On Stability Analysis of Car-Following Models with Various Discrete Operators

This paper investigates a car-following model that incorporates both classical and fractional discrete operators. While classical models have been extensively studied, the influence of discrete fractional operators on the stability of such systems has not yet been systematically analyzed. Stability conditions are derived and rigorously proven for systems employing three widely used fractional <i>h</i>-difference operators—Grünwald–Letnikov, Riemann–Liouville, and Caputo—as well as the classical <i>h</i>-difference operator. The analysis reveals that the established conditions are independent of the specific operator used. Furthermore, a comprehensive numerical study validates the theoretical findings and demonstrates that the fractional models can significantly extend the stability bound for the step size from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo><</mo><mn>6.67</mn></mrow></semantics></math></inline-formula> (classical case) to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo><</mo><mn>22.3</mn></mrow></semantics></math></inline-formula> (fractional case).