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Analysis of General Groundwater Flow Equation with Fractal Derivative
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
Mashudu Mathobo, Abdon Atangana
Recently Ramotsho and Atangana (2017) used fractal derivatives to derive the exact numerical solution of diffusion within a leaky aquifer. In their work, they used one of the fractals properties, self-similarity. Self-similarity occurs when a system replicate itself and shows iteration throughout. Ramotsho and Atangana (2017) derived the new groundwater flow equations within a self-similar leaky aquifer, with one showing a scenario of abstracting water from a self-similar leaky aquifer, and another showing a scenario of abstracting water from a leaky aquifer. Numerical solutions were derived using the newly developed Adams–Bashforth method by Atangana and Batogna. Groundwater models were also created. Their models took into account the geological formation of the system compared to models developed by classical formulas. Allwright and Atangana (2018) conducted research on groundwater transport in fractured aquifers with self-similarities. In their work, they highlighted that groundwater transport within a fractured aquifer with a fractal nature exhibiting self-similarity cannot be simulated accurately using the Fickian advection-dispersion transport equation. In order to close the gap in knowledge, Allwright and Atangana (2018) developed a fractal advection-dispersion groundwater transport equation and its integral with theorem and proof. Numerical simulations were developed and it was shown in their studies that by incorporating a fractal dimension, anomalous diffusion may be modeled in an effective and efficient way.
Macromolecular Crowding
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Adriana Isvoran, Laura Pitulice, Eudald Vilaseca, Isabel Pastor, Sergio Madurga, Francesc Mas
MC affects quantitatively and qualitatively the diffusion processes. The diffusion of a kind of molecule is hindered by the other molecules that are present in the diffusion environment and may result in anomalous diffusion (Saxton, 1987, 1989, 1990, 1992, 1993, 1994; Dix and Verkman, 2008). For anomalous diffusion, at least one of the validity conditions of the Einstein-Smolukovski equation is not fulfilled, and the mean squared displacement depends on time as follows: ()<r2(t)>=2dΓtα
Applications of Elzaki decomposition method to fractional relaxation-oscillation and fractional biological population equations
Published in Applied Mathematics in Science and Engineering, 2023
Lata Chanchlani, Mohini Agrawal, Rupakshi Mishra Pandey, Sunil Dutt Purohit, D. L. Suthar
In recent research, the concept of fractional generalization has been demonstrated to be a successful technique for modelling a variety of natural events. Various fractional diffusion models have been developed to predict the human brain’s response to external stimuli [27], the role of structural heterogeneity in repolarization dispersion in cardiac electrical propagation [28], charge transfer in dye-sensitized solar cells [29], dust aerosols floating in Mars’ atmosphere that cause attenuation of solar radiation traversing the atmosphere [30], and fluid transport through porous media [31], impact on raise of environmental pollution and occurrence in biological populations by presenting a mathematical model relating to incomplete H-function [32] and many others. In these models, fractional derivatives have been incorporated into partial differential equations, allowing for the description of anomalous diffusion within fractal spaces.
Numerical study and parameters estimation of anomalous diffusion process in porous media based on variable-order time fractional dual-phase-lag model
Published in Numerical Heat Transfer, Part A: Applications, 2023
Hossein Sobhani, Aziz Azimi, Aminreza Noghrehabadi, Milad Mozafarifard
The anomalous diffusion process has been observed in different cases, for example thermal response of porous media [1] where the normal diffusion based on Fourier’s law cannot accurately predict the thermal interactions between solid and fluid phases. Therefore, it is necessary to use non-Fourier models, such as the SPL or DPL to better elucidate the thermal response of a two-phase system. To address the shortage associated with the Fourier’s law, Cattaneo [2] and Vernotte [3] presented the C-V wave model, assuming that the thermal disturbance would propagate with finite speed. They considered a lagging time between temperature gradient and heat flux, and developed the heat conduction equation as below [2, 3], where is the heat flux, is the temperature and is the heat flux relaxation time. Additionally, Tzou [4, 5] proposed the SPL model by considering the relaxation time and applying the Taylor series expansion of Eq. (1) to and neglecting the higher-order terms (assuming a small amount of compared to the total time of the process),
Formulation of space-angular fractional radiative transfer equation in participating finite slab clumpy media
Published in Waves in Random and Complex Media, 2021
M. Sallah, R. Gamal, A. Elgarayhi, A. A. Mahmoud
Due to its parabolic nature, the classical (integer-order) diffusion model of radiation transfer predicts unphysical infinite particle speed. In addition, it is not applicable everywhere especially in heterogeneous and clumpy media. To insure the finite propagation velocity of the particle, Cattaneo proposed the hyperbolic normal diffusion equation [1]. The modeling of radiation transfer in a heterogeneous medium can be treated as anomalous diffusion (sub-diffusion or super-diffusion). Anomalous diffusion is different from the normal, Fickian diffusion and is characterized by features like: slower or faster movement of diffusing particles, non-Gaussian probability density function of particle concentration, departure of the particle velocities from the Maxwellian distribution, and the non-linear dependence of asymptotic mean-squared displacement on time, that is, . Condition indicates slower movement of particles, which is named as sub-diffusion, and indicates faster movement that is called super-diffusion [2–5]. It is now a well-established that processes involving anomalous diffusion can be better modeled using fractional-order models [6, 7]. In addition, fractional-order models are found to give a more realistic and compact representation of the natural and artificial systems [8–10].