fractal and fractional Article The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions Nicos Makris 1,2 1 Department of Civil and Environmental Engineering, Southern Methodist University, Dallas, TX 75276, USA; [email protected] 2 Office of Theoretical and Applied Mechanics, Academy of Athens, 10679 Athina, Greece Abstract: Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t) ∼ tq with q 2 + d d(t−0) 2 + q R , is proportional to the fractional derivative of the Dirac delta function, dtq with q R . + This leads to the finding that the inverse Laplace transform of sq for any q 2 R is the fractional dqd(t−0) derivative of the Dirac delta function, dtq . This result, in association with the convolution sq + theorem, makes possible the calculation of the inverse Laplace transform of sa∓l where a < q 2 R , a−1 a which is the fractional derivative of order q of the Rabotnov function #a−1(±l, t) = t Ea, a(±lt ). + The fractional derivative of order q 2 R of the Rabotnov function, #a−1(±l, t) 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. Keywords: generalized functions; laplace transform; anomalous relaxation; diffusion; fractional Citation: Makris, N. The Fractional calculus; mittag–leffler function Derivative of the Dirac Delta Function and Additional Results on MSC: 26A33; 30G99; 44A10; 33E12; 60K50 the Inverse Laplace Transform of Irrational Functions. Fractal Fract. 2021, 5, 18. https://doi.org/10.3390/ fractalfract5010018 1. Introduction 1 Academic Editor: Carlo Cattani The classical result for the inverse Laplace transform of the function F(s) = sq is [1] Received: 2 December 2020 −1 1 1 q−1 L q = t with q > 0 (1) Accepted: 22 February 2021 s G(q) Published: 25 February 2021 q > q = 1 = In Equation (1) the condition 0 is needed because when 0, the ratio G(0) 0 Publisher’s Note: MDPI stays neutral and the right-hand side of Equation (1) vanishes, except when t = 0, which leads to with regard to jurisdictional claims in a singularity. Nevertheless, within the context of generalized functions, when q = 0, published maps and institutional affil- the right-hand side of Equation (1) becomes the Dirac delta function [2] according to the iations. th Gel’fand and Shilov [3] definition of the n (n 2 N0) derivative of the Dirac delta function dnd(t − 0) 1 1 = = F−n(t) with n 2 f0, 1, 2 ...g (2) dtn G(−n) tn+1 Copyright: © 2021 by the author. 1 Licensee MDPI, Basel, Switzerland. with a proper interpretation of the quotient as a limit at t = 0. Thus, according to the tn+1 This article is an open access article Gel’fand and Shilov [3] definition expressed by Equation (2), Equation (1) can be extended distributed under the terms and for values of q 2 f0, − 1, − 2, −3 ...g, and in this way one can establish the following n conditions of the Creative Commons expression for the inverse Laplace transform of s with n 2 N0: Attribution (CC BY) license (https:// n creativecommons.org/licenses/by/ 1 1 d d(t − 0) L−1fsng = = n 2 f0, 1, 2 ...g (3) 4.0/). G(−n) tn+1 dtn Fractal Fract. 2021, 5, 18. https://doi.org/10.3390/fractalfract5010018 https://www.mdpi.com/journal/fractalfract Fractal Fract. 2021, 5, 18 2 of 14 For instance when n = 1, Equation (3) yields 1 1 dd(t − 0) L−1fsg = = (4) G(−1) t2 dt dd(t−0) which is the correct result, since the Laplace transform of dt is Z ¥ −st dd(t − 0) dd(t − 0) − d(e ) L = e stdt = − = −(−s) = s (5) − dt 0 dt dt t=0 Equation (5) is derived by making use of the property of the Dirac delta function and its higher-order derivatives Z ¥ dnd(t − 0) dn f (0) ( ) = (− )n 2 n f t dt 1 n with n f0, 1, 2 ...g (6) 0− dt dt In Equations (5) and (6), the lower limit of integration, 0− is a shorthand notation for Z ¥ dnd(t−0) lim , and it emphasizes that the entire singular function (n 2 ) is captured + tn N0 #!0 −# d by the integral operator. In this paper we first show that Equation (3) can be further extended for the case where the Laplace variable is raised to any positive real power; sq + with q 2 R . This generalization, in association with the convolution theorem, allows for the derivation of some new results on the inverse Laplace transform of irrational functions that appear in problems with fractional relaxation and fractional diffusion [4–11]. This work complements recent progress on the numerical and approximate Laplace-transform solutions of fractional diffusion equations [12–15]. Most materials are viscoelastic; they both dissipate and store energy in a way that depends on the frequency of loading. Their resistance to an imposed time-dependent shear t(w) (t) G( ) = deformation, g , is parametrized by the complex dynamic modulus w g(w) where Z ¥ Z ¥ t(w) = t(t)e− i wtdt and g(w) = g(t)e− i wtdt are the Fourier transforms of the −¥ −¥ output stress, t(t), and the input strain, g(t), histories. The output stress history, t(t), can be computed in the time domain with the convolution integral Z t t(t) = M(t − x)g(x)dx (7) 0− where M(t − x) is the memory function of the material [16–18] defined as the resulting stress at time t due to an impulsive strain input at time x(x < t), and it is the inverse Fourier transform of the complex dynamic modulus 1 Z ¥ M(t) = G(w)ei wtdw (8) 2p −¥ 2. The Fractional Derivative of the Dirac Delta Function Early studies on the behavior of viscoelastic materials, that their time-response func- tions follow power laws, have been presented by Nutting [4], who noticed that the stress response of several fluid-like materials to a step strain decays following a power law, t(t) ∼ t−q with 0 ≤ q ≤ 1. Following Nutting’s observation and the early work of Gemant [5,6] on fractional differentials, Scott Blair [19,20] pioneered the introduction of fractional calculus in viscoelasticity. With analogy to the Hookean spring, in which the stress is proportional to the zero-th derivative of the strain and the Newtonian dashpot, in which the stress is proportional to the first derivative of the strain, Scott Blair and his co- Fractal Fract. 2021, 5, 18 3 of 14 workers [19–21] proposed the springpot element—that is a mechanical element in-between a spring and a dashpot with constitutive law dqg(t) t(t) = m (9) q dtq where q is a positive real number, 0 ≤ q ≤ 1, mq is a phenomenological material parameter q ( ) [ ][ ]−1[ ]q−2 · q d g t with units M L T (say Pa sec ), and dtq is the fractional derivative of order q of the strain history, g(t). A definition of the fractional derivative of order q is given through the convolution integral Z t q 1 q−1 c I g(t) = (t − x) g(x)dx (10) G(q) c where G(q) is the Gamma function. When the lower limit, c = 0, the integral given by Equa- q tion (10) is often referred to as the Riemann–Liouville fractional integral 0 I g(t) [22–25]. The integral in Equation (10) converges only for q > 0, or in the case where q is a complex number, the integral converges for R(q) > 0. Nevertheless, by a proper analytic continua- tion across the line R(q) = 0, and provided that the function g(t) is n times differentiable, it can be shown that the integral given by Equation (10) exists for n − R(q) > 0 [26]. In this + case the fractional derivative of order q 2 R exists and is defined as dqg(t) 1 Z t g(x) = Dqg(t) = F (t) ∗ g(t) = dx, q 2 + (11) q 0 −q − q+ R dt G(−q) 0 (t − x) 1 where + is the set of positive real numbers, and F (t) = 1 1 is a generaliza- R −q G(−q) tq+1 + tion of the Gel’fand and Shilov kernel given by Equation (2) for q 2 R . Gorenflo and Mainardi [27] and subsequently Mainardi [28] concluded that within the context of gen- eralized functions, Equation (11) is indeed a formal definition of the fractional derivative + of order q 2 R of a sufficiently differentiable function by making use of the property of the Dirac delta function and its higher-order derivatives given by Equation (6) in as- sociation with the 1964 Gel’fand and Shilov [3] definition of the Dirac delta function and its higher-order derivatives given by Equation (2). Accordingly, the nth-order derivative, n 2 f0, 1, 2, ...g, of a sufficiently differentiable function g(t) is the convolution of g(t) with F−n(t) defined by Equation (2) dng(t) Z t dnd(t − x) 1 Z t g(x) = F (t) ∗ g(t) = g(x) dx = dx, t > 0 (12) n −n − n − n+ dt 0 dt G(−n) 0 (t − x) 1 + By replacing n 2 f0, 1, 2, ...g with q 2 R in Equation (12), Gorenflo and Mainardi [27] and Mainardi [28] explain that Equation (11) is a formal definition of the fractional deriva- + tive of order q 2 R and should be understood as the convolution of the function g(t) with the kernel F (t) = 1 1 as indicated in Equation (11) [22–24,27,28].