User:Egm6341.s10.team3.heejun/Meeting 40 transcript

 '''MTG 40: Thu. 8 Apr 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:57, 11 August 2010 (UTC)

[[media: Egm6341.s10.mtg40.djvu | Page 40-1]] 
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HW: Chaos Reproduce Fig.15.6 and 15.7 in King et. al. 2003. (p.455-456) Ref: Differential Equations: Amazon
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Hints for solution of logistic equation ([[media: Egm6341.s10.mtg39.djvu | HW p.39-1]])


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 * $$ \displaystyle \dot{x} = x(1-x) $$
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$$ \displaystyle \color{blue}{\int^{x}_{x_{0}=x(t_{0})}} \color{black}{\frac{dx}{x(1-x)}} = \color{blue}{\int^{t}_{t_{0}}} \color{black}{dt} $$
 * Seperation of variable:
 * Seperation of variable:
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$$ \displaystyle \frac{1}{x(1-x)} = \frac{a}{x} + \frac{b}{1-x} $$ a, b are constant to be determined
 * Partial fraction:
 * Partial fraction:
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Elementary stability FE(Forward Euler) + BE(Backward Euler) algorithms ([[media: Egm6341.s10.mtg39.djvu | p.39-2]])


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$$ \displaystyle \bar{x}_{i+1} = \underbrace{(1+hr)}_{\rho_{\color{blue}{F}}} \bar{x}_{i} = \rho_{\color{blue}{F}} (\rho_{\color{blue}{F}} \bar{x}_{i+1}) = \rho_{\color{blue}{F}}^{i+1} \bar{x}_{0} $$
 * FE:
 * FE:
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 * Fix $$ \displaystyle i $$, consider $$ \displaystyle h \to \infty $$
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 * $$ \displaystyle \rho_{\color{blue}{F}}^{i+1} \to \infty \Rightarrow \bar{x}_{i+1} \to \infty $$    unstable
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$$ \displaystyle \bar{x}_{i+1} = \underbrace{ \frac{1}{1-hr}}_{\rho_{\color{blue}{B}}} \bar{x}_{i} = \rho_{B}^{i+1} \bar{x}_{0} $$
 * BE:
 * BE:
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[[media: Egm6341.s10.mtg40.djvu | Page 40-2]] 


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 * Fix $$ \displaystyle i $$, let $$ \displaystyle h \to \infty \Rightarrow |\rho_{B}^{i+1}| \to 0 \Rightarrow |\bar{x}^{i+1}| \to 0 $$
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 * stable: numerical solution does not below up $$ \displaystyle \forall h $$
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HW: Integrate logistic equation  ([[media: Egm6341.s10.mtg39.djvu | HW p.39-1]]) Integrate logistic equation already found $$\hat{h}$$ such that Hermit-simpson algorithm yields error($$ 10^{-6}$$) 1) Run Hermit-Simpson with $$\displaystyle h= 2^k \hat{h}$$ 2) Develop and Run Forward Euler with $$\displaystyle h= 2^k \hat{h}$$ 3) Develop and Run Backward Euler with $$\displaystyle h= 2^k \hat{h}$$
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Equations of motion of aircraft: [[media: Egm6341.s10.mtg33.djvu | p.33-3]]


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 * $$ \displaystyle \underline{\dot{Z}} = \underline{f} (\underline{Z}, \underline{u}, \underline{t}) $$
 *   (3)  [[media: Egm6341.s10.mtg33.djvu | p.33-3]]
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 * $$ \displaystyle \int^{t_{i+1}}_{t_{i}} \underline{\dot{Z}} dt = \int^{t_{i+1}}_{t_{i}} \underline{f} (\underline{Z}, \underline{u}, \underline{t}) dt  $$
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 * $$ \displaystyle \underbrace{\underline{Z}_{i+1}}_{\color{red}{4 \times 1}} - \underbrace{\underline{Z}_{i}}_{\color{red}{4 \times 1}} = \frac{h/2}{3} [ \underline{f}_{i} + 4 \underline{f}_{i+ \frac{1}{2}} + \underline{f}_{i+1} ] $$
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 * $$ \displaystyle \Leftrightarrow \underbrace{\underline{F}}_{\color{red}{4 \times 1}} (\underline{Z}_{i+1}) = \underbrace{\underline{0}}_{\color{red}{4 \times 1}} $$
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[[media: Egm6341.s10.mtg40.djvu | Page 40-3]] 

NR algorithm [[media: Egm6341.s10.mtg38.djvu | p.38-1]]


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 * $$ \displaystyle \underbrace{\underline{Z}_{i+1}^{(k+1)}}_{\color{red}{4 \times 1}} = \underbrace{\underline{Z}_{i+1}^{(k)}}_{\color{red}{4 \times 1}} - \underbrace{[\frac{d \underline{F}(\underline{Z}_{i+1}^{(k)})}{d \underline{Z}}]^{-1}}_{\color{red}{4 \times 4}} \underbrace{\underline{F}}_{\color{red}{4 \times 1}} (\underline{Z}^{(k)}_{i+1}) $$
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 * $$ \displaystyle \underline{Z}^{(0)}_{i+1} = \underline{Z}_{i} $$
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 * Review: $$ \displaystyle \underbrace{\underline{f}}_{\color{red}{m \times 1}} (\underline{x}) = \left \lfloor f_{1}(\underline{x}), ..., f_{m}(\underline{x}) \right \rfloor ^{\color{red}{T}} $$
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 * $$ \displaystyle \underbrace{\underline{x}}_{\color{red}{n \times 1}} = \left \lfloor x_{1}, ..., x_{n} \right \rfloor ^{\color{red}{T}} $$
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 * $$ \displaystyle \underbrace{\frac{d \underline{f}(\underline{\hat{x}})}{d \underline{x}}}_{\color{red}{m \times n}} = \underbrace{[\frac{ df_{\color{red}{i}}(\underline{\hat{x}}) }{dx_{\color{red}{j}}}]}_{\color{red}{m \times n}}  $$
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$$ \displaystyle \color{red}{j:} $$   column index  $$ \displaystyle (j=1,...,n) $$
 * $$ \displaystyle \color{red}{i:} $$   row index  $$ \displaystyle (i=1,...,m) $$
 * $$ \displaystyle \color{red}{i:} $$   row index  $$ \displaystyle (i=1,...,m) $$
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HW: S+Z (2007) parameters for EOM(equation of motion) Table. I p. 317 S+Z (2007)
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