A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

Troparevsky, M. I.; Seminara, Silvia; Fabio, Marcela A.

doi:10.5772/intechopen.86273

Cited by 17 publications

(10 citation statements)

References 31 publications

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“…Almost all these problems were dealing with the numerical solution of a fractional differential equation (FDE) and related problem (boundary and/or initial). In recent years, various powerful methods have been suggested for the numerical approximate solution of the fractional differential equations, like, eg, the homotopy perturbation, the Adomian decomposition method (ADM), and wavelet method . A survey on some of the most efficient classes of numerical methods for fractional‐order problems is given in Garrappa .…”

Section: Introductionmentioning

confidence: 99%

Legendre spectral method for the fractional Bratu problem

Singh¹,

Ghassabzadeh

Tohidi

et al. 2020

Math Methods in App Sciences

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In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error.

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Section: Introductionmentioning

confidence: 99%

Legendre spectral method for the fractional Bratu problem

Singh¹,

Ghassabzadeh

Tohidi

et al. 2020

Math Methods in App Sciences

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“…Given that a reaction-diffusion process can depend not only on the previous time instance's concentrations but also on all the earlier stages of the concentrations with some weights [48,49,50,51,52,53], this idea is further developed in this paper. A time-fractional reaction-diffusion equation is used to analyze such processes in the AD context, which we call collectively the "memory effect".…”

Section: Introductionmentioning

confidence: 99%

Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

Pal¹,

Melnik²

2022

Preprint

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Alzheimer's disease (AD) is a common neurodegenerative disorder nowadays. Amyloid-beta (Aβ) and tau proteins are among the main contributor to the development or propagation of AD. In AD, Aβ proteins clump together to form plaques and disrupt cell functions. On the other hand, the abnormal chemical change in the brain helps to build sticky tau tangles that block the neuron's transport system. Astrocytes generally maintain a healthy balance in the brain by clearing the Aβ plaques (toxic Aβ). But, over-activated astrocytes release chemokines and cytokines in the presence of Aβ and also react to pro-inflammatory cytokines further increasing the production of Aβ. In this paper, we construct a mathematical model that can capture astrocytes' dual behaviour. Furthermore, we reveal that the disease propagation does not depend only on the current time instance; rather, it depends on the disease's earlier status, called the "memory effect". We consider a fractional order network mathematical model to capture the influence of such memory effect on AD propagation. We have integrated brain connectome data in the network model and studied the memory effect and the dual role of astrocytes together. Depending on toxic loads in the brain, we have also analyzed the neuronal damage in the brain. Based on the pathology, primary, secondary, and mixed tauopathies parameters have been used in the model. Due to the mixed tauopathy, different brain nodes or regions in the brain connectome accumulate different toxic concentrations of toxic Aβ and toxic tau proteins. Finally, we explain how the memory effect can slow down the propagation of such toxic proteins in the brain and hence decreases the rate of neuronal damage.

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“…In the last decades fractional calculus has been successfully used to model phenomena in different areas: diffusion problems, hydraulics, potential theory, control theory, electrochemistry, electromagnetism, viscoelasticity and nanotechnology (see for example [1,2,6,7,8,9,10,16,22,28,30,38] among others). Numerical schemes to calculate approximate solutions to fractional differential equations have also been introduced (see [5,24,35,39,40]).…”

Section: Introductionmentioning

confidence: 99%

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

Seminara

Troparevsky

Fabio

et al. 2022

TCAM

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In this work we approximate the source for a non homogeneous fractionaldiffusion equation in 1D, from measurements of the concentration at a finite number ofpoints. We use Caputo-Fabrizio time fractional derivative to model anomalous diffusion.Separating variables, we arrive to a linear system which provides approximate values forthe Fourier coefficients of the unknown source. Numerical examples show the efficiency ofthe method, as well as some of its practical limitations.

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A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

Cited by 17 publications

References 31 publications

Legendre spectral method for the fractional Bratu problem

Legendre spectral method for the fractional Bratu problem

Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

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