By T. Atanackovic, Steven Pilipovic, Bogoljub Stankovic, Dusan Zorica

This publication includes mathematical preliminaries within which uncomplicated definitions of fractional derivatives and areas are awarded. The critical a part of the ebook includes numerous functions in classical mechanics together with fields similar to: viscoelasticity, warmth conduction, wave propagation and variational Hamilton–type ideas. Mathematical rigor might be saw within the functions. The authors offer a few difficulties formulated within the classical environment and a few within the distributional environment. The strategies to those difficulties are awarded in analytical shape and those options are then analyzed numerically. Theorems at the lifestyles of options might be awarded for all examples mentioned. In utilizing a variety of constitutive equations the constraints following from the second one legislations of thermodynamics might be applied. ultimately, the actual implications of acquired suggestions can be mentioned in detail.

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0, b1 bN 0 ≤ α0 < α1 < . . < αN < β N +1 < . . < β M ≤ 1. 3. – Case III Suppose that N > M and αN −M +i = β i , i = 0, 1, . . e. 3]. ,M },i

37] For 0 < α < 1 α˜ L [0 Dα t f (t)] (s) = s f (s) − 1−α f 0 It (t) t=0 = sα f˜ (s) , Re s > s0 . 38] could be used for the (heuristic) deﬁnition of the fractional derivative. The Leibnitz rule for fractional derivatives does not hold in its usual form. It could be shown that for analytic functions, we have the following. – [SAM 93, p. 280] Suppose that f and g are analytic for t > 0 and α > 0. Then α a Dt ∞ α k (f (t) g (t)) = k=0 Dk g (t) α−k f a Dt (t) , t > a. 39], on the right-hand side we have integer-order derivatives of g and fractional-order derivatives of f .

70]. Both limits, if they exist, are equal almost everywhere. 2. 1. Fermat theorem for fractional derivative Let 0 < α < 1. As a motivation, following [SAM 93, p. 111], we start from α 0 Dt y (t) t y (t − τ ) dτ τα = d 1 Γ (1 − α) dt = y (0) 1 1 + Γ (1 − α) tα Γ (1 − α) = y (t) α + Γ (1 − α) tα Γ (1 − α) 0 t 0 t 0 y (1) (t − τ ) α t τ ξ −1−α dξ + y (t) − y (t − τ ) dτ , t > 0. 73] Basic Deﬁnitions and Properties of Fractional Integrals and Derivatives 39 Let 0 < α < 1. Similarly, for the Caputo derivative of an integrable function, we have C α 0 Dt y (t) = 0 Dα t y (t) − = y (0) Γ (1 − α) tα y (t) − y (0) α + Γ (1 − α) tα Γ (1 − α) t 0 y (t) − y (t − τ ) dτ , t > 0.