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13. Second-Order_IVP

13. Second-Order_IVP - 3.2 Second-Order I.V.P ex y y'2 y =...

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1/4 3.2 Second-Order I.V.P. ex. F dx dy = where v F = x y y y 2 2 ' ' ' = + () 0 0 = y , 1 0 ' = y G dx dv = y v x G 2 2 + = ( ) 0 0 = y ( ) 1 0 = v Explicit Euler Scheme i i i xF y y Δ + = + 1 i i v F = i i i xG v v Δ + = + 1 i i i i y v x G 2 2 + = Start at 0 0 = x where ICs give 0 0 = y , 1 0 = v Find slopes 0 F , 0 G Find 1 y , 1 v Find 1 F , 1 G Find 2 y , 2 v etc. See next page of the handout for Excel implementation. Second-Order Runge-Kutta Scheme s slope v F = s slope y v x G 2 2 + = + Δ + = + + 1 1 2 1 2 1 i i i i F F x y y + Δ + = + + 1 1 2 1 2 1 i i i i G G x v v Solution Process: [1] Calculate the slopes at i x i i i v x at slope y F = = i i i i i y v x x at slope v G 2 2 + = = [2] Use [1] to find y and v estimates at 1 + i x i i i xF y y Δ + = + 1 i i i xG v v Δ + = + 1 [3] Use [2] to estimate the slopes at 1 + i x + + + = = 1 1 1 i i i v x at estimate slope y F + + + + + + = = 1 1 1 1 1 2 2 i i i i i y v x x at estimate slope v G [4] Apply the Runge-Kutta formulas above to find 1 + i y and 1 + i v Implementation on Excel is straightforward.

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2/4 Example: Second Order IVP solution using (explicit) Euler method on Excel with 2 . 0 = Δ x Note : do a few iterations by hand to be sure you understand the process. AB C D E F G H 1 x i y i v i F i G i y exact error = y i y exact 2 001 1 . 0 0 0 1.000 0.000 0.000 3 0.2 0.200 0.800 0.800 0.000 0.186 0.014 4 0.4 0.360 0.800 0.800 0.720 0.367 0.007 5 0.6 0.520 0.944 0.944 1.296 0.572 0.052 6 0.8 0.709 1.203 1.203 1.814 0.825 0.116 7 1 0.949 1.566 1.566 2.333 1.151 0.201 8 1.2 1.263 2.033 2.033 2.893 1.575 0.312 9 1.4 1.669 2.611 2.611 3.527 2.125 0.456 10 1.6 2.191 3.317 3.317 4.266 2.833 0.641 11 1.8 2.855 4.170 4.170 5.140 3.736 0.881 12 2 3.689 5.198 5.198 6.180 4.880 1.191 13 2.2 4.728 6.434 6.434 7.423
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13. Second-Order_IVP - 3.2 Second-Order I.V.P ex y y'2 y =...

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