List of basic formulas for Euler and Runge-Kutta methods:

Euler’s method

Euler’s method is used for the numerical solution of ordinary differential equations (ODE) of the first order. The basic formula is:

  1. The general formula for Euler’s method is:
    [
    y_{n+1} = y_n + hf(t_n, y_n)
    ]
    where:
  • ( y_n ) — the value of the function at the point ( t_n ),
  • ( h ) — time step,
  • ( f(t_n, y_n) ) is a function describing the right-hand side of the ODE.

Method Runge-Kutty

The Runge-Kutta method is more accurate and is often used to solve ODEs. The most popular is the Runge-Kutta method of the fourth order (RK4):

  1. Formulas for the 4th-order Runge-Kutta method:
    [
    k_1 = hf(t_n, y_n)
    ]
    [
    k_2 = hf\left(t_n + \frac{h}{2}, y_n + \frac{k_1}{2}\ right)
    ]
    [
    k_3 = hf\left(t_n + \frac{h}{2}, y_n + \frac{k_2}{2}\right)
    ]
    [
    k_4 = hf(t_n + h, y_n + k_3)
    ]
    [
    y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)
    ]

Notes

  • ( t_n ) — current time value,
  • ( y_n ) — the current value of the function,
  • ( h ) — time step,
  • ( f(t, y) ) is a function describing the right-hand side of the ODE.

От Math

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