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∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,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.
Highlights
The classical result for the inverse Laplace transform of the function F (s) = s1q is [1] 1 q −1 −1 1 L = t with q > 0 (1) sq Γ(q)In Equation (1) the condition q > 0 is needed because when q = 0, the ratio Γ(10) = 0 and the right-hand side of Equation (1) vanishes, except when t = 0, which leads to a singularity
This result, in association with the convolution dtq q theorem, makes possible the calculation of the inverse Laplace transform of sαs∓λ where α < q ∈ R+, which is the fractional derivative of order q of the Rabotnov function εα−1 (±λ, t) = tα−1 Eα, α (±λtα )
With the introduction of the fractional derivative of the Dirac delta function expressed by Equations (17) or (21), the definition of the memory function given by Equation (8) offers a new and useful result regarding the Fourier transform of the function F (ω ) = (i ω )q with q ∈ R+
Summary
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 ∈ 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 [4,5,6,7,8,9,10,11]. Where M(t − ξ ) is the memory function of the material [16,17,18] defined as the resulting stress at time t due to an impulsive strain input at time ξ (ξ < t), and it is the inverse
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