The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions Nicos Makris1 Abstract Motivated from studies on anomalous diffusion, we show that the memory function M(t) of q + complex materials, that their creep compliance follows a power law, J(t) ∼ t with q 2 R , dqδ(t−0) + is the fractional derivative of the Dirac delta function, dtq with q 2 R . This leads q + to the finding that the inverse Laplace transform of s for any q 2 R is the fractional dqδ(t−0) derivative of the Dirac delta function, dtq . This result, in association with the con- sq volution theorem, makes possible the calculation of the inverse Laplace transform of sα∓λ + where α < q 2 R which is the fractional derivative of order q of the Rabotnov function α−1 α + "α−1(±λ, t) = t Eα, α(±λt ). The fractional derivative of order q 2 R of the Rabotnov function, "α−1(±λ, t) produces singularities which are extracted with a finite number of frac- tional 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. arXiv:2006.04966v1 [math-ph] 8 Jun 2020 Keywords: Generalized functions, Laplace transform, anomalous diffusion, fractional calculus, Mittag–Leffler function 1Department of Civil and Environmental Engineering, Southern Methodist University, Dallas, Texas, 75276
[email protected] Office of Theoretical and Applied Mechanics, Academy of Athens, 10679, Greece 1 1 Introduction 1 The classical result for the inverse Laplace transform of the function F(s) = sq is (Erdélyi 1954) 1 1 L−1 = tq−1 with q > 0 (1) sq Γ(q) 1 In Eq.