Methods for solving a semi-differential vibration equation

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 semi-derivative)