The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions - Nicos Makris (Fractal and Fractional, 2021)
Motivated from studies on anomalous relaxation and diffusion, we show that the memory function <semantics>M(t)</semantics> of complex materials, that their creep compliance follows a power law, <semantics>J(t)∼tq</semantics> with <semantics>q∈R+</semantics>, is proportional to the fractional derivative of the Dirac delta function, <semantics>dqδ(t−0)dtq</semantics> with <semantics>q∈R+</semantics>. This leads to the finding that the inverse Laplace transform of <semantics>sq</semantics> for any <semantics>q∈R+</semantics> is the fractional derivative of the Dirac delta function, <semantics>dqδ(t−0)dtq</semantics>. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of <semantics>sqsα∓λ</semantics> where <semantics>α<q∈R+</semantics>, which is the fractional derivative of order q of the Rabotnov function <semantics>ε</semantics><semantics>α−1(±λ,<mspace width="0.166667em"></mspace>t)=tα−1Eα,<mspace width="0.166667em"></mspace>α(±λtα)</semantics>. The fractional derivative of order <semantics>q∈R+</semantics> of the Rabotnov function, <semantics>ε</semantics><semantics>α−1(±λ,<mspace width="0.166667em"></mspace>t)</semantics> produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.