On a time–space fractional backward diffusion problem with inexact orders

Trong, Dang Duc; Hai, Dinh Nguyen Duy; Dien, Nguyen Minh

doi:10.1016/j.camwa.2019.03.014

Cited by 14 publications

(6 citation statements)

References 24 publications

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“…For convenience of implementation, our schemes Equations ( 42), (44), and ( 47) can be written in a general matrix form for the tablet and release medium as follows:…”

Section: The Dug Concentrationmentioning

confidence: 99%

“…Second, the spatial derivative is substituted by the fractional one. Other extensions have also been investigated including temporal and spatial FDs 41–44 . Finally, Fick's law is replaced by the so‐called “Maxwell–Cattaneo law” that is widely used in describing heat and mass transfer 45 .…”

Section: Introductionmentioning

confidence: 99%

“…Other extensions have also been investigated including temporal and spatial FDs. [41][42][43][44] Finally, Fick's law is replaced by the so-called "Maxwell-Cattaneo law" that is widely used in describing heat and mass transfer. 45 Cattaneo 46 modified Fourier's principle in 1948 by including a relaxation time term.…”

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confidence: 99%

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Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Khalaf

Elsaid

Hammad

et al. 2023

Numer Methods Biomed Eng

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Mathematical simulation of drug diffusion is a significant tool for predicting the bio‐transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi‐layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth‐order weighted‐shifted Grünwald–Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.

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“…For convenience of implementation, our schemes Equations ( 42), (44), and ( 47) can be written in a general matrix form for the tablet and release medium as follows:…”

Section: The Dug Concentrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Khalaf

Elsaid

Hammad

et al. 2023

Numer Methods Biomed Eng

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“…Secondly, the fractional derivative is employed to replace the spatial derivative. Additionally, different extensions have been explored, incorporating fractional derivatives with respect to time and space [29,30]. Lastly, Fickʼs law is substituted by the "Maxwell-Cattaneo law," which is frequently employed to describe heat and mass transfer phenomena [31].…”

Section: Introductionmentioning

confidence: 99%

Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Khalaf,

Elsaid,

Hammad

et al. 2024

Phys. Scr.

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Although memory-dependent derivative (MDD) has recently been the subject of extensive study, only one numerical approximation has been reported in the literature. Hence, this study introduces a novel approximation for MDD. Moreover, a new form of the kernel function is presented. The convergence order of our approximation is O(h^(3-S) ),0<S<1, where (1-S) is the exponent of the kernel function. The proposed approach is used to numerically solve the memory-dependent advection-diffusion problem, and the numerical scheme’s stability and convergence are discussed. Also, memory-based models have been described to study drug delivery and its diffusion from multi-layer capsules/tablets. These models are based on the fractional derivative and the MDD to solve the paradox of the unphysical feature of the infinite propagation speed of the published Fickian-based models. The theoretical analysis is validated with numerical examples by investigating the convergence order that is (3-S) in time. To illustrate the validity and efficiency of the proposed models, profiles of concentration and drug mass are compared with the observed in vivo data. It is observed to be highly accurate and compatible with the in vivo data when compared with Fick’s model.

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“…Furthermore, numerical computations are not allowed if this continuity does not hold. Motivated from [7,31], this paper studies the continuity of the solution of both the initial problem (1)-(2) and the inverse initial value problems (1),(3) with respect to the fractional order α. Namely, this work focuses on the question Does u αn → u α in an appropriate sense as n → ∞?…”

mentioning

confidence: 99%

Continuity with respect to fractional order of the time fractional diffusion-wave equation

Tuan¹,

O’Regan²,

Ngoc³

2020

Evolution Equations &Amp; Control Theory

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This paper studies a time-fractional diffusion-wave equation with a linear source function. First, some stability results on parameters of the Mittag-Leffler functions are established. Then, we focus on studying the continuity of the solution of both the initial problem and the inverse initial value problems corresponding to the fractional-order in our main results. One of the difficulties encounteblack comes from estimating all constants independently of the fractional orders. Finally, we present some numerical results to confirm the effectiveness of our methods. 2020 Mathematics Subject Classification. 35R11, 35B65, 26A33. Key words and phrases. Initial value problem, inverse initial value problem, final value problem, time fractional wave equation, continuity/stability on fractional orders.We note that, for any > 0, there always exists C > 0 such that 1 − e −y ≤ C y for all y > 0. Therefore, we obtainCase 2. z ≥ 1. We see thatUsing the inequality 1 − e −y ≤ C y for any > 0 and y > 0, we obtainCombining the above cases completes the proof.CONTINUITY WITH RESPECT TO FRACTIONAL ORDER 777 Lemma 3.3. Let 1 < ν 0 < α < α < η 0 < 2 and > 0. Then there exists a positive constant D 2 (ν 0 , η 0 , , β, T ) such thatfor any 0 ≤ β ≤ 1 and 0 < t ≤ T .Proof. First, it is easy to see that

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On a time–space fractional backward diffusion problem with inexact orders

Cited by 14 publications

References 24 publications

Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Fractional modeling of drug diffusion from cylindrical tablets based on Fickian and relaxed approaches with in vivo validation

Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Continuity with respect to fractional order of the time fractional diffusion-wave equation

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