By Francesco Mainardi

This monograph presents a entire review of the author's paintings at the fields of fractional calculus and waves in linear viscoelastic media, inclusive of his pioneering contributions at the functions of distinctive services of the Mittag-Leffler and Wright varieties. it really is meant to function a common advent to the above-mentioned parts of mathematical modeling. the reasons within the ebook are distinct sufficient to trap the curiosity of the curious reader, and whole sufficient to supply the required historical past fabric had to delve additional into the topic and discover the learn literature given within the large common bibliography. This ebook might be of curiosity to utilized scientists and engineers.

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**Example text**

4. As a matter of fact, each of the two representations can assume one of the four canonic forms, which are obtained by cutting out one, both, or none of the two single elements which have appeared besides the branches. 1). 8) in the Laplace domain. March 1, 2010 19:27 36 World Scientific Book - 9in x 6in fmws Fractional Calculus and Waves in Linear Viscoelasticity Fig. 4 The mechanical representations of the compound Voigt model (top) and compound Maxwell model (bottom). 23) Gn (s τσ,n ) ⎪ ⎪ + + G s .

27a) sJ(s) = m + b1 s We note that the condition (0+ ) = 0 is surely satisﬁed for any reasonable stress history since Jg = 0, but is not valid for any reasonable strain history; in fact, if we consider the relaxation test for which (t) = Θ(t) we have (0+ ) = 1. This fact may be understood recalling that for the Voigt model we have Jg = 0 and Gg = ∞ (due to the delta contribution in the relaxation modulus). For the Maxwell model we get σ(s) + a1 sσ(s) − σ(0+ ) = b1 s (s) − (0+ ) , so, for any causal stress and strain histories it would be 1 a1 ⇐⇒ a1 σ(0+ ) = b1 (0+ ) .

49) u(t) = Hα,β (t) := t(1−β)(α−1) Eα,α+β(1−α) (−tα ) , t ≥ 0 . For plots of the Hilfer function Hα,β (t) we refer to [Seybold and Hilfer (2005)]. 4 Fractional integrals and derivatives with support in IR Choosing −∞ as the lower limit in the fractional integral, we have the so-called Liouville-Weyl fractional integral. 51) −∞ Dt f (t) := Dt ◦ −∞ It with m ∈ IN, namely ⎧ 1 dm t f (τ ) dτ ⎪ ⎪ , m − 1 < α < m, ⎪ ⎪ ⎨ Γ(m − α) dtm −∞ (t − τ )α+1−m α −∞ Dt f (t) := ⎪ ⎪ ⎪ dm ⎪ ⎩ f (t) , α = m. 18). 50) to converge is f (t) = O |t|−α− , > 0 , t → −∞ .