Research

11 pages, 563 KiB

Open AccessArticle

An Efficient Numerical Scheme for Fractional Order Mathematical Model of Cytosolic Calcium Ion in Astrocytes

by Devendra Kumar, Hunney Nama, Jagdev Singh and Jitendra Kumar

Fractal Fract. 2024, 8(4), 184; https://doi.org/10.3390/fractalfract8040184 - 23 Mar 2024

Viewed by 600

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of [...] Read more.

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of the considered fractional model using the Laguerre collocation method (LCM) and Jacobi collocation method (JCM). An operational matrix of the fractional order derivative in the Caputo sense is needed for the recommended approach. The computational scheme converts fractional differential equations (FDEs) into an algebraic set of equations using the collocation method. The technique is used more quickly and successfully than in other existing schemes. A comparison between LCM and JCM is also presented in the form of figures. We obtained very good results with a great agreement between both the schemes. Additionally, an error analysis of the suggested procedures is provided. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

13 pages, 314 KiB

Open AccessArticle

Analytical Solutions for a Generalized Nonlinear Local Fractional Bratu-Type Equation in a Fractal Environment

by Ghaliah Alhamzi, Ravi Shanker Dubey, Badr Saad T. Alkahtani and G. L. Saini

Fractal Fract. 2024, 8(1), 15; https://doi.org/10.3390/fractalfract8010015 - 23 Dec 2023

Viewed by 1107

Abstract

In the context of fractal space, this study presents a higher-order nonlinear local fractional Bratu-type equation and thoroughly examines this generalized nonlinear equation. Additional analysis and identification of particular special situations of the generalized local fractional Bratu equation is performed. Finally, the Adomian [...] Read more.

In the context of fractal space, this study presents a higher-order nonlinear local fractional Bratu-type equation and thoroughly examines this generalized nonlinear equation. Additional analysis and identification of particular special situations of the generalized local fractional Bratu equation is performed. Finally, the Adomian decomposition method is utilized to derive that solution for the generalized Bratu equation of local fractional type. This study contributes to a deeper understanding of these equations and provides a practical computational approach to their solutions. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

19 pages, 376 KiB

Open AccessArticle

Impulsive Controllers Design for the Practical Stability Analysis of Gene Regulatory Networks with Distributed Delays

by Jinde Cao, Trayan Stamov, Gani Stamov and Ivanka Stamova

Fractal Fract. 2023, 7(12), 847; https://doi.org/10.3390/fractalfract7120847 - 29 Nov 2023

Viewed by 754

Abstract

This paper studies gene regulatory networks (GRNs) with distributed delays. The essential concept of practical stability of the genes is introduced. We investigate the problems of practical stability and global practical exponential stability of the GRN model under an impulsive control. New practical [...] Read more.

This paper studies gene regulatory networks (GRNs) with distributed delays. The essential concept of practical stability of the genes is introduced. We investigate the problems of practical stability and global practical exponential stability of the GRN model under an impulsive control. New practical stability criteria are proposed by designing appropriate impulsive controllers via the Lyapunov functions approach. In the design of the impulsive controller, we consider the effect of impulsive perturbations at fixed times and distributed delays on the stability of the considered GRNs. Several numerical examples are also presented to justify the proposed criteria. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

20 pages, 691 KiB

Open AccessArticle

An Analysis of the Effects of Lifestyle Changes by Using a Fractional-Order Population Model of the Overweight/Obese Diabetic Population

by Kholoud Saad Albalawi, Kuldeep Malik, Badr Saad T. Alkahtani and Pranay Goswami

Fractal Fract. 2023, 7(12), 839; https://doi.org/10.3390/fractalfract7120839 - 27 Nov 2023

Viewed by 827

Abstract

Unbalanced lifestyles and other underlying medical conditions are responsible for the worrying pace at which diabetes mellitus is becoming a global health crisis. Recent studies suggest that placing a diabetic patient into remission through a rigorous lifestyle change program can normalize blood glucose [...] Read more.

Unbalanced lifestyles and other underlying medical conditions are responsible for the worrying pace at which diabetes mellitus is becoming a global health crisis. Recent studies suggest that placing a diabetic patient into remission through a rigorous lifestyle change program can normalize blood glucose levels. This research focuses on fractional order derivative-based mathematical modeling and analysis of the diabetes mellitus model with remission parameters. Firstly, the existence and uniqueness of the solution of the diabetes mellitus model are discussed. Non-negativity and boundedness are also examined. Afterward, the concept of the Jacobian matrix is used to investigate the stability of the model’s equilibrium points. The Daftardar-Gejji and Jafari Method has finally been applied to approximate the solutions. The conclusions drawn from numerical simulations of the diabetic model with fractional-order derivatives show a clear dependence on the remission parameters and fractional-order derivative. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

19 pages, 354 KiB

Open AccessArticle

Domains of Quasi Attraction: Why Stable Processes Are Observed in Reality?

by Vassili N. Kolokoltsov

Fractal Fract. 2023, 7(10), 752; https://doi.org/10.3390/fractalfract7100752 - 12 Oct 2023

Viewed by 856

Abstract

From the very start of modelling with power-tail distributions, concerns were expressed about the actual applicability of distributions with infinite expectations to real-world distributions, which usually have bounded ranges. Here, we suggest resolving this issue by shifting the analysis from the true convergence [...] Read more.

From the very start of modelling with power-tail distributions, concerns were expressed about the actual applicability of distributions with infinite expectations to real-world distributions, which usually have bounded ranges. Here, we suggest resolving this issue by shifting the analysis from the true convergence in various CLTs to some kind of quasi convergence, where a stable approximation to, say, normalised sums of n i.i.d. random variables (or more generally, in a functional setting, to the processes of random walks), holds for large n, but not “too large” n. If the range of “large n” includes all imaginable applications, the approximation is practically indistinguishable from the true limit. This approach allows us to justify a stable approximation to random walks with bounded jumps and, moreover, it leads to some kind of cascading (quasi) asymptotics, where for different ranges of a small parameter, one can have different stable or light-tail approximations. The author believes that this development might be relevant to all applications of stable laws (and thus of fractional equations), say, in Earth systems, astrophysics, biological transport and finances. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

16 pages, 360 KiB

Open AccessArticle

Subdiffusion–Superdiffusion Random-Field Transition

by Alexander Iomin

Fractal Fract. 2023, 7(10), 745; https://doi.org/10.3390/fractalfract7100745 - 10 Oct 2023

Viewed by 844

Abstract

A contaminant spreading affected by a random field at boundaries in the comb geometry is considered. The physical effect of the random boundary conditions results in increasing a transport exponent such that the mean squared displacement increases with time from

t^{\frac{1}{2}}

[...] Read more.

A contaminant spreading affected by a random field at boundaries in the comb geometry is considered. The physical effect of the random boundary conditions results in increasing a transport exponent such that the mean squared displacement increases with time from

t^{\frac{1}{2}}

to

t^{\frac{1}{2} + 5 α / 2}

for real

0 \leq α \leq 1

. This stochastic acceleration due to these space-time-dependent boundary conditions leads to a transition from subdiffusion to superdiffusion. Experimentally, it can be realized by controlling the boundary conditions of

2 D

diffusion in the comb geometry. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

18 pages, 12453 KiB

Open AccessArticle

Modeling and Initialization of Nonlinear and Chaotic Fractional Order Systems Based on the Infinite State Representation

by Jean-Claude Trigeassou and Nezha Maamri

Fractal Fract. 2023, 7(10), 713; https://doi.org/10.3390/fractalfract7100713 - 27 Sep 2023

Cited by 2 | Viewed by 840

Abstract

Based on the infinite state representation, any linear or nonlinear fractional order differential system can be modelized by a finite-dimension set of integer order differential equations. Consequently, the recurrent issue of the Caputo derivative initialization disappears since the initial conditions of the fractional [...] Read more.

Based on the infinite state representation, any linear or nonlinear fractional order differential system can be modelized by a finite-dimension set of integer order differential equations. Consequently, the recurrent issue of the Caputo derivative initialization disappears since the initial conditions of the fractional order system are those of its distributed integer order differential system, as proven by the numerical simulations presented in the paper. Moreover, this technique applies directly to fractional-order chaotic systems, like the Chen system. The true interest of the fractional order approach is to multiply the number of equations to increase the complexity of the chaotic original system, which is essential for the confidentiality of coded communications. Moreover, the sensitivity to initial conditions of this augmented system generalizes the Lorenz approach. Determining the Lyapunov exponents by an experimental technique and with the G.S. spectrum algorithm provides proof of the validity of the infinite state representation approach. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

14 pages, 345 KiB

Open AccessArticle

An Efficient Approach to Solving the Fractional SIR Epidemic Model with the Atangana–Baleanu–Caputo Fractional Operator

by Lakhdar Riabi, Mountassir Hamdi Cherif and Carlo Cattani

Fractal Fract. 2023, 7(8), 618; https://doi.org/10.3390/fractalfract7080618 - 11 Aug 2023

Viewed by 932

Abstract

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an [...] Read more.

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

13 pages, 610 KiB

Open AccessArticle

Discrete q-Exponential Limit Order Cancellation Time Distribution

by Vygintas Gontis

Fractal Fract. 2023, 7(8), 581; https://doi.org/10.3390/fractalfract7080581 - 28 Jul 2023

Viewed by 646

Abstract

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best fitting model. Long-range memory and self-similarity estimators, commonly used for this purpose, can [...] Read more.

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best fitting model. Long-range memory and self-similarity estimators, commonly used for this purpose, can yield inconsistent parameter values, as they are tailored to specific time series models. In our previous work, we explored order disbalance time series from the broader perspective of fractional L’evy stable motion, revealing a stable anti-correlation in the financial market order flow. However, a more detailed analysis of empirical data indicates the need for a more specific order flow model that incorporates the power-law distribution of limit order cancellation times. When considering a series in event time, the limit order cancellation times follow a discrete probability mass function derived from the Tsallis q-exponential distribution. The combination of power-law distributions for limit order volumes and cancellation times introduces a novel approach to modeling order disbalance in the financial markets. Moreover, this proposed model has the potential to serve as an example for modeling opinion dynamics in social systems. By tailoring the model to incorporate the unique statistical properties of financial market data, we can improve the accuracy of our predictions and gain deeper insights into the dynamics of these complex systems. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

► Show Figures

Figure 1

12 pages, 293 KiB

Open AccessArticle

Fractional Factor Model for Data Transmission in Real-Time Monitoring Network

by Juxiang Zhou, Wei Gao and Hainan Zhang

Fractal Fract. 2023, 7(7), 493; https://doi.org/10.3390/fractalfract7070493 - 22 Jun 2023

Viewed by 606

Abstract

Modeling data transmission problems in graph theory is internalized to the existence of fractional flows, and thus can be surrogated to be characterized by a fractional factor in diversified settings. We study the fractional factor framework in the network environment when some sites [...] Read more.

Modeling data transmission problems in graph theory is internalized to the existence of fractional flows, and thus can be surrogated to be characterized by a fractional factor in diversified settings. We study the fractional factor framework in the network environment when some sites are damaged. The setting we focus on refers to the lower and upper fractional degrees described by two functions on the vertex set. It is determined that G is fractional

(g, f, n)

critical if

δ (G) \geq ⌊ \frac{a^{2} + b^{2} + 2 a b + 2 a + 2 b - 3}{4 a} ⌋ + n

and

I (G) > \frac{n + ⌊ \frac{{(a + b - 1)}^{2}}{2 a} + \frac{2 b - 1}{a} ⌋}{2}

, where

1 \leq a \leq b

and

b \geq 2

. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

16 pages, 342 KiB

Open AccessArticle

A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation

by Ravshan Ashurov, Oqila Mukhiddinova and Sabir Umarov

Fractal Fract. 2023, 7(6), 490; https://doi.org/10.3390/fractalfract7060490 - 20 Jun 2023

Cited by 1 | Viewed by 837

Abstract

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition

u (x, T) = β u (x, 0) + φ (x)

, where [...] Read more.

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition

u (x, T) = β u (x, 0) + φ (x)

, where

β

is an arbitrary real number, is proposed instead of the initial condition. If

β = 0

, then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If

β = 1

, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter

β

, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if

β \geq 1,

or

β < 0

, then the problem is well-posed; if

β \in (0, 1)

, then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

Journal Menu

Journal Browser

Feature Papers for Mathematical Physics Section

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Published Papers (11 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI