>~~
	~~:<math> p'(t_0+h) = f(t_0+h, p(t_0+h))~~. ~~\, </math>~~

	~~There are three conditions, so ''p'' should be a polynomial of degree 2~~. ~~Write ''p'' in the form~~

	~~:<math> p(t) = \alpha (t-t_0)^2 + \beta (t-t_0) + \gamma \, </math>~~

	~~to simplify the computations. Then the collocation conditions can be solved to give the coefficients~~

	~~:<math>~~
	~~\begin{align}~~
	~~\alpha &= \frac{1}{2h} \Big( f(t_0+h, p(t_0+h)) - f(t_0, p(t_0)) \Big), \\~~
	~~\beta &= f(t_0, p(t_0)), \\~~
	~~\gamma &= y_0~~.
	~~\end{align}~~
	~~</math>~~

	~~The collocation method is now given (implicitly) by~~

	~~:<math> y_1 = p(t_0 + h) = y_0 + \frac12h \Big (f(t_0+h, y_1) + f(t_0,y_0) \Big), \, <~~/~~math>~~

	~~where ''y''<sub>1<~~/~~sub>~~ = ~~''p''(''t''<sub>0</sub> +~~&~~nbsp;''h'') is the approximate solution at ''t''~~ = ~~''t''<sub>0</sub> + ''h''.~~

	~~This method is known as the "[[trapezoidal rule (differential equations)\|trapezoidal rule~~]~~]" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as~~

	~~:<math> y(t) = y(t_0) + \int_{t_0}^t f(\tau, y(\tau)) \,\textrm{d}\tau, \, </math>~~

	and ~~approximating the integral on the~~ right~~-hand side by the [[trapezoidal rule]] for integrals~~.

	~~=== Other examples ===~~

	~~The [[Gauss–Legendre method]]s use the points of [[Gauss–Legendre quadrature]]~~ as ~~collocation points. The Gauss–Legendre method based on ''s'' points has order 2''s''~~.<~~ref~~>~~{{harvnb\|Iserles\|1996\|pp=47}}~~<~~/ref~~> ~~All Gauss–Legendre methods are [[A-stability\|A-stable]]~~.~~<ref>{{harvnb\|Iserles\|1996\|pp=63}}</ref>~~

	~~== Notes ==~~
	~~{{Reflist}}~~

	~~== References ==~~
	* {{Citation \| last1=Ascher \| first1=Uri M. ~~\| last2=Petzold \| first2=Linda R~~. ~~\| title=Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations \| publisher=~~[~~[Society for Industrial and Applied Mathematics]] \| location=Philadelphia \| isbn=978-0-89871-412-8 \| year=1998}}~~.
	* {{Citation \| last1=Hairer \| first1=Ernst \| last2=Nørsett \| first2=Syvert Paul \| last3=Wanner \| first3=Gerhard \| title=Solving ordinary differential equations I: Nonstiff problems \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-3-540-56670-0 \| year=1993}}.
	* {{Citation \| last1=~~Iserles \| first1~~=~~Arieh \| author1-link=Arieh Iserles \| title=A First Course in the Numerical Analysis of Differential Equations \| publisher=[[Cambridge University Press]] \| isbn~~=~~978-0-521-55655-2 \| year=1996}}.~~

	~~{{Numerical PDE}}~~

	~~{{DEFAULTSORT:Collocation Method}}~~
	~~[[Category:Numerical differential equations]~~]

en>Skatche: /* Formal definition */ Second condition was falsely stated (take e.g. E=X).

2014-01-22T07:35:30Z

Formal definition: Second condition was falsely stated (take e.g. E=X).

← Older revision		Revision as of 09:35, 22 January 2014
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	~~Greetings! I am~~ [~~http://Www~~.~~whitman~~-~~walker~~.~~org/page.aspx?pid~~=~~435 Marvella~~] ~~and I feel comfortable when people use home~~ [~~http:~~//~~www~~.~~sex-porn-tube~~.ch/~~blog~~/~~36870 std home test] test kit~~ the ~~full name~~. ~~Managing individuals has been his day occupation for~~ a ~~while~~. It's not ~~a common thing but what she likes doing is foundation jumping~~ and ~~now she is trying to make cash with it~~. ~~California is exactly where [http~~://~~www~~.~~hotporn123~~.~~com~~/~~user~~/~~RWhiteman over~~ the ~~counter std test] I~~'~~ve always been~~ [~~http~~:/~~/celebs~~.~~Answers~~.~~com~~/~~scandals/shocking~~-~~celebrities~~-~~with-stis residing~~] ~~and I love every working day living here~~.<br><br>~~Feel free to visit my weblog~~ ... ~~std testing at home (~~[~~http://www~~.~~articlecube~~.~~com/profile/Nydia-Lyons/532548 site~~])		In mathematics, a '''collocation method''' is a method for the [[numerical analysis\|numerical]] solution of [[ordinary differential equation]]s, [[partial differential equation]]s and [[integral equation]]s. The idea is to choose a finite-dimensional space of candidate solutions (usually, [[polynomial]]s up to a certain degree) and a number of points in the domain (called ''collocation points''), and to select that solution which satisfies the given equation at the collocation points.

			== Ordinary differential equations ==

			Suppose that the [[ordinary differential equation]]
			:<math> y'(t) = f(t,y(t)), \quad y(t_0)=y_0, </math>
			is to be solved over the interval [''t''<sub>0</sub>, ''t''<sub>0</sub> + ''h'']. Choose 0 ≤ ''c''<sub>1</sub>< ''c''<sub>2</sub>< … < ''c''<sub>''n''</sub> ≤ 1.

			The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition ''p''(''t''<sub>0</sub>) = ''y''<sub>0</sub>, and the differential equation ''p''<nowiki>'</nowiki>(''t'') = ''f''(''t'',''p''(''t'')) at all points, called the '''collocation points,''' ''t'' = ''t''<sub>0</sub> + ''c''<sub>''k''</sub>''h'' where ''k'' = 1, …, ''n''. This gives ''n'' + 1 conditions, which matches the ''n'' + 1 parameters needed to specify a polynomial of degree ''n''.

			All these collocation methods are in fact implicit [[Runge–Kutta methods]]. The coefficient ''c''<sub>''k''</sub> in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.
			<ref>{{harvnb\|Ascher\|Petzold\|1998}}; {{harvnb\|Iserles\|1996\|pp=43–44}}</ref>

			=== Example: The trapezoidal rule ===

			Pick, as an example, the two collocation points ''c''<sub>1</sub> = 0 and ''c''<sub>2</sub> = 1 (so ''n'' = 2). The collocation conditions are

			:<math> p(t_0) = y_0, \, </math>
			:<math> p'(t_0) = f(t_0, p(t_0)), \, </math>
			:<math> p'(t_0+h) = f(t_0+h, p(t_0+h)). \, </math>

			There are three conditions, so ''p'' should be a polynomial of degree 2. Write ''p'' in the form

			:<math> p(t) = \alpha (t-t_0)^2 + \beta (t-t_0) + \gamma \, </math>

			to simplify the computations. Then the collocation conditions can be solved to give the coefficients

			:<math>
			\begin{align}
			\alpha &= \frac{1}{2h} \Big( f(t_0+h, p(t_0+h)) - f(t_0, p(t_0)) \Big), \\
			\beta &= f(t_0, p(t_0)), \\
			\gamma &= y_0.
			\end{align}
			</math>

			The collocation method is now given (implicitly) by

			:<math> y_1 = p(t_0 + h) = y_0 + \frac12h \Big (f(t_0+h, y_1) + f(t_0,y_0) \Big), \, </math>

			where ''y''<sub>1</sub> = ''p''(''t''<sub>0</sub> + ''h'') is the approximate solution at ''t'' = ''t''<sub>0</sub> + ''h''.

			This method is known as the "[[trapezoidal rule (differential equations)\|trapezoidal rule]]" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as

			:<math> y(t) = y(t_0) + \int_{t_0}^t f(\tau, y(\tau)) \,\textrm{d}\tau, \, </math>

			and approximating the integral on the right-hand side by the [[trapezoidal rule]] for integrals.

			=== Other examples ===

			The [[Gauss–Legendre method]]s use the points of [[Gauss–Legendre quadrature]] as collocation points. The Gauss–Legendre method based on ''s'' points has order 2''s''.<ref>{{harvnb\|Iserles\|1996\|pp=47}}</ref> All Gauss–Legendre methods are [[A-stability\|A-stable]].<ref>{{harvnb\|Iserles\|1996\|pp=63}}</ref>

			== Notes ==
			{{Reflist}}

			== References ==
			* {{Citation \| last1=Ascher \| first1=Uri M. \| last2=Petzold \| first2=Linda R. \| title=Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations \| publisher=[[Society for Industrial and Applied Mathematics]] \| location=Philadelphia \| isbn=978-0-89871-412-8 \| year=1998}}.
			* {{Citation \| last1=Hairer \| first1=Ernst \| last2=Nørsett \| first2=Syvert Paul \| last3=Wanner \| first3=Gerhard \| title=Solving ordinary differential equations I: Nonstiff problems \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-3-540-56670-0 \| year=1993}}.
			* {{Citation \| last1=Iserles \| first1=Arieh \| author1-link=Arieh Iserles \| title=A First Course in the Numerical Analysis of Differential Equations \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-55655-2 \| year=1996}}.

			{{Numerical PDE}}

			{{DEFAULTSORT:Collocation Method}}
			[[Category:Numerical differential equations]]

71.106.171.27 at 05:29, 9 August 2012

2012-08-09T05:29:32Z

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