Runge-Kutta method: Difference between revisions
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</pre> |
</pre> |
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=={{header|J}}== |
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'''Solution:''' |
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<lang j>NB.*rk4 a Solve function using Runge-Kutta method |
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NB. y is: y(ta) , ta , tb , tstep |
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NB. eg: fyp rk4 1 0 10 0.1 |
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rk4=: 1 : 0 |
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'Y0 a b h'=. 4{. y |
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T=. a + i.@>:&.(%&h) b - a |
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Y=. Yt=. Y0 |
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for_t. }: T do. |
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ty=. t,Yt |
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k1=. h * u ty |
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k2=. h * u ty + -: h,k1 |
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k3=. h * u ty + -: h,k2 |
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k4=. h * u ty + h,k3 |
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Y=. Y, Yt=. Yt + (%6) * 1 2 2 1 +/@:* k1, k2, k3, k4 |
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end. |
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T ,. Y |
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)</lang> |
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'''Example:''' |
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<lang j> fy=: (%16) * [: *: 4 + *: NB. f(t,y) |
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fyp=: (* %:)/ NB. f'(t,y) |
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report_whole=: (10 * i. >:10)&{ NB. report at whole-numbered t values |
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report_err=: (, {: - [: fy {.)"1 NB. report errors |
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report_err report_whole fyp rk4b 1 0 10 0.1 |
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0 1 0 |
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1 1.5625 _1.45722e_7 |
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2 4 _9.19479e_7 |
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3 10.5625 _2.90956e_6 |
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4 25 _6.23491e_6 |
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5 52.5625 _1.08197e_5 |
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6 100 _1.65946e_5 |
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7 175.562 _2.35177e_5 |
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8 289 _3.15652e_5 |
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9 451.562 _4.07232e_5 |
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10 676 _5.09833e_5</lang> |
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=={{header|Mathematica}}== |
=={{header|Mathematica}}== |
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<lang Mathematica>DSolve[{y'[t] == t * Sqrt[y[t]], y[0] == 1, y'[0] == 0}, y[t], t]//Simplify |
<lang Mathematica>DSolve[{y'[t] == t * Sqrt[y[t]], y[0] == 1, y'[0] == 0}, y[t], t]//Simplify |
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y[t] -> 1/16*(4+t^2)^2 /. t -> Range[10] |
y[t] -> 1/16*(4+t^2)^2 /. t -> Range[10] |
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->y[{1,2,3,4,5,6,7,8,9,10}]->{25/16, 4, 169/16, 25, 841/16, 100, 2809/16, 289, 7225/16, 676}</lang> |
->y[{1,2,3,4,5,6,7,8,9,10}]->{25/16, 4, 169/16, 25, 841/16, 100, 2809/16, 289, 7225/16, 676}</lang> |
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=={{header|Python}}== |
=={{header|Python}}== |
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