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Program 1

1st Order ODE (RK4 Method)

An object in free fall under the influence of gravity, with constant acceleration: $a = 9.81 \, \text{m/s}^2$

$$ \frac{dv}{dt}=a ;\quad v(t=0)=0 $$

The following code implements the Runge-Kutta 4th order (RK4) method to calculate the velocity of the object over a time period.

Output 1

The RK4 method is applied to a differential equation of the form:

$$ \frac{dy}{dt} = f(t, y) $$

with an initial condition $ y(t_0) = y_0 $ over a time interval $[t_0, t_{\text{end}}]$. The method proceeds as follows:

For each time step $ h $, the RK4 method calculates four slopes:

Compute $ k_1 $: This is the initial slope, calculated at the beginning of the interval.

$$ k_1 = f(t, y) $$
Compute $ k_2 $: This slope is calculated at the midpoint, using $ k_1 $ to approximate the halfway position.

$$ k_2 = f\left(t + \frac{h}{2}, y + \frac{h}{2} \cdot k_1\right) $$
Compute $ k_3 $: Similar to $ k_2 $, but uses the slope from $ k_2 $ to further refine the midpoint estimate.

$$ k_3 = f\left(t + \frac{h}{2}, y + \frac{h}{2} \cdot k_2\right) $$
Compute $ k_4 $: This final slope is calculated at the end of the interval.

$$ k_4 = f(t + h, y + h \cdot k_3) $$

The next value, $ y_{\text{next}} $, is computed as a weighted average of these slopes:

$$ y_{\text{next}} = y + \frac{h}{6} \cdot (k_1 + 2 \cdot k_2 + 2 \cdot k_3 + k_4) $$

Ordinary Differential Equation-2