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Diffstat (limited to 'mne')
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| -rw-r--r-- | mne/n1.lyx | 462 | ||||
| -rw-r--r-- | mne/n2.lyx | 994 |
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diff --git a/mne/n.lyx b/mne/n.lyx new file mode 100644 index 0000000..9a9e5d7 --- /dev/null +++ b/mne/n.lyx @@ -0,0 +1,198 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize 10 +\spacing single +\use_hyperref false +\papersize a5paper +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 0.2cm +\topmargin 0.7cm +\rightmargin 0.2cm +\bottommargin 0.7cm +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style swiss +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle empty +\listings_params "basicstyle={\ttfamily}" +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Métodos Numéricos de las Ecuaciones Diferenciales +\end_layout + +\begin_layout Date +\begin_inset Note Note +status open + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +def +\backslash +cryear{2020} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "../license.lyx" + +\end_inset + + +\end_layout + +\begin_layout Standard +Bibliografía: +\end_layout + +\begin_layout Itemize +F. + Esquembre (2020). + Notas de clase. +\end_layout + +\begin_layout Itemize + +\lang english +Wikipedia, the Free Encyclopedia +\lang spanish + ( +\begin_inset Flex URL +status open + +\begin_layout Plain Layout + +https://en.wikipedia.org/ +\end_layout + +\end_inset + +). + +\emph on +\lang english +Runge-Kutta methods +\emph default +\lang spanish +. +\end_layout + +\begin_layout Chapter +Introducción +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n1.lyx" + +\end_inset + + +\end_layout + +\begin_layout Chapter +Métodos de paso fijo +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n2.lyx" + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/mne/n1.lyx b/mne/n1.lyx new file mode 100644 index 0000000..bc4abd2 --- /dev/null +++ b/mne/n1.lyx @@ -0,0 +1,462 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +Un +\series bold +problema de valores iniciales +\series default + real es uno de la forma +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =f(t,x),\\ +x(t_{0}) & =x_{0}, +\end{aligned} +\right. +\] + +\end_inset + +dado por +\begin_inset Formula $f:\Omega\subseteq\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ +\end_inset + + con +\begin_inset Formula $\Omega$ +\end_inset + + abierto y +\begin_inset Formula $(t_{0},x_{0})\in\Omega$ +\end_inset + +, y donde +\begin_inset Formula $x:I\subseteq\mathbb{R}\to\mathbb{R}^{n}$ +\end_inset + + es la incógnita, siendo +\begin_inset Formula $I$ +\end_inset + + un entorno de +\begin_inset Formula $t_{0}$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +El problema está +\series bold +bien planteado +\series default + en un intervalo +\begin_inset Formula $[a,b]\subseteq I$ +\end_inset + + si tiene solución única en +\begin_inset Formula $[a,b]$ +\end_inset + + y para todo +\begin_inset Formula $\varepsilon>0$ +\end_inset + + existe un +\begin_inset Formula $\delta>0$ +\end_inset + + tal que si +\begin_inset Formula $c\in(-\varepsilon,\varepsilon)$ +\end_inset + + y +\begin_inset Formula $e:[a,b]\to\mathbb{R}^{n}$ +\end_inset + + es tal que +\begin_inset Formula $|e(t)|<\varepsilon$ +\end_inset + + para todo +\begin_inset Formula $t\in[a,b]$ +\end_inset + +, entonces el +\series bold +problema perturbado +\series default + +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{z} & =f(t,z)+e(t),\\ +z(t_{0}) & =x_{0}+c, +\end{aligned} +\right. +\] + +\end_inset + +tiene solución única. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $D:=[a,b]\times\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $t_{0}\in[a,b]$ +\end_inset + + y +\begin_inset Formula $f:D\to\mathbb{R}$ +\end_inset + + es continua y lipschitziana en la segunda variable en todo +\begin_inset Formula $D$ +\end_inset + +, entonces +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =f(t,x),\\ +x(t_{0}) & =x_{0} +\end{aligned} +\right. +\] + +\end_inset + +está bien planteado. +\end_layout + +\begin_layout Standard +En adelante supondremos que el dominio de +\begin_inset Formula $x$ +\end_inset + + incluye un intervalo +\begin_inset Formula $[a,b]$ +\end_inset + + y +\begin_inset Formula $t_{0}=a$ +\end_inset + +. + No siempre se puede resolver un problema de valores iniciales de forma + analítica, por lo que usamos métodos de resolución numérica, que aproximan + +\begin_inset Formula $x|_{[a,b]}$ +\end_inset + + creando una partición +\begin_inset Formula $a=t_{0}<\dots<t_{n}=c$ +\end_inset + +, para un cierto +\begin_inset Formula $c\geq b$ +\end_inset + + con +\begin_inset Formula $[a,c]$ +\end_inset + + en el dominio de +\begin_inset Formula $x$ +\end_inset + +, y obtienen una secuencia +\begin_inset Formula $(\omega_{i})_{i=0}^{n}$ +\end_inset + + que aproxima +\begin_inset Formula $(x(t_{i}))_{i=0}^{n}$ +\end_inset + + con un error +\begin_inset Formula $\max_{i=0}^{n}\Vert x(t_{i})-\omega_{i}\Vert$ +\end_inset + + aceptable. + Los valores de +\begin_inset Formula $x$ +\end_inset + + en el resto de puntos de +\begin_inset Formula $[a,b]$ +\end_inset + + se obtienen por interpolación. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{CN} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Si el polinomio interpolador de +\begin_inset Formula $f$ +\end_inset + + en +\begin_inset Formula $x_{0},\dots,x_{n}$ +\end_inset + + es +\begin_inset Formula $P(x)=a_{n}x^{n}+\dots+a_{0}$ +\end_inset + +, llamamos +\begin_inset Formula $f[x_{0},\dots,x_{n}]:=a_{n}$ +\end_inset + +. + [...] +\begin_inset Formula $f[x]=f(x)$ +\end_inset + +. + [...] Para +\begin_inset Formula $n\geq1$ +\end_inset + +, +\begin_inset Formula +\[ +f[x_{0},\dots,x_{n}]=\frac{f[x_{1},\dots,x_{n}]-f[x_{0},\dots,x_{n-1}]}{x_{n}-x_{0}}. +\] + +\end_inset + +[...] Una forma de hallar +\begin_inset Formula $f[x_{0},\dots,x_{n}]$ +\end_inset + + es usando una tabla triangular que se va llenando por columnas, donde la + primera columna contiene los +\begin_inset Formula $x_{k}$ +\end_inset + +, la segunda, los +\begin_inset Formula $f(x_{k})$ +\end_inset + +, [...], y la +\begin_inset Formula $j$ +\end_inset + +-ésima, los +\begin_inset Formula $f[x_{k-j+1},\dots,x_{k}]$ +\end_inset + +, llegando a +\begin_inset Formula $f[x_{0},\dots,x_{n}]$ +\end_inset + + en la +\begin_inset Formula $(n+1)$ +\end_inset + +-ésima fila. +\end_layout + +\begin_layout Standard +[...] +\series bold +Forma de Newton +\series default + del polinomio interpolador, [...] +\begin_inset Formula +\[ +f[x_{0}]+f[x_{0},x_{1}](x-x_{0})+\dots+f[x_{0},\dots,x_{n}](x-x_{0})\cdots(x-x_{n-1}). +\] + +\end_inset + +Para cálculo computacional es más apropiada la forma anidada, +\begin_inset Formula +\[ +f[x_{0}]+(x-x_{0})(f[x_{0},x_{1}]+(x-x_{1})(f[x_{0},x_{1},x_{2}]+\dots)). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +[...] Un +\series bold +problema de Hermite +\series default + consiste en hallar un polinomio +\begin_inset Formula $P$ +\end_inset + + de grado +\begin_inset Formula $N$ +\end_inset + + tal que, para +\begin_inset Formula $k\in\{0,\dots,m\}$ +\end_inset + + y +\begin_inset Formula $x\in S_{k}$ +\end_inset + +, +\begin_inset Formula $P^{(k)}(x)=f^{(k)}(x)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Existe un único polinomio de grado máximo +\begin_inset Formula $\sum_{k=0}^{m}|S_{k}|$ +\end_inset + + que cumpla las condiciones, y podemos hallarlo mediante +\series bold +diferencias divididas generalizadas +\series default +. + Si +\begin_inset Formula $x_{0}\leq\dots\leq x_{n}$ +\end_inset + +, [...] +\begin_inset Formula +\[ +f[x_{0},\dots,x_{n}]:=\begin{cases} +\frac{f[x_{1},\dots,x_{n}]-f[x_{0},\dots,x_{n-1}]}{x_{n}-x_{0}}, & x_{0}<x_{n};\\ +\frac{f^{(n)}(x_{0})}{n!}, & x_{0}=\dots=x_{n}. +\end{cases} +\] + +\end_inset + +Creamos una tabla de diferencias divididas en la que cada elemento +\begin_inset Formula $x\in S_{0}$ +\end_inset + + aparece tantas veces como conjuntos de entre +\begin_inset Formula $S_{0},\dots,S_{m}$ +\end_inset + + lo contienen, y expresamos el polinomio resultante [...] en forma de Newton. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document diff --git a/mne/n2.lyx b/mne/n2.lyx new file mode 100644 index 0000000..8442b0d --- /dev/null +++ b/mne/n2.lyx @@ -0,0 +1,994 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +Dado un problema de valores iniciales +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =f(t,x),\\ +x(a) & =x_{0}, +\end{aligned} +\right. +\] + +\end_inset + +los +\series bold +métodos de paso fijo +\series default + toman un +\series bold +paso +\series default + +\begin_inset Formula $h>0$ +\end_inset + + y particionan +\begin_inset Formula $[a,b]$ +\end_inset + + con +\begin_inset Formula $t_{i}:=a+hi$ +\end_inset + +, aunque esto se suele calcular como +\begin_inset Formula $t_{0}=a$ +\end_inset + + y +\begin_inset Formula $t_{i}=t_{i-1}+h$ +\end_inset + + para +\begin_inset Formula $i\geq1$ +\end_inset + +. +\end_layout + +\begin_layout Section +Método de Euler +\end_layout + +\begin_layout Standard +El +\series bold +método de Euler +\series default + viene dado por +\begin_inset Formula $\omega_{0}:=x_{0}$ +\end_inset + + y +\begin_inset Formula $\omega_{i+1}:=\omega_{i}+hf(t_{i},\omega_{i})$ +\end_inset + +. + Para obtener el valor para un +\begin_inset Formula $t\in(t_{i-1},t_{i})$ +\end_inset + +, podemos interpolar con el propio método, lo que si se hace desde +\begin_inset Formula $t_{i-1}$ +\end_inset + + equivale a una interpolación lineal o una interpolación de Newton en los + puntos +\begin_inset Formula $(t_{i-1},\omega_{i-1})$ +\end_inset + + y +\begin_inset Formula $(t_{i},\omega_{i})$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de convergencia del método de Euler: +\series default + Sean +\begin_inset Formula $a,b\in\mathbb{R}$ +\end_inset + + con +\begin_inset Formula $a<b$ +\end_inset + +, +\begin_inset Formula $D\subseteq\mathbb{R}^{2}$ +\end_inset + + un abierto conexo, +\begin_inset Formula $x_{0}\in\mathbb{R}$ +\end_inset + + con +\begin_inset Formula $(a,x_{0})\in D$ +\end_inset + +, +\begin_inset Formula $f:D\to\mathbb{R}$ +\end_inset + + lipschitziana en +\begin_inset Formula $D$ +\end_inset + + en la segunda variable con constante de lipschitzianidad +\begin_inset Formula $K\geq0$ +\end_inset + +, +\begin_inset Formula $n\in\mathbb{N}^{*}$ +\end_inset + +, +\begin_inset Formula $h:=\frac{b-a}{n}$ +\end_inset + +, +\begin_inset Formula $x:[a,b]\to\mathbb{R}$ +\end_inset + + una solución de +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{x} & =f(t,x),\\ +x(a) & =x_{0} +\end{aligned} +\right. +\] + +\end_inset + +con +\begin_inset Formula $\ddot{x}(D)$ +\end_inset + + acotada por un +\begin_inset Formula $C\geq0$ +\end_inset + +, +\begin_inset Formula $(t_{i},\omega_{i})_{i=0}^{n}$ +\end_inset + + los puntos dados por el método de Euler con paso +\begin_inset Formula $h$ +\end_inset + + para dicho problema con redondeo, dado por +\begin_inset Formula +\[ +\left\{ \begin{aligned}\omega_{0} & :=x_{0}+\delta_{0},\\ +\omega_{i+1} & :=\omega_{i}+hf(t_{i},\omega_{i})+\delta_{i+1}, +\end{aligned} +\right. +\] + +\end_inset + +con cada +\begin_inset Formula $|\delta_{i}|<\delta$ +\end_inset + + para un cierto +\begin_inset Formula $\delta\geq0$ +\end_inset + +, y +\begin_inset Formula $x_{i}:=x(t_{i})$ +\end_inset + + para cada +\begin_inset Formula $i$ +\end_inset + +, entonces +\begin_inset Formula +\[ +\max_{0\leq i\leq n}|x_{i}-\omega_{i}|\leq e^{(b-a)K}\delta+\left(\frac{e^{(b-a)K}-1}{K}\right)\left(\frac{1}{2}Ch+\frac{\delta}{h}\right). +\] + +\end_inset + +En particular, sin redondeo el método de Euler tiene una precisión de +\begin_inset Formula $O(h)$ +\end_inset + +. + +\series bold +Demostración: +\series default + Por Taylor, +\begin_inset Formula +\[ +x_{i+1}=x(t_{i}+h)=x(t_{i})+h\dot{x}(t_{i})+\frac{1}{2}h^{2}\ddot{x}(\xi_{i})=x_{i}+hf(t_{i},x_{i})+\frac{1}{2}h^{2}\ddot{x}(\xi_{i}) +\] + +\end_inset + + para algún +\begin_inset Formula $\xi_{i}\in[t_{i},t_{i+1}]$ +\end_inset + +. + Queremos ver que, para +\begin_inset Formula $i\in\{0,\dots,n\}$ +\end_inset + +, +\begin_inset Formula +\[ +|x_{i}-\omega_{i}|\leq(1+hK)^{i}\delta+\sum_{j=0}^{i-1}(1+hK)^{j}\left(\frac{1}{2}Ch^{2}+\delta\right). +\] + +\end_inset + +Para +\begin_inset Formula $i=0$ +\end_inset + + esto es obvio, y supuesto esto probado para un +\begin_inset Formula $i\in\{0,\dots,n-1\}$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +|y_{i+1}-\omega_{i+1}| & =\left|(y_{i}-\omega_{i})+h(f(t_{i},y_{i})-f(t_{i},\omega_{i}))+\frac{1}{2}h^{2}\ddot{y}(\xi_{i})-\delta_{i+1}\right|\\ + & \leq|y_{i}-\omega_{i}|+h|f(t_{i},y_{i})-f(t_{i},\omega_{i})|+\frac{1}{2}h^{2}|\ddot{y}(\xi_{i})|+|\delta_{i+1}|\\ + & \leq(1+hK)|y_{i}-\omega_{i}|+\frac{1}{2}Ch^{2}+\delta\\ + & \leq(1+hK)\left((1+hK)^{i}\delta+\sum_{j=0}^{i-1}(1+hK)^{j}\left(\frac{1}{2}Ch^{2}+\delta\right)\right)+\frac{1}{2}Ch^{2}+\delta\\ + & =(1+hK)^{i+1}\delta+\sum_{j=0}^{i}(1+hK)^{j}\left(\frac{1}{2}Ch^{2}+\delta\right). +\end{align*} + +\end_inset + +Ahora bien, +\begin_inset Formula $\sum_{j=0}^{i}(1+hK)^{j}=\frac{(1+hK)^{i+1}-1}{(1+hK)-1}=\frac{(1+hK)^{i+1}-1}{hK}$ +\end_inset + +, y para +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + +, existe +\begin_inset Formula $\xi$ +\end_inset + + con +\begin_inset Formula $e^{t}=1+t+\frac{t^{2}}{2}e^{\xi}\geq1+t$ +\end_inset + + y por tanto +\begin_inset Formula $(1+t)^{n}\leq e^{nt}$ +\end_inset + +, luego en particular +\begin_inset Formula $(1+hK)^{i+1}\leq(1+hK)^{n}\leq e^{hKn}=e^{(b-a)K}$ +\end_inset + + y +\begin_inset Formula +\[ +|y_{i}-\omega_{i}|\leq e^{(b-a)K}+\frac{e^{(b-a)K}-1}{hK}\left(\frac{1}{2}Ch^{2}+\delta\right). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $x:[a,b]\to\mathbb{R}$ +\end_inset + + es de clase +\begin_inset Formula ${\cal C}^{3}$ +\end_inset + +, +\begin_inset Formula $\frac{\partial f}{\partial x}$ +\end_inset + + y +\begin_inset Formula $\frac{\partial^{2}f}{\partial x^{2}}$ +\end_inset + + son continuas y +\begin_inset Formula $(t_{i},\omega_{i})_{i=0}^{n}$ +\end_inset + + son los puntos dados por el método de Euler en +\begin_inset Formula $[a,b]$ +\end_inset + + con paso +\begin_inset Formula $h$ +\end_inset + +, +\begin_inset Formula $x_{i}-\omega_{i}=hD(t_{i})+O(h^{2})$ +\end_inset + +, donde +\begin_inset Formula $D$ +\end_inset + + es la solución del problema +\begin_inset Formula +\[ +\left\{ \begin{aligned}\dot{D}(t) & =\frac{\partial f}{\partial x}(t,x(t))D(t)+\frac{1}{2}\ddot{x}(t),\\ +D(t_{0}) & =0. +\end{aligned} +\right. +\] + +\end_inset + +De aquí, si +\begin_inset Formula $(t_{i},\xi_{i})_{i=0}^{2n}$ +\end_inset + + son los puntos dados por el método de Euler en +\begin_inset Formula $[a,b]$ +\end_inset + + con paso +\begin_inset Formula $\frac{h}{2}$ +\end_inset + +, +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $|x_{i}-\xi_{2i}|=(\xi_{2i}-\omega_{i})+O(h^{2})$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $x_{i}-\omega_{i}=hD(t)+O(h^{2})$ +\end_inset + + y +\begin_inset Formula $x_{i}-\xi_{2i}=\frac{h}{2}D(t)+O(h^{2})$ +\end_inset + +, despejando, +\begin_inset Formula $x_{i}-2\xi_{i}+\omega_{i}=O(h^{2})$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $y_{i}:=2\xi_{2i}-\omega_{i}$ +\end_inset + + es un método de paso fijo +\begin_inset Formula $h$ +\end_inset + + de orden +\begin_inset Formula $O(h^{2})$ +\end_inset + +. +\end_layout + +\begin_layout Section +Métodos de Taylor +\end_layout + +\begin_layout Standard +El +\series bold +método de Taylor +\series default + de orden +\begin_inset Formula $p\in\mathbb{N}^{*}$ +\end_inset + + es el dado por +\begin_inset Formula $\omega_{0}=x_{0}$ +\end_inset + + y +\begin_inset Formula +\[ +\omega_{i+1}=\omega_{i}+h\left(f(t_{i},\omega_{i})+\frac{h}{2}f'(t_{i},\omega_{i})+\dots+\frac{h^{p-1}}{p!}f^{(p-1)}(t_{i},\omega_{i})\right), +\] + +\end_inset + +donde +\begin_inset Formula $f^{(p)}(t_{i},\omega_{i})$ +\end_inset + + se define como +\begin_inset Formula $x^{(p+1)}(t_{i})$ +\end_inset + + en el problema con la misma e.d.o. + pero condición inicial +\begin_inset Formula $x(t_{i})=\omega_{i}$ +\end_inset + +. + Por ejemplo, +\begin_inset Formula +\begin{align*} +f'(t_{i}) & =\ddot{x}(t_{i})=\frac{\partial f}{\partial t}(t,x(t))+\frac{\partial f}{\partial x}(t,x(t))\dot{x}(t)=\frac{\partial f}{\partial t}(t_{i},\omega_{i})+\frac{\partial f}{\partial x}(t_{i},\omega_{i})f(t_{i},\omega_{i}), +\end{align*} + +\end_inset + +El método de Euler es el método de Taylor de orden 1. +\end_layout + +\begin_layout Standard +Dado un método de paso fijo de la forma +\begin_inset Formula $\omega_{0}:=\alpha$ +\end_inset + +, +\begin_inset Formula $\omega_{i+1}:=\omega_{i}+hØ(t_{i},\omega_{i})$ +\end_inset + +, llamamos +\series bold +error local de truncamiento +\series default + en +\begin_inset Formula $i\in\{1,\dots,n\}$ +\end_inset + + a +\begin_inset Formula +\[ +\tau_{i}(h):=\frac{x(t_{i})-x(t_{i-1})}{h}-Ø(t_{i-1},x_{i-1}). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, si +\begin_inset Formula $x\in{\cal C}^{(p+1)}[a,b]$ +\end_inset + +, el error local de truncamiento del método de Taylor de orden +\begin_inset Formula $p$ +\end_inset + + es +\begin_inset Formula $O(h^{p})$ +\end_inset + +. + +\series bold +Demostración: +\series default + +\begin_inset Formula +\[ +x(t_{i+1})=x(t_{i}+h)=x(t_{i})+h\dot{x}(t_{i})+\dots+\frac{h^{p}}{p!}x^{(p)}(t_{i})+\frac{h^{p+1}}{(p+1)!}x^{(p+1)}(\xi_{i}) +\] + +\end_inset + +para un cierto +\begin_inset Formula $\xi_{i}\in[t_{i},t_{i+1}]$ +\end_inset + +, luego +\begin_inset Formula +\[ +\tau_{i+1}(h)=\frac{x(t_{i+1})-x(t_{i})}{h}-\left(\dot{x}(t_{i})+\frac{h}{2}\ddot{x}(t_{i})+\dots+\frac{h^{p-1}}{p!}x^{(p)}(t_{i})\right)=\frac{h^{p}}{(p+1)!}x^{(p+1)}(\xi_{i}), +\] + +\end_inset + +pero +\begin_inset Formula $[a,b]$ +\end_inset + + es compacto y por tanto +\begin_inset Formula $x^{(p+1)}([a,b])$ +\end_inset + + es acotado, digamos, por +\begin_inset Formula $M$ +\end_inset + +, por lo que +\begin_inset Formula $|\tau_{i+1}(h)|\leq\frac{M}{(p+1)!}h^{p}=O(h^{p})$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Decimos que un método de paso fijo es de orden +\begin_inset Formula $p$ +\end_inset + + si su error local de truncamiento con +\begin_inset Formula $f\in{\cal C}^{\infty}$ +\end_inset + + es +\begin_inset Formula $O(h^{p})$ +\end_inset + +. +\end_layout + +\begin_layout Section +Métodos de Runge-Kutta +\end_layout + +\begin_layout Standard +Los métodos de Taylor tienen mucha precisión, pero requieren trabajo previo + y son difíciles de reutilizar, por lo que intentamos +\begin_inset Quotes cld +\end_inset + +imitar +\begin_inset Quotes crd +\end_inset + + la precisión de estos con operaciones que no requieran derivar +\begin_inset Formula $f$ +\end_inset + +. + Los +\series bold +métodos de Runge-Kutta +\series default + tienen la forma +\begin_inset Formula +\begin{align*} +\omega_{i+1} & :=\omega_{i}+h\sum_{j=1}^{s}b_{j}k_{j}, & k_{1} & :=f(t_{i},\omega_{i}), & k_{j>1} & :=(t_{i}+c_{j}h,\omega_{i}+h(a_{j,1}k_{1}+\dots+a_{j,j-1}k_{j-1})), +\end{align*} + +\end_inset + +para ciertos +\begin_inset Formula $s\in\mathbb{N}$ +\end_inset + + y +\begin_inset Formula $(a_{ij})_{1\leq j<i\leq s},(b_{j})_{j=1}^{s},(c_{i})_{i=2}^{s}$ +\end_inset + + reales. + Estos métodos se pueden representar con una +\series bold +tabla de Butcher +\series default +: +\begin_inset Formula +\[ +\begin{array}{c|ccccc} +c_{2} & a_{21}\\ +c_{3} & a_{31} & a_{32}\\ +\vdots & \vdots & \vdots & \ddots\\ +c_{s} & a_{s1} & a_{s2} & \cdots & a_{s,s-1}\\ +\hline & b_{1} & b_{2} & \cdots & b_{s-1} & b_{s} +\end{array} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +El +\series bold +método del punto medio +\series default + tiene tabla +\begin_inset Formula +\[ +\begin{array}{c|cc} +\frac{1}{2} & \frac{1}{2}\\ +\hline & 0 & 1 +\end{array}, +\] + +\end_inset + + y es de orden 2. + +\series bold +Demostración: +\series default + Por Taylor, +\begin_inset Formula +\begin{align*} +f(t+\tfrac{h}{2},x+\tfrac{h}{2}f(t,x))= & f(t,x)+\frac{h}{2}\frac{\partial f}{\partial t}(t,x)+\frac{h}{2}f(t,x)\frac{\partial f}{\partial x}(t,x)+\\ + & +\frac{h^{2}}{8}\frac{\partial^{2}f}{\partial t^{2}}(\xi_{1},\mu_{1})+\frac{h^{2}}{4}f(t,x)\frac{\partial^{2}f}{\partial t\partial x}(\xi_{2},\mu_{2})+\frac{h^{2}}{8}f(t,x)^{2}\frac{\partial^{2}f}{\partial x^{2}}(\xi_{3},\mu_{3}). +\end{align*} + +\end_inset + +El método de Taylor de orden 2 viene dado por +\begin_inset Formula $\omega_{i+1}=\omega_{i}+h(f(t_{i},\omega_{i})+\frac{h}{2}f'(t_{i},\omega_{i}))$ +\end_inset + +, pero +\begin_inset Formula +\[ +f(t,x)+\frac{h}{2}f'(t,x)=f(t,x)+\frac{h}{2}\frac{\partial f}{\partial t}(t,x)+\frac{h}{2}f(t,x)\frac{\partial f}{\partial x}(t,x), +\] + +\end_inset + +luego como las dobles derivadas parciales son continuas y por tanto su imagen + por +\begin_inset Formula $[a,b]$ +\end_inset + + es compacta, la diferencia de error local entre ambos métodos es +\begin_inset Formula $O(h^{2})$ +\end_inset + +, que se suma al error de +\begin_inset Formula $O(h^{2})$ +\end_inset + + del método de Taylor de orden 2. +\end_layout + +\begin_layout Standard +No existe un método de Runge-Kutta de orden 3 con solo 2 evaluaciones de + +\begin_inset Formula $f$ +\end_inset + +. + +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Como +\begin_inset Formula +\begin{align*} +f' & =\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x},\\ +f'' & =\frac{\partial f'(t,x(t))}{\partial t}=\frac{\partial^{2}f}{\partial t^{2}}+f\frac{\partial^{2}f}{\partial x\partial t}+\left(\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x}\right)\frac{\partial f}{\partial x}+f\left(\frac{\partial^{2}f}{\partial x\partial t}+f\frac{\partial^{2}f}{\partial x^{2}}\right)\\ + & =\frac{\partial^{2}f}{\partial t^{2}}+\frac{\partial f}{\partial t}\frac{\partial f}{\partial x}+f\left(\frac{\partial f}{\partial x}\right)^{2}+2f\frac{\partial^{2}f}{\partial x\partial t}+f^{2}\frac{\partial^{2}f}{\partial x^{2}}, +\end{align*} + +\end_inset + +el método de Taylor de orden 3 es +\begin_inset Formula $\omega_{i+1}=\omega_{i}+hØ(t_{i},\omega_{i})$ +\end_inset + +, con +\begin_inset Formula +\begin{align*} +Ø(t,x) & =f(t,x)+\frac{h}{2}f'(t,x)+\frac{h^{2}}{6}f''(t,x)\\ + & =f+\frac{h}{2}\left(\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x}\right)+\frac{h^{2}}{6}\left(\frac{\partial^{2}f}{\partial t^{2}}+\frac{\partial f}{\partial t}\frac{\partial f}{\partial x}+f\left(\frac{\partial f}{\partial x}\right)^{2}+2f\frac{\partial^{2}f}{\partial x\partial t}+f^{2}\frac{\partial^{2}f}{\partial x^{2}}\right), +\end{align*} + +\end_inset + +pero los métodos de 2 evaluaciones tienen la forma +\begin_inset Formula +\begin{multline*} +b_{1}f(t,x)+b_{2}f(t+c_{2},x+a_{21}f(t,x))=\\ +=b_{1}f+b_{2}f+b_{2}c_{2}\frac{\partial f}{\partial t}+b_{2}a_{21}f\frac{\partial f}{\partial x}+b_{2}\frac{a_{21}^{2}}{2}\frac{\partial^{2}f}{\partial t^{2}}+b_{2}c_{2}a_{21}f\frac{\partial^{2}f}{\partial t\partial x}+b_{2}\frac{a_{21}^{2}}{2}f^{2}\frac{\partial^{2}f}{\partial x^{2}}+O(h^{3}). +\end{multline*} + +\end_inset + +Para que ambas coincidieran en los términos hasta el orden 2, la última + fórmula debería tener un término proporcional a +\begin_inset Formula $f\left(\frac{\partial f}{\partial x}\right)^{2}$ +\end_inset + +, pero no lo tiene. +\end_layout + +\begin_layout Standard +Otros métodos son el +\series bold +método de Euler modificado +\series default +, con tabla +\begin_inset Formula +\[ +\begin{array}{c|cc} +1 & 1\\ +\hline & \frac{1}{2} & \frac{1}{2} +\end{array}, +\] + +\end_inset + +y el +\series bold +método de Heun +\series default +, con tabla +\begin_inset Formula +\[ +\begin{array}{c|cc} +\frac{2}{3} & \frac{2}{3}\\ +\hline & \frac{1}{4} & \frac{3}{4} +\end{array}, +\] + +\end_inset + +ambos de orden 2. +\end_layout + +\begin_layout Standard +El método de Runge-Kutta más usado es el de orden 4 ( +\series bold +RK4 +\series default +): +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\begin{array}{c|cccc} +\frac{1}{2} & \frac{1}{2}\\ +\frac{1}{2} & 0 & \frac{1}{2}\\ +1 & 0 & 0 & 1\\ +\hline & \frac{1}{6} & \frac{1}{3} & \frac{1}{3} & \frac{1}{6} +\end{array} +\] + +\end_inset + +La siguiente tabla muestra el máximo orden alcanzable con métodos de Runge-Kutta + en función del número de evaluaciones de +\begin_inset Formula $f$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\align center +\begin_inset Tabular +<lyxtabular version="3" rows="2" columns="5"> +<features tabularvalignment="middle"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<column alignment="center" valignment="top"> +<row> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +Evaluaciones ( +\begin_inset Formula $s$ +\end_inset + +) +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\leq4$ +\end_inset + + +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +5–7 +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +8–9 +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\geq10$ +\end_inset + + +\end_layout + +\end_inset +</cell> +</row> +<row> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +Mejor orden +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s$ +\end_inset + + +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s-1$ +\end_inset + + +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s-2$ +\end_inset + + +\end_layout + +\end_inset +</cell> +<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $s-3$ +\end_inset + + +\end_layout + +\end_inset +</cell> +</row> +</lyxtabular> + +\end_inset + + +\end_layout + +\end_body +\end_document |