An inverse fractional source problem in a space of periodic spatial distributions

Lopushansky, Andrzej; Lopushanska, Halyna; Myaus, Olga

doi:10.7153/fdc-06-17

Cited by 5 publications

(5 citation statements)

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“…A [a] or A ̺,a , respectively) is injective and M ∈ CM † , where CM † = {M ∈ CM : the Laplace transform of M cannot be meromorphically extended to the whole complex plane}. (10) Moreover, suppose that f | (t 0 ,T )×Ω and u| (t 0 ,T )×Ω are given, where t 0 is some number in (0, T ). Then history of f and u is uniquely recovered in (0, t 0 ) × Ω [8].…”

Section: Physical Background Formulation Of Problems and Overview Of ...mentioning

confidence: 99%

“…Let us give an overview of other studies in the field of inverse source problems for fractional diffusion equations. Several papers [9,10,11,12] deal with reconstruction of time-dependent components of the sources making use of different measurements over time. The paper [13] is concerned with the reconstruction of the source function that depends on time and part of spatial variables from boundary measurements over the time.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation

Janno¹,

Kasemets²

2017

Inverse Problems &Amp; Imaging

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Section: Physical Background Formulation Of Problems and Overview Of ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation

Janno¹,

Kasemets²

2017

Inverse Problems &Amp; Imaging

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“…Çàóâàaeèìî, ùî ç âèêîðèñòàííÿì iíòåãðàëüíî¨çà ÷àñîì óìîâè ïåðåâèçíà÷åííÿ ó ðiçíèõ ôóíêöiéíèõ ïðîñòîðàõ îäåðaeàíî îäíîçíà÷íó ðîçâ'ÿçíiñòü îáåðíåíî¨çàäà÷i äëÿ ðiâíÿííÿ äðîáîâî¨äèôóçi¨ç íåâiäîìîþ, çàëåaeíîþ âiä ÷àñó, êîìïîíåíòîþ ïðàâî¨÷àñòèíè ðiâíÿííÿ (ó [14,15]), ç íåâiäîìèì ìîëîäøèì êîåôiöi¹íòîì (ó [10]), íåâiäîìèìè ïî÷àòêîâèìè äàíèìè ðîçâ'ÿçêó (ó [16]). 1 Îçíà÷åííÿ é äîïîìiaeíi ôàêòè Âèêîðèñòîâó¹ìî ïîçíà÷åííÿ:…”

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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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“…Inverse problems in fractional diffusion models have been intensively studied over the last decade. Results have been obtained for problems to recover terms related to medium properties [3,10,11,17,20,26], source terms [6,9,14,15,22,27,28] and initial states [14,16,24] (see also [13] for an overview of methods used in analysis of these problems).…”

Section: Introductionmentioning

confidence: 99%

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

Janno

Kinash

2018

Inverse Problems

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An inverse fractional source problem in a space of periodic spatial distributions

Cited by 5 publications

References 0 publications

Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation

Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation

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

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

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