Solving Differential Equations for Real-world Problems

1. Follow-the-leader model

$$ \dot{x}_i = v_i $$

$$ \dot{v}_i = \beta\cfrac{\Delta v}{s_i^2} $$

$$ s_i = x_{i-1} - x_i - L $$

$$ \Delta v_i = v_{i-1} - v_{i} $$

Reference: https://www.annualreviews.org/content/journals/10.1146/annurev-control-030123-015145 (Page 3 of the PDF)

Note: Page 3 has a typo. $\Delta v_i = v_i - v_{i-1}$ should be $\Delta v_i = v_{i-1} - v_{i}$.