Mathematical methods in engineering by K Taş; J A Tenreiro Machado; D Baleanu

By K Taş; J A Tenreiro Machado; D Baleanu

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R. Astr. : A new dissipation model based on memory mechanism. : Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento (Ser. : Abel Integral Equations: Analysis and Applications. : Fractional Integrals and Derivatives: Theory and Applications. : An Introduction to the Fractional Calculus and Fractional Differential Equations. : Theory and Applications of Fractional Differential Equations. : Physics of Fractal Operators.

T ≥ 0 . 3) of normal diffusion. 21) noting that σ 2 (t) = − ∂2 u(κ = 0, t) . 4 Graphical representation of the fundamental solutions Let us consider the time-fractional diffusion equation of a single order β = β0 , whose fundamental solution has the peculiar property to be self-similar according to the similarity variable x/tβ0 /2 . For this reason it is sufficient to consider the fundamental solution for t = 1, namely the reduced Green function U (x), given by Eq. 19) in terms of the special function Mβ0 /2 (x) of the Wright type.

217-224, we find, in addition to the functions of the Mittag-Leffler and Wright type, the function ∞ ta+τ dτ , a > −1 . 15) ν(t, a) = Γ (a + τ + 1) 0 Such special function is relevant for our purposes because of the Laplace transform pair, see [EMOT55], Eq. 16) Definition: We call a (measurable) positive function a(y), defined in a right neighbourhood of zero, slowly varying at zero if a(cy)/a(y) → 1 with y → 0 for every c > 0. We call a (measurable) positive function b(y), defined in a neighbourhood of infinity, slowly varying at infinity if b(cy)/a(y) → 1 with y → ∞ for every c > 0.

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