Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

Feng, Pengbin; Karimov, Erkinjon

doi:10.1515/jiip-2014-0022

Cited by 24 publications

(9 citation statements)

References 9 publications

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“…Насколько нам известно, подобные обратные задачи для уравнений смешанного типа изучаются впервые. Однако, следует отметить, что начально-краевые задачи и обратные задачи по определению правой части в уравнениях смешанного типа с участием дробных производных были исследованы в работах [9], [27], [28]. В качестве эллиптической части в этих работах рассматривались обыкновенные дифференциальные выражения и изучалось существование слабого решения.…”

Section: Introductionunclassified

Inverse problem for determining the order of fractional derivative in mixed-type equations

Ashurov¹,

Zunnunov²

2021

Preprint

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In this paper the inverse problem of determining the fractional orders in mixed-type equations is considered. In one part of the domain the considered equation is the subdiffusion equation with a fractional derivative in the sense of Gerasimov-Caputo of the order α ∈ (0, 1), and in the other part -a wave equation with a fractional derivative of the order β ∈ (1, 2) . The elliptic part of the equation is a second-order operator, considered in a N -dimensional domain Ω. Assuming the parameters α and β to be unknown, additional conditions are found that provide an unambiguous determination of the required parameters.

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Section: Introductionunclassified

Inverse problem for determining the order of fractional derivative in mixed-type equations

Ashurov¹,

Zunnunov²

2021

Preprint

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“…To our knowledge, in the field of inverse problems for time fractional diffusion-wave equations, very few works have been presented. In [9] and [11] authors considered some inverse source problems for time-fractional mixed parabolic-hyperbolic equations. Also in [15] authors investigated an inverse problem of determining diffusion coefficient in the diffusion-wave equation.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method Based on the Jacobi Polynomials to Reconstruct an Unknown Source Term in a Time Fractional Diffusion-wave Equation

Nemati¹,

Babaei²

2019

Taiwanese J. Math.

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In this paper, we consider an inverse problem of identifying an unknown time dependent source function in a time-fractional diffusion-wave equation. First, some basic properties of the shifted Jacobi polynomials (SJPs) are presented. Then, the analytical solution of the direct problem is given and used to obtain an approximation of the unknown source function in a series of SJPs. Due to ill-posedness of this inverse problem, the Tikhonov regularization method with Morozov's discrepancy principle criterion is applied to find a stable solution. After that, an error bound is obtained for the approximation of the unknown source function. Finally, some numerical examples are provided to show effectiveness and robustness of the proposed algorithm.

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“…Feng and Karimov [5] used eigenfunctions to analyze an inverse source problem for a fractional mixed parabolic hyperbolic equation. They formulated the problem into an optimization problem and then used semismooth Newton algorithm to solve it.…”

Section: Introductionmentioning

confidence: 99%

An inverse source problem for a two-parameter anomalous diffusion with local time datum

Furati¹,

Iyiola²,

Mustapha³

2017

Computers & Mathematics with Applications

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We determine the space-dependent source term for a two-parameter fractional diffusion problem subject to nonlocal non-self-adjoint boundary conditions and two local time-distinct datum. A bi-orthogonal pair of bases is used to construct a series representation of the solution and the source term. The two local time conditions spare us from measuring the fractional integral initial conditions commonly associated with fractional derivatives. On the other hand, they lead to delicate 2 × 2 linear systems for the Fourier coefficients of the source term and of the fractional integral of the solution at t = 0. The asymptotic behavior and estimates of the generalized Mittag-Leffler function are used to establish the solvability of these linear systems, and to obtain sufficient conditions for the existence of our construction. Analytical and numerical examples are provided.

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Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

Cited by 24 publications

References 9 publications

Inverse problem for determining the order of fractional derivative in mixed-type equations

Inverse problem for determining the order of fractional derivative in mixed-type equations

A Numerical Method Based on the Jacobi Polynomials to Reconstruct an Unknown Source Term in a Time Fractional Diffusion-wave Equation

An inverse source problem for a two-parameter anomalous diffusion with local time datum

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