Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
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Cites background from "Initial value/boundary value proble..."
..., −1 < q ≤ 0, is very common in inverse problems and optimal control [14,32]; see also [5,13] for the parabolic counterpart....
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...However, it differs considerably from the latter in the sense that, due to the presence of the nonlocal fractional derivative term, it has limited smoothing property in space and slow asymptotic decay in time [32], which in turn also impacts related numerical analysis [12] and inverse problems [14,32]....
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"Initial value/boundary value proble..." refers background or methods in this paper
...22 in [37]: d dt (t α−1 Eα,α(−λtα)) = tα−2 Eα,α−1(−λntα)....
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...22 of [37]) and λ −2 n C ′ 9n with γ1 > 1 by γ > 4 + 1 and λn C ′ 8n 2 d , we have ∥∥∂t u(·, t)∥∥2D((−L)−γ ) = ∞ ∑ n=1 1 λ 2γ n ∣∣∣∣ t ∫...
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...33–34) in [37], for any p ∈ N, we have u(x, t) = − ∞ ∑ k=1 mk ∑ j=1 p ∑ =1 (−1) (1 − α )μ kt (a,φkj)φkj(x) + ∞ ∑ k=1 mk ∑ j=1 O ( 1 μ p+1 k t α(p+1) ) (a,φkj)φkj(x) as t → ∞....
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...140 in [37], by means of the Laplace transform, we can see that...
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...Then there exists a constant C1 = C1(α,β,μ) > 0 such that∣∣Eα,β(z)∣∣ C1 1 + |z| , μ ∣∣arg(z)∣∣ π. (3.1) The proof can be found on p. 35 in Podlubny [37]....
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"Initial value/boundary value proble..." refers background in this paper
..., [14,18]), we have ∂ t u(x, t) = ∞ ∑ n=1 {−λn(a,φn)Eα,1(−λntα) − λn(b,φn)t Eα,2(−λntα)} (3....
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...19) are given in [14,18,37] for example, and we can formally obtain the expansions....
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..., Chapter 3 in [18], [37]), we obtain that un(t) = 0, n = 1,2,3, ....
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...As source books related with fractional derivatives, see Samko, Kilbas and Marichev [44] which is an encyclopedic treatment of the fractional calculus and also Gorenflo and Mainardi [14], Kilbas, Srivastava and Trujillo [18], Mainardi [29], Miller and Ross [31], Oldham and Spanier [35], Podlubny [37]....
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...[18] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006....
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