Runge-Kutta method (4th-order,2nd-derivative) Calculator
Calculates the solution y=f(x) of the ordinary differential equation y'=F(x,y) using Runge-Kutta fourth-order method. |
The initial condition is y0=f(x0), y'0=p0=f'(x0) and the root x is calculated within the range of from x0 to xn. \(\normalsize \\
y''=F(x,y,y')\hspace{30px} y_0=f(x_0),\ y'_0=f'(x_0) \rightarrow\ y=f(x)\\\) |
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\(\normalsize Runge-Kutta\ method\\
(1)\ y''=F(x,y,y')\hspace{30px} p=y',\ y_0=f(x_0),\ y'_0=f'(x_0) \rightarrow\ y=f(x)\\
(2)\ p_{n+1}=p_n+{\large\frac{1}{6}}(j_1+2j_2+2j_3+j_4)+{\small O}(h^5)\\\\
\hspace{25px} y_{n+1}=y_n+{\large\frac{1}{6}}(k_1+2k_2+2k_3+k_4)+{\small O}(h^5)\\\\
\hspace{25px} j_1=hF(x_n,\ y_n,\ p_n)\\
\hspace{25px} k_1=h \cdot p_n \\
\hspace{25px} j_2=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_1}{2}},\ p_n+{\large\frac{j_1}{2}})\\
\hspace{25px} k_2=h ( p_n + {\large\frac{j_1}{2}} )\\
\hspace{25px} j_3=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_2}{2}},\ p_n+{\large\frac{j_2}{2}})\\
\hspace{25px} k_3=h ( p_n + {\large\frac{j_2}{2}} )\\
\hspace{25px} j_4=hF(x_n+h,\ y_n+k_3,\ p_n+j_3)\\
\hspace{25px} k_4=h ( p_n+ j_3)\\
\) |
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Runge-Kutta method (4th-order,2nd-derivative)
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[1] 2019/03/12 07:15 Male / 30 years old level / A teacher / A researcher / Very /
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