Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements

Janno, Jaan; Kinash, Nataliia

doi:10.1088/1361-6420/aaa0f0

Cited by 58 publications

(42 citation statements)

References 26 publications

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“…The kernels (k1)-(k7) satisfy (5), (6). Moreover, it is evident that the kernels (k1), (k2), (k4), (k5), (k7) satisfy (7), because Laplace transforms of these functions have branch points.…”

Section: Uniqueness Resultsmentioning

confidence: 92%

“…Quite often in the inverse source problem, the goal is to determine a source that is either a spaceor time-dependent function. The space-dependent source term is usually reconstructed based on the final time overdetermination condition [6][7][8][9][10][11]. The time-dependent source term can be recovered from additional boundary measurements [7] or from integral conditions [12,13].…”

Section: Introductionmentioning

confidence: 99%

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An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

Kinash

Janno

2019

Mathematics

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In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.

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Section: Uniqueness Resultsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

Kinash

Janno

2019

Mathematics

Self Cite

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“…Combining the equation (19) and the estimates (21) and (22), there exists a positive constant C 1 (ν 0 , η 0 , , β, T ) such that…”

mentioning

confidence: 92%

“…If we try to establish the same estimate as in (25) for α, α ∈ [α 0 , α 1 ] ⊂ (1, 2), and apply it instead of (24), then we require h ∈ H 2γ1 (Ω) to estimate the output error on L 2 (Ω). In [19] (see Lemma 5), the authors gave the following estimate by using solutions of two ordinary differential equations…”

mentioning

confidence: 99%

Continuity with respect to fractional order of the time fractional diffusion-wave equation

Tuan¹,

O’Regan²,

Ngoc³

2020

Evolution Equations &Amp; Control Theory

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This paper studies a time-fractional diffusion-wave equation with a linear source function. First, some stability results on parameters of the Mittag-Leffler functions are established. Then, we focus on studying the continuity of the solution of both the initial problem and the inverse initial value problems corresponding to the fractional-order in our main results. One of the difficulties encounteblack comes from estimating all constants independently of the fractional orders. Finally, we present some numerical results to confirm the effectiveness of our methods. 2020 Mathematics Subject Classification. 35R11, 35B65, 26A33. Key words and phrases. Initial value problem, inverse initial value problem, final value problem, time fractional wave equation, continuity/stability on fractional orders.We note that, for any > 0, there always exists C > 0 such that 1 − e −y ≤ C y for all y > 0. Therefore, we obtainCase 2. z ≥ 1. We see thatUsing the inequality 1 − e −y ≤ C y for any > 0 and y > 0, we obtainCombining the above cases completes the proof.CONTINUITY WITH RESPECT TO FRACTIONAL ORDER 777 Lemma 3.3. Let 1 < ν 0 < α < α < η 0 < 2 and > 0. Then there exists a positive constant D 2 (ν 0 , η 0 , , β, T ) such thatfor any 0 ≤ β ≤ 1 and 0 < t ≤ T .Proof. First, it is easy to see that

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“…Although there are many works on direct problem, but the results on inverse problem for fractional diffusion are scarce. We can list some papers of M. Yamamoto and his group see [36,48,70,54,51,49], of B. Kaltenbacher et al [5,6] , of W. Rundell et al [44,45], of J. Janno see [33,34], etc.…”

Section: Introductionmentioning

confidence: 99%

On existence and regularity of a terminal value problem for the time fractional diffusion equation

et al. 2020

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In this paper we consider a final value problem for a diffusion equation with timespace fractional differentiation on a bounded domain D of R k , k ≥ 1, which includes the fractional power L β , 0 < β ≤ 1, of a symmetric uniformly elliptic operator L defined on L 2 (D). A representation of solutions is given by using the Laplace transform and the spectrum of L β . We establish some existence and regularity results for our problem in both the linear and nonlinear case.where ϕ is a given function. Here J is the interval (0, T ), The notation c D α t for 0 < α < 1 represents the left Caputo fractional derivative of order α which is defined by c D α t v(t) := I 1−α t (N.H. Tuan) Applied

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Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements

Cited by 58 publications

References 26 publications

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

Continuity with respect to fractional order of the time fractional diffusion-wave equation

On existence and regularity of a terminal value problem for the time fractional diffusion equation

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