University of Florida/Egm6341/s11.team1.Gong/Mtg38

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page38-1 $(1)p.37-3Z^{{}^{\prime }}=h\stackrel{\bullet }{z}(1)$ $z(s)=\sum _{i=1}^4\overline{N_i}(s)\overline{d_i}(2)$ $z(t)=\sum _{i=1}^4{N}_i(t)d_i(3)$ Template:Font $Collocationat{t}_i\to (5)p.36-4$ $Collocationat{t}_{i+1}\to (6)p.36-4$ Template:Font $Collocationat{t}_{i+\frac{1}{2}}\Rightarrow$ ${\stackrel{\bullet }{z}}_{i+\frac{1}{2}}=f_{i+\frac{1}{2}}=f(z_{i+\frac{1}{2}},t_{i+\frac{1}{2}})(4)$ $z_{i+\frac{1}{2}}=z(s=\frac{1}{2})$ $\stackrel{H{W}^{*}6.6}{=}\frac{1}{2}(z_i+z_{i+1})+\frac{h}{8}(f_i-f_{i+1})(5)$ page38-2 ${(1)p.38-3\stackrel{\bullet }{z}}_{i+\frac{1}{2}}=z_{i+\frac{1}{2}}^{{}^{\prime }}\frac{1}{h}(1)$ ${\stackrel{\bullet }{z}}_{i+\frac{1}{2}}={z^{}}^{\prime }(s=\frac{1}{2})\stackrel{H{W}^{*}6.6\{\begin{array}{cc}& (1)p.37-2\\ & (1)p.37-3\end{array}}{=}-\frac{3}{2}(z_i-z_{i+1})-\frac{1}{4}(z_i^{{}^{\prime }}+z_{i+1}^{{}^{\prime }})(2)$ $(1){}_{}(2):$ ${\stackrel{\bullet }{z}}_{i+\frac{1}{2}}=\frac{-3}{2h}(z_i-z_{i+1})-\frac{1}{4}(f_i+f_{i+1})(3)$ ${\stackrel{\bullet }{z}}_{i+\frac{1}{2}}\ne f_{i+\frac{1}{2}}ingeneral$ $Gap=\mathrm{\Delta }{\stackrel{\bullet }{z}}_{i+\frac{1}{2}}-f_{i+\frac{1}{2}}(4)$ $Collocationat{t}_{i+\frac{1}{2}}\Rightarrow \mathrm{\Delta }=0(5)$ $\underset{\_}{Goal:}Find(z_i,z_{i+1})st\mathrm{\Delta }=0(6)$ page38-3 $\mathrm{\Delta }=0\Rightarrow z_{i+1}\stackrel{(1)}{=}\underset{Simpso{n}^{\prime }srule!(2)p.7-4}{\underbrace{z_i+\frac{h/2}{3}[f_i+4f_{i+\frac{1}{2}}+f_{i+1}]}}$ $\stackrel{\bullet }{z}=f(z,t)(2)$ $\underset{z_{i+1}-z_i}{\underbrace{\underset{t_i}{\overset{t_{i+1}}{\int }}\stackrel{\bullet }{z}dt}}=\underset{applysimpso{n}^{\prime }srule\Rightarrow (1)}{\underbrace{\underset{t_i}{\overset{t_{i+1}}{\int }}dt}}(3)$ Template:FontTemplate:Font{zi, i=1,2,...,n} Template:Fontsolved by NLPTemplate:Font $IVP(\underset{(4)z(t_o)=z_0}{\underbrace{InitialValue}}pb):Int.\underset{(2)}{\underbrace{nonlinearODEs}}$ page38-4 Template:Font Template:Font Need: $1)f_i=f(z_i,t_i)cancomp.$ $2)f_{i+1}=f(\underset{unknown}{\underbrace{z_{i+1}}},\underset{known}{\underbrace{t_{i+1}}})unknown$ $3)f_{i+\frac{1}{2}}=f(\underset{unknown}{\underbrace{z_{i+\frac{1}{2}}}},\underset{known}{\underbrace{t_{i+\frac{1}{2}}}})unknown$ $t_{i+\frac{1}{2}}=t_i+\frac{h}{2}$ Template:Font $z_{i+\frac{1}{2}}=g(z_i,z_{i+1})(1)$ Template:Font $z_{i+1}=z_i+\frac{h/2}{3}[f_i+4f(g(z_i,z_{i+1}),t_{i+\frac{1}{2}})+f_{i+1}](2)$ $\iff F(z_{i+1})\stackrel{(3)}{=}\underset{nonlinearalg.eq.}{\underbrace{0}}$ $\Rightarrow Newton-Raphson-Simpson$