Runge-Kutta methods for solving ordinary differential equations

8.1 Runge-Kutta methods for solving ordinary differential equations

Runge-Kutta methods are used for solving ordinary differential equations (Cauchy problems). Thisapplet is intended for solving ordinary differential equations using Euler, First and Second Euler modification and Runge-Kutta method. Using this methods for solution of Cauchy problem in node x, solution in node x+h is calculated.

This applet is intended for solving ordinary differential equations given:

u'(x) = f(x, u(x)) , u(x₀) = u₀

This applet calulate solution in node x+h:

For n = 1 in formula above Euler algortihym is used for calculating results.

For n = 2 and 3 First(1) and Second(2) Euler modification algorithym is used for calculating results.

(1) u(x+h) = u(x) + h/2 * (f(x, u(x)) + f(x+h, u(x) + h * f(x, u(x)))

(2) u(x+h) = u(x) + h * f(x+ h/2, u(x) + h/2 * f(x, u(x)))

For n = 4 Runge-Kutta algorithym is used for calculating results.