Identification of a kernel in an evolutionary integral equation occurring in subdiffusion

Janno, Jaan; Kasemets, Kairi

doi:10.1515/jiip-2016-0082

Cited by 19 publications

(29 citation statements)

References 22 publications

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“…However, alone condition (15) does not ensure an essential requirement for the kernel k of any fractional derivative, namely, the condition that this kernel should have a singularity at the origin, see [38][39][40]. In [19], an important class of the kernels that satisfy the condition (15) and possess an integrable singularity of a power law type at the origin was introduced.…”

Section: Gfd Of Arbitrary Order In the Riemann-liouville Sense And So...mentioning

confidence: 99%

“…). Let the functions κ and k satisfy the condition (15) with n ∈ N and the inclusions κ ∈ C −1 (0, +∞) and k ∈ C −1,0 (0, +∞) hold true, where…”

Section: Definition 1 ([19]mentioning

confidence: 99%

“…Further results regarding the ordinary and partial time-fractional differential equations with the GFD (11) were derived in [9][10][11][12][13][14]. For results regarding inverse problems for the fractional differential equations involving the GFDs and their applications, we refer to [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

Luchko

2022

Mathematics

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In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals.

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Section: Gfd Of Arbitrary Order In the Riemann-liouville Sense And So...mentioning

confidence: 99%

“…). Let the functions κ and k satisfy the condition (15) with n ∈ N and the inclusions κ ∈ C −1 (0, +∞) and k ∈ C −1,0 (0, +∞) hold true, where…”

Section: Definition 1 ([19]mentioning

confidence: 99%

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Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

Luchko

2022

Mathematics

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“…We are solving problems with a generalized fractional derivative. This concept has been used in [2][3][4][5]. We utilize D {k},n a as a unified notation that stands for the generalized fractional derivatives in…”

Section: Generalized Fractional Derivativesmentioning

confidence: 99%

“…In addition to classical fractional derivatives, several generalizations have been introduced to better match the models to the reality in different situations. In this paper, we work with generalized fractional derivatives of Riemann-Liouville and Caputo type where the power-type kernel (fractional derivative case) is replaced by an arbitrary function k. Such a generalization was previously used in [2][3][4][5] and covers many specific cases that are important in applications (see Section 2.1).…”

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

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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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General Fractional Calculus Operators with the Sonin kernels and Some of Their Applications

Luchko

2024

IFAC-PapersOnLine

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Identification of a kernel in an evolutionary integral equation occurring in subdiffusion

Abstract: An inverse problem to determine a kernel in an evolutionary integral equation occurring in modeling of subdiffusion is considered. The existence, uniqueness and stability of a solution of the inverse problem are proved in an abstract setting. The results are global in time.

Cited by 19 publications

References 22 publications

Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

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

General Fractional Calculus Operators with the Sonin kernels and Some of Their Applications

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