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, Nataliia; Janno, Jaan

doi:10.3390/math7121138

Cited by 50 publications

(43 citation statements)

References 32 publications

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“…where λ 2 1 = 4π 2 is the smallest positive eigenvalue and the function S(t; λ) is defined in (26). In this way, (60) is established with C 1 = q 0 (1 − S(T − T 1 ; λ 2 1 )) > 0.…”

Section: Uniqueness Of Solution and Stability Estimates In Sobolev Spacesmentioning

confidence: 99%

“…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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“…where λ 2 1 = 4π 2 is the smallest positive eigenvalue and the function S(t; λ) is defined in (26). In this way, (60) is established with C 1 = q 0 (1 − S(T − T 1 ; λ 2 1 )) > 0.…”

Section: Uniqueness Of Solution and Stability Estimates In Sobolev Spacesmentioning

confidence: 99%

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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“…Óìîâè êëàñè÷íî¨ðîçâ'ÿçíîñòi çàäà÷i Êîøi i êðàéîâèõ çàäà÷ äëÿ ðiâíÿíü iç äðîáîâèìè ïîõiäíèìè îäåðaeàíi â [17,21,26,13,4,5,23,19,18,24,25] òà iíøèõ ïðàöÿõ. Íàéáiëüøå ðîáiò ïî îáåðíåíèõ çàäà÷àõ äëÿ òàêèõ ðiâíÿíü, ÿê i äëÿ ðiâíÿíü iç ÷àñòèííèìè ïîõiäíèìè öiëèõ ïîðÿäêiâ (äèâ., íàïðèêëàä, [24,1,11,12,29,27,28,30,22] i áiáëiîãðàôiþ) ïðèñâÿ-÷åíî çàäà÷àì iç íåâiäîìèìè ïðàâèìè ÷àñòèíàìè ó ðiâíÿííÿõ.…”

Section: âñòóïunclassified

Regular Solution of the Inverse Problem With Integral Condition for a Time-Fractional Equation

Lopushanska¹,

Lopushansky²

2020

BMJ

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Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob\-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Bbb R^n\] where $u$ is the unknown solution of the Cauchy problem, $\eta_1$ and $\Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.

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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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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

Cited by 50 publications

References 32 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

Regular Solution of the Inverse Problem With Integral Condition for a Time-Fractional Equation

General Fractional Dynamics

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