Inverse Problems of a Fractional Differential Equation with Bessel Operator

Al-Musalhi, Fatma; Al-Salti, Nasser; Kerbal, Sebti

doi:10.1051/mmnp/201712310

Cited by 15 publications

(11 citation statements)

References 9 publications

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“…At present, the expressions of fractional differential mainly include Riemann-Liouville, Grünwald-Letnikov, and Caputo [16][17][18], and the most commonly used expressions are Grünwald-Letnikov (G-L) expressions. The G-L differential is defined by:…”

Section: Fractional Derivativementioning

confidence: 99%

Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content

Xiong

Tian

2018

Mathematics

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The study of field spectra based on fractional-order differentials has rarely been reported, and traditional integer-order differentials only perform the derivative calculation for 1st-order or 2nd-order spectrum signals, ignoring the spectral transformation details between 0th-order to 1st-order and 1st-order to 2nd-order, resulting in the problem of low-prediction accuracy. In this paper, a spectral quantitative analysis model of soil-available phosphorus content based on a fractional-order differential is proposed. Firstly, a fractional-order differential was used to perform a derivative calculation of original spectral data from 0th-order to 2nd-order using 0.2-order intervals, to obtain 11 fractional-order spectrum data. Afterwards, seven bands with absolute correlation coefficient greater than 0.5 were selected as sensitive bands. Finally, a stepwise multiple linear regression algorithm was used to establish a spectral estimation model of soil-available phosphorus content under different orders, then the prediction effect of the model under different orders was compared and analyzed. Simulation results show that the best order for a soil-available phosphorus content regression model is a 0.6 fractional-order, the coefficient of determination (), root mean square error (RMSE), and ratio of performance to deviation (RPD) of the best model are 0.7888, 3.348878, and 2.001142, respectively. Since the RPD value is greater than 2, the optimal fractional model established in this study has good quantitative predictive ability for soil-available phosphorus content.

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Section: Fractional Derivativementioning

confidence: 99%

Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content

Xiong

Tian

2018

Mathematics

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“…Let now f (x, t) = g(x) be an unknown function. In this case, the inverse source problem for equation (1) can be formulated as follows:…”

Section: Formulation Of Problemsmentioning

confidence: 99%

“…In [10], the initial inverse problem for heat equation with Bessel operator was investigated. In a recent work By Fatma Al-Musalhi et al [1], the authors studied inverse initial and inverse source problems for time-fractional diffusion equation with zero order Bessel operator. For review of different questions for equations with integer order derivatives we can refer to [12].…”

Section: Introductionmentioning

confidence: 99%

On Boundary-Value Problems for a Partial Differential Equation with Caputo and Bessel Operators

Agarwal

Karimov

Мамчуев

et al. 2017

Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science

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“…Generalized Tikhonov regularization method is used to construct the solution of the inverse source problem for time-fractional diffusion equation. 23 AL-Musalhi 33 discussed the recovery of initial distribution for differential equation involving Bessel operator.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Some inverse problems for time‐fractional diffusion equation with nonlocal Samarskii‐Ionkin type condition

Ali

Aziz

2020

Math Methods in App Sciences

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Two inverse problems for time-fractional diffusion equation having a family of nonlocal boundary conditions are discussed. In first inverse problem, initial distribution is determined provided that the data at final temperature t = T is given.Second inverse problem addresses the recovery of temporal component of source term whenever total energy of the system is known. A bi-orthogonal system of functions is used to write the series solution by Fourier's method. The classical nature of the solution of both inverse problems is established by using the estimates of Mittag-Leffler function and by imposing some regularity conditions on given datum.

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Inverse Problems of a Fractional Differential Equation with Bessel Operator

Cited by 15 publications

References 9 publications

Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content

Fractional Modeling for Quantitative Inversion of Soil-Available Phosphorus Content

On Boundary-Value Problems for a Partial Differential Equation with Caputo and Bessel Operators

Some inverse problems for time‐fractional diffusion equation with nonlocal Samarskii‐Ionkin type condition

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