Fractional models of falling object with linear and quadratic frictional forces considering Caputo derivative

Abstract

Distinct fractional models of falling object with linear and quadratic air resistive forces are explored under Caputo fractional derivative. Analytical solutions to each model are extracted from which the motion of fall can be vividly observed. The contribution of nonlinearity to the fractional model is particularly conceived from the solutions. Unlike the traditional results from an integer derivative model, a rich phenomenon is present under the proposed fractional models, which are shown to be truly converging back to the traditional one. The short time perturbation and large time asymptotic formulae are also derived. Instead of attaining a terminal speed, the solutions under some fractional models suggest that objects travel either at an increased speed overwhelming the gravitational force or decelerate to stop asymptotically. In the case of quadratic air resistance, a straightforward power series solution around the initial time having a short radius of convergence under Caputo fractional derivative and a powerful asymptotic series solution possessing an infinite convergence radius under the infinite base fractional differentiation are also presented. It can be concluded from analysis of the ideal falling object motion that even though novel physical implications are observed from the modified fractional models, the fractional model as well as the definition of fractional derivative literally affect the physical motion in-depth, which may require proper justification.