Notes for COT 11-23-09

Notes for COT 11-23-09 - HW Help A↔k-1k1B→k2C...

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Unformatted text preview: HW Help A↔k-1k1B→k2C dCAdt=-k1CA+k-1CB dCBdt=k1CA-k2+k-1CB dCCdt=k2CB detA-λI=0=-λ λλ2+bλ+c=0 ←do not expand xt+∆t=xt+∆tf(xt,t) Implicit Euler x=fxt,t Implicit Euler takes the tangent line of a graph calculated at time, t, and extrapolates for the value for x at time, t+∆t Slope at t+∆t=xt+∆t =f(xt+∆t,t+∆t) ximplicit=t+∆t-xt∆=f(xt+∆t,t+∆t) →xt+∆t=xt+∆tf(x(t+∆t),t+∆t) Implicit Euler Of the format: gxt+∆t=xt+∆t-xt-∆tfxt+∆t,t+∆t=0 Another way to Obtain Implicit Euler x≅xt-xt-∆t∆t x=fxt,t→xt-xt-∆t∆t=f(xt,t) xt=xt-∆t+∆tf(xt,t) for all t xτ=xτ-∆t+∆tfxτ,τ for all τ Use τ=t+∆t xt+∆t=xt+∆tf(xt+∆t,t+∆t) CA=x1=-x1+x2 CB=x2=x1-3x2 x1(t+∆t)x2(t+∆t)=x1(t)x2(t)+∆t-x1t+∆t+x2(t+∆t)x1t+∆t3x2(t+∆t) xt+∆t xt + ∆t f(xt,t) x1t+∆t+∆tx1t+∆t-∆tx2t+∆t=x1(t) x2t+∆t-∆tx1t+∆t+3∆tx2t+∆t=x2t C xt+∆t=bt => xt+∆t=C-1b 1+∆t-∆t-∆t(1+3∆t)x1(t+∆t)x2(t+∆t)=x1(t)x2(t) => x1(t+∆t)x2(t+∆t)=11+∆t1+3∆t-∆t2 1+3∆t∆t∆t1+∆tx1(t)x2(t) x1t+∆t=1+3∆tx1t+∆tx2t1+∆t1+3∆t-∆t2 x2t+∆t=∆tx1t+1+∆tx2t1+∆t1+3∆t-∆t2 Luke’s Method xt+∆t=xt+∆tfxt,t+fxt+∆t,t+∆t2 Heun Method (Single Predictor) Predictor: xpt+∆t=xt+∆tf(xt,t) Corrector: xt+∆t=xt+∆tfxt,t+fxpt+∆t,t+∆t2 Stephanie’s Method xpt+∆t2=xt+∆t2f(xt,t) xt+∆t=xt+∆tf(xpt+∆t2,t+∆t2) ...
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This note was uploaded on 01/08/2010 for the course COT 3502 taught by Professor Hawkins during the Fall '08 term at University of Florida.

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Notes for COT 11-23-09 - HW Help A↔k-1k1B→k2C...

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