Methods for solving a semi-differential vibration equation

Abstract

Fractional-order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980's, and they grow more and more popular nowadays. However, their efficient numerical calculation is nontrivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty. In the followings, some of the proposed methods will be examined for a derivative of order 1 2 (that is sometimes called a semiderivative).

Efficient solution of a vibration equation involving fractional derivatives

“…However, Schmidt and Gaul [15] have shown that in some cases, this method loses the advantages of fractional calculus over integer-order calculus. Later, this has also been checked by the present author [16] for the field of interest of this paper.…”