Mild Solution for the Time-Fractional Navier–Stokes Equation Incorporating MHD Effects

In this article, we derive a new numerical method to solve fractional differential equations containing Caputo-Fabrizio derivatives. The fundamental concepts of fractional calculus, numerical analysis, and fixed point theory form the basis of this study. Along with the derivation of the algorithm of the proposed method, error and stability analyses are performed briefly. To explore the validity and effectiveness of the proposed method, several examples are simulated, and the new solutions are compared with the outputs of the previously published two-step Adams-Bashforth method.

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“…Using definition of global error: e n = g(t n ) − g n and subtracting equation (16) from equation (15), we get…”

Section: Error Estimationmentioning

confidence: 99%

“…In [14], a computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators is derived. In [15], the authors explored the mild solution for the time-fractional Navier-Stokes equation incorporating MHD effects.…”

Section: Introductionmentioning

confidence: 99%

A new numerical method to solve fractional differential equations in terms of Caputo-Fabrizio derivatives

Mahatekar¹,

Scindia²,

Kumar

2023

Phys. Scr.

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“…Joshi H et al have modeled for the COVID-19 outbreak to examine real-life data [22]. Yavuz M. et al studied the time fractional approach for Naiver Stokes for magnetohydrodynamics [23]. Magnetohydrodynamics of the boundary layer is examined with fractional differential equations for maxwell fluid by Yavuz M. [24].…”

Section: Introductionmentioning

confidence: 99%

Modeling and analysis of fractional order calcium distribution model with Amyloid beta and Orai channel

Vora,

Jha,

Singh

2024

Preprint

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Calcium is a ubiquitous signaling component of neuronal functions. Calcium regulation is a complex phenomenon that is controlled by cellular organelles. Abnormality created by amyloid plaques is a crucial phenomenon in which the [Ca2+] flux enters through the plasma membrane. Entry of such flux is managed by intracellular homeostasis. In the neuronal calcium homeostasis, any alteration could lead to neurodegener-ation. In this paper, the effect of various parameters like SERCA, IPR, Leak, VGCC, etc. has been investigated. A mathematical model has been developed by considering a time-fractional differential approach to investigate the non-local impact on cytosolic [Ca2+]. An analytic solution was obtained by applying the Laplace transform, Fourier cosine transform, and Green’s function. The significant implication of amyloid beta, endoplas-mic reticulum flux, and plasma membrane flux on calcium signaling was observed. It is also observed that Amyloid beta created pores are toxic to the neuron. The effect of S100B, EGTA, and BAPTA with diffusion coefficient observed neuroprotective properties for calcium signaling. From all this simulation therapeutic insight from different buffers and amyloid plaques can be observed which is a better fit as per realistic physiology.

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“…Other memory effect modeling approach is taken by Joshi et al [ 32 ] for covid 19 outbreak. Shafqat et al [ 33 ] explored Naiver Stokes with fractional order. Magnetohydrodynamics is also explored with fractional order for maxwell fluid by Fayz-Al-Asad et al [ 34 ].…”

Section: Introductionmentioning

confidence: 99%

To study the effect of ER flux with buffer on the neuronal calcium

Vatsal,

Jha,

Singh

2023

Eur. Phys. J. Plus

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Calcium signaling is decisive for cellular functions. This calcium random walk stipulates neuronal functions. Calcium concentration could provoke gene transcription, apoptosis, neuronal plasticity, etc. A malformation in calcium could change the neuron’s intracellular behavior. Calcium concentration balancing is a complex cellular mechanism. This occurrence can be handled with the Caputo fractional reaction–diffusion equation. In this mathematical modeling, we have included the STIM-Orai mechanism and Endoplasmic Reticulum (ER) flux, Inositol Triphosphate Receptor (IPR), SERCA, plasma membrane flux, voltage-gated calcium entry, and different buffer interactions. A hybrid integral transform and Green’s function approach were taken to solve the initial boundary problem. A closed-form solution of a Mittag-Leffler family function plotted using MATLAB software. Different parameters impact changes in the spatiotemporal behavior of the calcium concentration. Specific roles of organelles involved in Alzheimer’s disease-affected neurons are computed. Ethylene glycol tetraacetic acid (EGTA), 1,2-bis(o-aminophenoxy)ethane N,N,N,N-tetraacetic acid (BAPTA), and S100B protein effects are also observed. In all simulations, we can say S100B and the STIM-Orai effect cannot be neglected. This model lights up the different approaches for calcium signaling pathway simulation. As a consequence, we determine that a generalized reaction–diffusion approach is a better fit realistic model.

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Mild Solution for the Time-Fractional Navier–Stokes Equation Incorporating MHD Effects

Cited by 17 publications

References 27 publications

A new numerical method to solve fractional differential equations in terms of Caputo-Fabrizio derivatives

A new numerical method to solve fractional differential equations in terms of Caputo-Fabrizio derivatives

Modeling and analysis of fractional order calcium distribution model with Amyloid beta and Orai channel

To study the effect of ER flux with buffer on the neuronal calcium

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