COT - December 8 - December 8, 2009 Error ∅ Δt 2 Δt 3...

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Unformatted text preview: December 8, 2009 Error ∅ Δt 2 Δt 3 Δt 4 Method Explicit & Implicit Euler 2nd order RK 4th order RK December 8, 2009 x=f(t)Ralston has error Δt 4 “Classical” 4th order RK xt+∆t=xt+∆tΦ Φ=16k1+13k2+13k3+16kn k1=f(xt,t) k2=f(xt+12∆tk1, t+∆t2) k3=f(xt+12∆tk2, t+∆t2) k4=f(xt+∆tk3, t+∆t) ***Go to programming xcel file for code Example 1 x1+2x1x2+sinx2x1=0 x2+2x1+x2x2+cosx1x2=0 I.C. x10=4 x20=3 x10=7 x20=5 New variables y1=x1 y2=x2 y3=x1↔y3=y1 y4=x2↔y4=y2 y3+2y3y2+siny2y1=0 y4+2y2y3+2y2y4+cosy1y2=0 December 8, 2009 Example 2 x1+2x1x2+sinx2x1=0 x2+2x2x1+x2x2+cosx1x2=0 I.C. x10=4 x20=3 x10=7 x20=5 New variables y1=x1 y2=x2 y3=x1↔y3=y1 y4=x2↔y4=y2 f=y3+2y3y2+siny2y1=0→y3-2y3y2-siny2y1 f=y4+2y2y3+2y2y4+cosy1y2=0→y4+2y2y3=-y2y4-cosy1y2 102y21y3y4=-2y3y2-siny2y1-2y2y4-cosy1y2 2y2y4-cosy1y2 → y3y4=10-2y21-2y3y2-siny2y1- December 8, 2009 Exam 2004 #1 x+3x+2=0 x0=0 x0=0 Part a y1=x y2=x→y1=y2 →y2+3y2+2=0 →y2=-3y2-2 y1=y2 y1=y2 y2=-3y2-2 y=010-3y1y2+0-2 A=010-3 b=0-2 y0=y1(0)y2(0)=x(0)x(0)= 00 λ1=0λ3=-3 w1:0-(0)103-(0)w11w21= 00= w21=0-3w21=0w11=C→w1=10 w2:0-(-3)10-3-(-3)w12w22= 0→ Σw12+w22=00=0w12=Cw22=-3C→w2=1-3 W=1103 W-1=1-3-3-101=11/30-1/3 ξt=W-1b=11/30-1/30-2=-2/32/3 z0=W-1x0=11/30-1/300=00 z1t=zi0eλit-ξiλi1-eλit=0-(-23)01-1→z1t=-23t z2t=0+(2/3)-31-e-3t=291-e-3t REVIEW SESSION ON TUESDAY AT 4PM ...
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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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