Systems of equations

In many engineering problems we will have a system of equations which depend on a single parameter:

\[\begin{split}\begin{align} \frac{dy_1}{dt} &= f_1(x, y_1, y_2, ..., y_j) \\ \frac{dy_2}{dt} &= f_2(x, y_1, y_2, ..., y_j) \\ \vdots \\ \frac{dy_j}{dt} &= f_j(x, y_1, y_2, ..., y_j) \\ \end{align}\end{split}\]

Happily, extension of the Runge-Kutta methods is straightforward! Collecting functions as a vector, we get the vector RK form:

\[ \vec{y}_{i+1} \approx \vec{y}_i + h \sum_{n=1}^s a_n \vec{k}_n\]

with

\[ \vec{k}_n = \vec{f}(x_i + p_n, \vec{y}_i + \sum_{m=1}^s q_{nm}\vec{k}_m) \]