Inverse source problems for a space–time fractional differential equation

Ali, Muhammad; Aziz, Sara; Malik, Salman A.

doi:10.1080/17415977.2019.1597079

Cited by 26 publications

(14 citation statements)

References 43 publications

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“…In [13,14] (see also [2,15]), using the theorem of Matsaev and Palant, it was established that the system of eigenfunctions of the operator A ρ is complete in L 2 (0, 1). And this fact used by M. Ali, S. Aziz and S.A. Malik in their paper [16].…”

Section: Resultsmentioning

confidence: 97%

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On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

Aleroev

2019

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The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used.

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Section: Resultsmentioning

confidence: 97%

“…in L 2 (0, 1) (but the system of these eigenfunctions, unfortunately, is not orthogonal, therefore, for solving inverse problems, and in Reference [16] the corresponding biorthogonal system was used).…”

Section: Corollarymentioning

confidence: 99%

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

Aleroev

2019

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“…Suppose the numerator and denominator are approximated as polynomials, and the two are regarded as the numerator and denominator of the pulse transfer function of a filter. In that case, this filter is a discrete FOC filter [10]. Furthermore, we can use the Maclaurin series expansion of the binomial power function to ensure convergence, and the weight coefficient of the expansion term can be quickly and recursively obtained.…”

Section: Convergence and Recursive Realizationmentioning

confidence: 99%

Precision algorithms in second-order fractional differential equations

Liu

2021

Applied Mathematics and Nonlinear Sciences

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The discretization of fractional-order differential operators is the key to the digital realization of fractional-order controllers. This paper proposes an improved second-order fractional differential equation operation method based on power series expansion. The algorithm's operation speed and accuracy performance are analyzed. The research found that the algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.

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“…In [4], for a Space-Time Fractional Diffusion Equation (STFDE) inverse problem of determining a temporal component in the source term from the total energy of the system is considered and the recovery of a space dependent source term from final data is discussed in [3]. Inverse problems of identifying the space and time dependent source terms for a STFDE are considered in [5]. Recovering order of fractional derivatives from multi-term TFDE with constant coefficients from boundary measurements is discussed in [18].…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Determination of a Source Term and Diffusion Concentration for a Multi-Term Space-Time Fractional Diffusion Equation

Malik

Ilyas

Samreen

2021

Mathematical Modelling and Analysis

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An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

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Inverse source problems for a space–time fractional differential equation

Cited by 26 publications

References 43 publications

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

Precision algorithms in second-order fractional differential equations

Simultaneous Determination of a Source Term and Diffusion Concentration for a Multi-Term Space-Time Fractional Diffusion Equation

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