Inverse Problems for a Generalized Subdiffusion Equation With Final Overdetermination

Janno, Jaan; Kinash, Nataliia

doi:10.3846/mma.2019.016

Cited by 40 publications

(10 citation statements)

References 29 publications

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“…For λ > 0 this follows from (30). If λ = 0 then (31) reduces to (1 * l)(t) > 0, see (27). Since l(t) ∈ CMF then (1 * l)(t) ∈ BF .…”

Section: Generalized Relaxation Equationmentioning

confidence: 97%

“…Identification of a space-dependent source factor h(x) in a source function of the form F(x, t) = h(x)q(x, t) from final overdetermination are studied in [17,19,[23][24][25], where different assumptions on the known source factor q(x, t) are discussed. Concerning the generalized subdiffusion equation, various types of inverse problems for such equations are studied in [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

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An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

Bazhlekova

2021

Fractal Fract

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An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

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“…For λ > 0 this follows from (30). If λ = 0 then (31) reduces to (1 * l)(t) > 0, see (27). Since l(t) ∈ CMF then (1 * l)(t) ∈ BF .…”

Section: Generalized Relaxation Equationmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

Bazhlekova

2021

Fractal Fract

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“…Integral equations of the first kind with kernels from the Sonin set and the GFI and GFD of the Liouville and Marchaud type are described in [34,35]. The partial differential equations containing GFD and GFI are considered in [36,37]. Some applications of GFC are described in recent published works (see [31][32][33]38,39] and references therein).…”

Section: Introductionmentioning

confidence: 99%

General Fractional Dynamics

Tarasov

2021

Mathematics

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General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

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“…Operational calculus for equations with general fractional derivatives was proposed in [60]. The general fractional calculus was also developed in works [61][62][63][64][65][66][67][68][69][70][71][72] devoted to mathematical aspects and some applications in classical physics.…”

Section: Introductionmentioning

confidence: 99%

General Non-Markovian Quantum Dynamics

Tarasov

2021

Entropy

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A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.

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Inverse Problems for a Generalized Subdiffusion Equation With Final Overdetermination

Cited by 40 publications

References 29 publications

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

General Fractional Dynamics

General Non-Markovian Quantum Dynamics

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