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Runge-Kutta method

Calculates the solution y=f(x) of the linear ordinary differential equation y'=F(x,y) using Runge-Kutta method.

The initial condition is y0=f(x0), and the root x is calculated within the range of from x0 to xn. $\normal \\\vspace{10} y'=F(x,y)\hspace{30} y_0=f(x_0)\rightarrow\ y=f(x)\\$

F(x,y)

x₀

initial condition

y₀

= f(x₀)

x_n

x₀≦x≦x_n

[ partition n

]

x	y=f(x)

$\normal Runge-Kutta\ method\\[10pt] (1)\ y'=F(x,y),\hspace{30} y_0=f(x_0)\rightarrow\ y=f(x)\\ (2)\ y_{n+1}=y_n+{\large\frac{1}{6}}(k_1+2k_2+2k_3+k_4)+{\small O}(h^5)\\\vspace{10} \hspace{30} k_1=hF(x_n,\ y_n)\\ \hspace{30} k_2=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_1}{2}})\\ \hspace{30} k_3=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_2}{2}})\\ \hspace{30}k_4=hF(x_n+h,\ y_n+k_3)\\ $

Runge-Kutta method

Calculates the solution y=f(x) of the linear ordinary differential equation y'=F(x,y) using Runge-Kutta method.

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