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Article citations


Saxena, R.K., Mathai, A.M. and Haubold, H.J. (2002) On Fractional Kinetic Equations. Astrophysics and Space Science, 282, 281-287.

has been cited by the following article:

  • TITLE: The Equilibrium of Fractional Derivative and Second Derivative: The Mechanics of a Power-Law Visco-Elastic Solid

    AUTHORS: Franz Konstantin Fuss

    KEYWORDS: Fractional Derivative, Non-Linear Visco-Elasticity, Power Law, Mittag-Leffler Function, Underdamping, Overdamping, Innovative Design of Material Damping

    JOURNAL NAME: Applied Mathematics, Vol.7 No.16, October 14, 2016

    ABSTRACT: This paper investigates the equilibrium of fractional derivative and 2nd derivative, which occurs if the original function is damped (damping of a power-law viscoelastic solid with viscosities η of 0 ≤ η ≤ 1), where the fractional derivative corresponds to a force applied to the solid (e.g. an impact force), and the second derivative corresponds to acceleration of the solid’s centre of mass, and therefore to the inertial force. Consequently, the equilibrium satisfies the principle of the force equilibrium. Further-more, the paper provides a new definition of under- and overdamping that is not exclusively disjunctive, i.e. not either under- or over-damped as in a linear Voigt model, but rather exhibits damping phases co-existing consecutively as time progresses, separated not by critical damping, but rather by a transition phase. The three damping phases of a power-law viscoelastic solid—underdamping, transition and overdamping—are characterized by: underdamping—centre of mass oscillation about zero line; transition—centre of mass reciprocation without crossing the zero line; overdamping—power decay. The innovation of this new definition is critical for designing non-linear visco-elastic power-law dampers and fine-tuning the ratio of under- and overdamping, considering that three phases—underdamping, transition, and overdamping—co-exist consecutively if 0