Semi-Local Approximation of Fractional Derivatives in FDTD Modeling of Cole–Cole Media: A Questionable Approach

Recently, a novel finite-difference time-domain (FDTD) scheme has been proposed for modeling wave propagation in Cole-Cole dispersive media (Abdullah et al., IEEE Antennas Wireless Propag. Lett., vol. 11, pp. 281-284, 2012). The scheme is based on the approximation of the fractional derivative, which appears in the polarization relation, by means of a semi-local method. This approach ends up to an approximate time-domain polarization relation given by a first-order differential equation with nonconstant coefficients. As a result, this scheme is very attractive because it has very limited storage demands compared to all the previously reported schemes. However, in this letter, it is shown that the proposed scheme is actually untrustworthy. This criticism is based on both analytical and numerical evidences and illustrates that the semi-local approximation of fractional derivatives is very poor.