Fractal and Fractional

18 pages, 461 KiB

Open AccessArticle

An Energy Conserving Numerical Scheme for the Klein–Gordon Equation with Cubic Nonlinearity

by Lewa’ Alzaleq and Valipuram Manoranjan

Fractal Fract. 2022, 6(8), 461; https://doi.org/10.3390/fractalfract6080461 - 22 Aug 2022

Cited by 1 | Viewed by 1136

In this paper, we develop a numerical scheme that conserves the discrete energy for solving the Klein-Gordon equation with cubic nonlinearity. We prove theoretically that our scheme conserves not just discrete energy, but also other energy-like discrete quantities. In addition, we prove the [...] Read more.

In this paper, we develop a numerical scheme that conserves the discrete energy for solving the Klein-Gordon equation with cubic nonlinearity. We prove theoretically that our scheme conserves not just discrete energy, but also other energy-like discrete quantities. In addition, we prove the convergence and the stability of the scheme. Finally, we present some numerical simulations to demonstrate the performance of our energy-conserving scheme. Full article

(This article belongs to the Special Issue Numerical and Analytical Methods for Differential Equations and Systems)

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16 pages, 16461 KiB

Open AccessArticle

Fractal Analysis of Porous Alumina and Its Relationships with the Pore Structure and Mechanical Properties

by Xufu Wang, Chang Chen and Yubin Wang

Fractal Fract. 2022, 6(8), 460; https://doi.org/10.3390/fractalfract6080460 - 21 Aug 2022

Cited by 7 | Viewed by 1381

Abstract

Porous alumina was prepared by the sacrificial template approach using 30 vol.%, 50 vol.%, and 70 vol.% of carbon fibers and graphite as pore formers. In order to determine the pore size distribution, porosity, most probable pore size, and median pore size, a [...] Read more.

Porous alumina was prepared by the sacrificial template approach using 30 vol.%, 50 vol.%, and 70 vol.% of carbon fibers and graphite as pore formers. In order to determine the pore size distribution, porosity, most probable pore size, and median pore size, a mercury intrusion porosimeter (MIP) was used. The surface fractal dimensions (D_s) of porous alumina with various pore formers were assessed based on MIP data. The findings revealed that the pore size distribution of the prepared porous alumina was either bimodal or trimodal at 50 vol.% of the pore formers, and unimodal at 30 vol.% and 70 vol.% of the pore formers in the raw materials. The porous alumina’s pore structure and morphology varied depending on the volume content of the pore formers and their shapes. The porosity and pore size of the porous alumina increased with the increase in carbon fiber content because the carbon fiber was unfavorable to the densification of the initial billet before sintering. After sintering, there were no residual pore formers other than alumina in the samples. The pore structure of the porous alumina samples showed prominent fractal characteristics, and its D_S decreased with the increase in the pore former content. The samples’ D_s was highly negatively correlated with the pore structure parameters, and was positively correlated with the flexural strength. Full article

(This article belongs to the Special Issue Investigation and Application of Fractals in Civil Engineering Materials)

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16 pages, 7460 KiB

Open AccessArticle

A Heterogeneous Duopoly Game under an Isoelastic Demand and Diseconomies of Scale

by Xiaoliang Li and Li Su

Fractal Fract. 2022, 6(8), 459; https://doi.org/10.3390/fractalfract6080459 - 21 Aug 2022

Cited by 4 | Viewed by 1188

Abstract

In this paper, we investigate a duopolistic market with heterogeneous firms under the assumptions of an isoelastic demand and quadratic costs. We obtain the sufficient and necessary condition of the local stability of the Cournot–Nash equilibrium and analytically compare it with that of [...] Read more.

In this paper, we investigate a duopolistic market with heterogeneous firms under the assumptions of an isoelastic demand and quadratic costs. We obtain the sufficient and necessary condition of the local stability of the Cournot–Nash equilibrium and analytically compare it with that of the analogue model under linear rather than quadratic costs. By approaches of symbolic computation, we prove that diseconomies of scale have an effect of stabilizing the game provided that the cost parameters are large enough. Moreover, by means of numerical simulations, we find that our model loses its stability only through a period-doubling bifurcation, which is different from its analogue having two possible routes to chaotic dynamics. Full article

(This article belongs to the Section Complexity)

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17 pages, 6977 KiB

Open AccessArticle

Fractal Analysis and Time Series Application in ZY-4 SEM Micro Fractographies Evaluation

by Maria-Alexandra Paun, Vladimir-Alexandru Paun and Viorel-Puiu Paun

Fractal Fract. 2022, 6(8), 458; https://doi.org/10.3390/fractalfract6080458 - 21 Aug 2022

Cited by 3 | Viewed by 1209

Abstract

SEM microfractographies of Zircaloy-4 are studied by fractal analysis and the time-series method. We first develop a computer application that associates the fractal dimension and lacunarity to each SEM micrograph picture, and produce a nonlinear analysis of the data acquired from the quantitatively [...] Read more.

SEM microfractographies of Zircaloy-4 are studied by fractal analysis and the time-series method. We first develop a computer application that associates the fractal dimension and lacunarity to each SEM micrograph picture, and produce a nonlinear analysis of the data acquired from the quantitatively evaluated time series. Utilizing the phase space-embedding technique to reconstruct the attractor and to compute the autocorrelation dimension, the fracture surface of the Zircaloy-4 samples is investigated. The fractal analysis method manages to highlight damage complications and provide a description of morphological parameters of various fractures by calculating the fractal dimension and lacunarity. Full article

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17 pages, 1897 KiB

Open AccessArticle

A Mathematical Investigation of Sex Differences in Alzheimer’s Disease

by Corina S. Drapaca

Fractal Fract. 2022, 6(8), 457; https://doi.org/10.3390/fractalfract6080457 - 21 Aug 2022

Cited by 1 | Viewed by 1476

Abstract

Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than [...] Read more.

Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than in aged-matched male patients. Currently, there is no cure for AD. Mathematical models can help us to understand mechanisms of AD onset, progression, and therapies. However, existing models of AD do not account for sex differences. In this paper a mathematical model of AD is proposed that uses variable-order fractional temporal derivatives to describe the temporal evolutions of some relevant cells’ populations and aggregation-prone amyloid-

β

fibrils. The approach generalizes the model of Puri and Li. The variable fractional order describes variable fading memory due to neuroprotection loss caused by AD progression with age which, in the case of post-menopausal females, is more aggressive because of fast estrogen decrease. Different expressions of the variable fractional order are used for the two sexes and a sharper decreasing function corresponds to the female’s neuroprotection decay. Numerical simulations show that the population of surviving neurons has decreased more in post-menopausal female patients than in males at the same stage of the disease. The results suggest that if a treatment that may include estrogen replacement therapy is applied to female patients, then the loss of neurons slows down at later times. Additionally, the sooner a treatment starts, the better the outcome is. Full article

(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)

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15 pages, 405 KiB

Open AccessArticle

A New Incommensurate Fractional-Order Discrete COVID-19 Model with Vaccinated Individuals Compartment

by Amer Dababneh, Noureddine Djenina, Adel Ouannas, Giuseppe Grassi, Iqbal M. Batiha and Iqbal H. Jebril

Fractal Fract. 2022, 6(8), 456; https://doi.org/10.3390/fractalfract6080456 - 21 Aug 2022

Cited by 16 | Viewed by 1326

Abstract

Fractional-order systems have proved to be accurate in describing the spread of the COVID-19 pandemic by virtue of their capability to include the memory effects into the system dynamics. This manuscript presents a novel fractional discrete-time COVID-19 model that includes the number of [...] Read more.

Fractional-order systems have proved to be accurate in describing the spread of the COVID-19 pandemic by virtue of their capability to include the memory effects into the system dynamics. This manuscript presents a novel fractional discrete-time COVID-19 model that includes the number of vaccinated individuals as an additional state variable in the system equations. The paper shows that the proposed compartment model, described by difference equations, has two fixed points, i.e., a disease-free fixed point and an epidemic fixed point. A new theorem is proven which highlights that the pandemic disappears when an inequality involving the percentage of the population in quarantine is satisfied. Finally, numerical simulations are carried out to show that the proposed incommensurate fractional-order model is effective in describing the spread of the COVID-19 pandemic. Full article

(This article belongs to the Special Issue Theories and Applications of Fractional Order Bio-Mathematics in Medicine and Biology)

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19 pages, 345 KiB

Open AccessArticle

Best Proximity Point Theorems for the Generalized Fuzzy Interpolative Proximal Contractions

by Khalil Javed, Maha M. A. Lashin, Muhammad Nazam, Hamed H. Al Sulami, Aftab Hussain and Muhammad Arshad

Fractal Fract. 2022, 6(8), 455; https://doi.org/10.3390/fractalfract6080455 - 21 Aug 2022

Viewed by 1208

Abstract

The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy [...] Read more.

The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy complete metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing

(Ψ, Φ)

-fuzzy proximal contractions and

(Ψ, Φ)

-fuzzy proximal interpolative contractions. The obtained results improve and generalize many best proximity point theorems published earlier. Moreover, we provide many nontrivial examples to validate our best proximity point theorem. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

25 pages, 358 KiB

Open AccessArticle

Asymptotic Behavior on a Linear Self-Attracting Diffusion Driven by Fractional Brownian Motion

by Litan Yan, Xue Wu and Xiaoyu Xia

Fractal Fract. 2022, 6(8), 454; https://doi.org/10.3390/fractalfract6080454 - 20 Aug 2022

Cited by 1 | Viewed by 1103

Abstract

Let

B^{H} = {B_{t}^{H}, t \geq 0}

be a fractional Brownian motion with Hurst index

\frac{1}{2} \leq H < 1

. In this paper, we consider the linear self-attracting diffusion: [...] Read more.

Let

B^{H} = {B_{t}^{H}, t \geq 0}

be a fractional Brownian motion with Hurst index

\frac{1}{2} \leq H < 1

. In this paper, we consider the linear self-attracting diffusion:

d X_{t}^{H} = d B_{t}^{H} + σ X_{t}^{H} d t - θ (\int_{0}^{t} (X_{s}^{H} - X_{u}^{H}) d s) d t

+ ν d t

with

X_{0}^{H} = 0

, where

θ > 0

and

σ, ν \in R

are three parameters. The process is an analogue of the self-attracting diffusion (Cranston and Le Jan, Math. Ann.303 (1995), 87–93). Our main aim is to study the large time behaviors. We show that the solution

{(t - \frac{σ}{θ})}^{H} (X_{t}^{H} - X_{\infty}^{H})

converges in distribution to a normal random variable, as t tends to infinity, and obtain two strong laws of large numbers associated with the solution

X^{H}

. Full article

(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)

14 pages, 318 KiB

Open AccessArticle

Split-Plot Designs with Few Whole Plot Factors Containing Clear Effects

by Yuna Zhao

Fractal Fract. 2022, 6(8), 453; https://doi.org/10.3390/fractalfract6080453 - 20 Aug 2022

Cited by 1 | Viewed by 1225

Abstract

Fractional factorial split-plot designs are widely used when it is impractical to perform fractional factorial experiments in a completely random order. When there are too many subplots per whole plot, or too few whole plots, fractional factorial split-plot designs with replicated settings of [...] Read more.

Fractional factorial split-plot designs are widely used when it is impractical to perform fractional factorial experiments in a completely random order. When there are too many subplots per whole plot, or too few whole plots, fractional factorial split-plot designs with replicated settings of the whole plot factors are preferred. However, such an important study is undeveloped in the literature. This paper considers fractional factorial split-plot designs with replicated settings of the WP factors from the viewpoint of clear effects. We investigate the sufficient and necessary conditions for this class of designs to have clear effects. An algorithm is proposed to generate the desirable designs which have the most clear effects of interest. The fractional factorial split-plot design with replicated settings of the WP factors is analysed and the results are discussed. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)

12 pages, 1276 KiB

Open AccessArticle

A Numerical Strategy for the Approximate Solution of the Nonlinear Time-Fractional Foam Drainage Equation

by Fenglian Liu, Jinxing Liu and Muhammad Nadeem

Fractal Fract. 2022, 6(8), 452; https://doi.org/10.3390/fractalfract6080452 - 19 Aug 2022

Cited by 3 | Viewed by 1086

Abstract

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy [...] Read more.

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions. Full article

(This article belongs to the Special Issue Numerical and Analytical Methods for Differential Equations and Systems)

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15 pages, 717 KiB

Open AccessArticle

Local Stabilization of Delayed Fractional-Order Neural Networks Subject to Actuator Saturation

by Yingjie Fan, Xia Huang and Zhen Wang

Fractal Fract. 2022, 6(8), 451; https://doi.org/10.3390/fractalfract6080451 - 19 Aug 2022

Cited by 4 | Viewed by 1174

Abstract

This paper investigates the local stabilization problem of delayed fractional-order neural networks (FNNs) under the influence of actuator saturation. First, the sector condition and dead-zone nonlinear function are specially introduced to characterize the features of the saturation phenomenon. Then, based on the fractional-order [...] Read more.

This paper investigates the local stabilization problem of delayed fractional-order neural networks (FNNs) under the influence of actuator saturation. First, the sector condition and dead-zone nonlinear function are specially introduced to characterize the features of the saturation phenomenon. Then, based on the fractional-order Lyapunov method and the estimation technique of the Mittag–Leffler function, an LMIs-based criterion is derived to guarantee the local stability of closed-loop delayed FNNs subject to actuator saturation. Furthermore, two corresponding convex optimization schemes are proposed to minimize the actuator costs and expand the region of admissible initial values, respectively. At last, two simulation examples are developed to demonstrate the feasibility and effectiveness of the derived results. Full article

(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)

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24 pages, 406 KiB

Open AccessArticle

Generalized Space-Time Fractional Stochastic Kinetic Equation

by Junfeng Liu, Zhigang Yao and Bin Zhang

Fractal Fract. 2022, 6(8), 450; https://doi.org/10.3390/fractalfract6080450 - 18 Aug 2022

Viewed by 1214

Abstract

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in

R^{d}

with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving [...] Read more.

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in

R^{d}

with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving fractional derivatives in time and fractional Laplacian in space. We firstly give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. Furthermore, we also study various properties of the solution, such as Hölder regularity, the upper bound of second moment, and the stationarity with respect to the spatial variable in the case of linear additive noise. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

24 pages, 1440 KiB

Open AccessArticle

Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions

by Gerd Baumann and Norbert Südland

Fractal Fract. 2022, 6(8), 449; https://doi.org/10.3390/fractalfract6080449 - 18 Aug 2022

Viewed by 1444

Abstract

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on [...] Read more.

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time. Full article

(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)

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22 pages, 2715 KiB

Open AccessArticle

Optimizing the Maximum Lyapunov Exponent of Fractional Order Chaotic Spherical System by Evolutionary Algorithms

by Vincent-Ademola Adeyemi, Esteban Tlelo-Cuautle, Francisco-Javier Perez-Pinal and Jose-Cruz Nuñez-Perez

Fractal Fract. 2022, 6(8), 448; https://doi.org/10.3390/fractalfract6080448 - 18 Aug 2022

Cited by 5 | Viewed by 1834

Abstract

The main goal of this work is to optimize the chaotic behavior of a three-dimensional chaotic-spherical-attractor-generating fractional-order system and compare the results with its novel hyperchaotic counterpart. The fractional-order chaotic system is a smooth system perturbed with a hyperbolic tangent function. There are [...] Read more.

The main goal of this work is to optimize the chaotic behavior of a three-dimensional chaotic-spherical-attractor-generating fractional-order system and compare the results with its novel hyperchaotic counterpart. The fractional-order chaotic system is a smooth system perturbed with a hyperbolic tangent function. There are two major contributions in this investigation. First, the maximum Lyapunov exponent of the chaotic system was optimized by applying evolutionary algorithms, which are meta-heuristics search algorithms, namely, the differential evolution, particle swarm optimization, and invasive weed optimization. Each of the algorithms was populated with one hundred individuals, the maximum generation was five hundred, and the total number of design variables was eleven. The results show a massive increase of over 5000% in the value of the maximum Lyapunov exponent, thereby leading to an increase in the chaotic behavior of the system. Second, a hyperchaotic system of four dimensions was constructed from the inital chaotic system. The dynamics of the optimized chaotic and the new hyperchaotic systems were analyzed using phase portraits, time series, bifurcation diagrams, and Lyapunov exponent spectra. Finally, comparison between the optimized chaotic systems and the hyperchaotic states shows an evidence of more complexity, ergodicity, internal randomness, and unpredictability in the optimized systems than its hyperchaotic counterpart according to the analysis of their information entropies and prediction times. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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56 pages, 23318 KiB

Open AccessReview

Integer and Fractional-Order Sliding Mode Control Schemes in Wind Energy Conversion Systems: Comprehensive Review, Comparison, and Technical Insight

by Irfan Sami, Shafaat Ullah, Laiq Khan, Ahmed Al-Durra and Jong-Suk Ro

Fractal Fract. 2022, 6(8), 447; https://doi.org/10.3390/fractalfract6080447 - 17 Aug 2022

Cited by 7 | Viewed by 2241

Abstract

The technological development in wind energy conversion systems (WECSs) places emphasis on the injection of wind power into the grid in a smoother and robust way. Sliding mode control (SMC) has proven to be a popular solution for the grid-connected WECS due to [...] Read more.

The technological development in wind energy conversion systems (WECSs) places emphasis on the injection of wind power into the grid in a smoother and robust way. Sliding mode control (SMC) has proven to be a popular solution for the grid-connected WECS due to its robust nature. This paper reviews the enhancement trends in the integer-order SMC (IOSMC) and fractional-order SMC (FOSMC) schemes reported in reputed journals over the last two decades. This work starts with a mathematical description of the wind turbine, generators, grid, and SMC and its variants available in literature. A comprehensive literature review is tabulated that includes the proposed errors, sliding surfaces, typologies, and major outcomes. Moreover, a comparative analysis of the integer-order and fractional-order SMC and its variants is also presented in this paper. This paper will provide insight for the researcher working in the WECS and will serve them in the selection and exploration of the most appropriate control schemes for quality wind power extraction. The concise mathematical proofs of the IOSMC, FOSMC and their variants will also serve the researchers in selecting the relevant sliding surfaces control laws for their research tasks. This paper also provides a comparative analysis of IOSMC, FOSMC, and fuzzy-FOSMC in terms of chattering reduction, robustness, and computational complexities using mathematical theories, simulation carried out in Matlab/Simulink, and a processor in the loop (PIL)-based experimental environment. Full article

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15 pages, 5541 KiB

Open AccessArticle

A Theoretical and Numerical Study on Fractional Order Biological Models with Caputo Fabrizio Derivative

by Mati ur Rahman, Ali Althobaiti, Muhammad Bilal Riaz and Fuad S. Al-Duais

Fractal Fract. 2022, 6(8), 446; https://doi.org/10.3390/fractalfract6080446 - 17 Aug 2022

Cited by 10 | Viewed by 1262

Abstract

This article studies a biological population model in the context of a fractional Caputo-Fabrizio operator using double Laplace transform combined with the Adomian method. The conditions for the existence and uniqueness of solution of the problem under consideration is established with the use [...] Read more.

This article studies a biological population model in the context of a fractional Caputo-Fabrizio operator using double Laplace transform combined with the Adomian method. The conditions for the existence and uniqueness of solution of the problem under consideration is established with the use of the Banach principle and some theorems from fixed point theory. Furthermore, the convergence analysis is presented. For the accuracy and validation of the technique, some applications are presented. The numerical simulations present the obtained approximate solutions with a variety of fractional orders. From the numerical simulations, it is observed that when the fractional order is large, then the population density is also large; on the other hand, population density decreases with the decrease in the fractional order. The obtained results reveal that the considered technique is suitable and highly accurate in terms of the cost of computing, and can be used to analyze a wide range of complex non-linear fractional differential equations. Full article

(This article belongs to the Special Issue Mathematical Modelling of Real Phenomena Based on Fractional Derivatives)

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18 pages, 11037 KiB

Open AccessArticle

On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media

by Kamran, Shahzad Khan, Sharifah E. Alhazmi, Fahad M. Alotaibi, Massimiliano Ferrara and Ali Ahmadian

Fractal Fract. 2022, 6(8), 445; https://doi.org/10.3390/fractalfract6080445 - 17 Aug 2022

Cited by 4 | Viewed by 1102

Abstract

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in [...] Read more.

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method. Full article

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13 pages, 1544 KiB

Open AccessArticle

Novel Precise Solitary Wave Solutions of Two Time Fractional Nonlinear Evolution Models via the MSE Scheme

by Zillur Rahman, Alrazi Abdeljabbar, Harun-Or-Roshid and M. Zulfikar Ali

Fractal Fract. 2022, 6(8), 444; https://doi.org/10.3390/fractalfract6080444 - 17 Aug 2022

Cited by 16 | Viewed by 1358

Abstract

We construct soliton solutions of the complex time fractional Schrodinger model (tFSM), as well as the space–time fractional differential model (stFDM), leading wave spread through electrical transmission lines model (ETLM) in low pass with the help of modified simple equation scheme. The approach [...] Read more.

We construct soliton solutions of the complex time fractional Schrodinger model (tFSM), as well as the space–time fractional differential model (stFDM), leading wave spread through electrical transmission lines model (ETLM) in low pass with the help of modified simple equation scheme. The approach provides us with generalized rational exponential function solutions with some free parameters. A few well-known solitary wave resolutions are derived, starting from the generalized rational solutions selecting specific values of the free constants. The precise solutions acquired via the technique signify that the scheme is comparatively easier to execute and attractive in view of the results. No auxiliary equation is needed to solve any nonlinear fractional models by the scheme. Additionally, we observed that the numerical results are very encouraging for researchers conducting further research on stFDMs in mathematics and physics. Full article

(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)

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23 pages, 362 KiB

Open AccessArticle

An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels

by Xiaoxuan Jiang, Xuehua Yang, Haixiang Zhang and Qingqing Tian

Fractal Fract. 2022, 6(8), 443; https://doi.org/10.3390/fractalfract6080443 - 15 Aug 2022

Cited by 1 | Viewed by 1112

Abstract

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by [...] Read more.

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by using the Galerkin method based on piecewise linear test functions. The Riemann–Liouvile fractional integral terms are treated by convolution quadrature. In order to obtain a fully discrete method, the standard central difference approximation is used to discretize the spatial derivative. The stability and convergence are rigorously proved by the discrete energy method. In addition, the existence and uniqueness of numerical solutions for nonlinear systems are proved strictly. Additionally, we introduce and compare the Besse relaxation algorithm, the Newton iterative method, and the linearized iterative algorithm for solving the nonlinear systems. Numerical results confirm the theoretical analysis and show the effectiveness of the method. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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3 pages, 193 KiB

Open AccessEditorial

Editorial for Special Issue “Fractional Calculus Operators and the Mittag–Leffler Function”

by Maja Andrić

Fractal Fract. 2022, 6(8), 442; https://doi.org/10.3390/fractalfract6080442 - 14 Aug 2022

Viewed by 958

Abstract

Among the numerous applications of the theory of fractional calculus in almost all applied sciences, applications in numerical analysis and various fields of physics and engineering stand out [...] Full article

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)

18 pages, 978 KiB

Open AccessArticle

Asymptotic Synchronization of Fractional-Order Complex Dynamical Networks with Different Structures and Parameter Uncertainties

by Xiliang He, Tianzeng Li and Dehui Liu

Fractal Fract. 2022, 6(8), 441; https://doi.org/10.3390/fractalfract6080441 - 14 Aug 2022

Cited by 1 | Viewed by 1267

Abstract

This paper deals with the asymptotic synchronization of fractional-order complex dynamical networks with different structures and parameter uncertainties (FCDNDP). Firstly, the FCDNDP model is proposed by the Riemann–Liouville (R-L) fractional derivative. According to the property of fractional calculus and the Lyapunov direct method, [...] Read more.

This paper deals with the asymptotic synchronization of fractional-order complex dynamical networks with different structures and parameter uncertainties (FCDNDP). Firstly, the FCDNDP model is proposed by the Riemann–Liouville (R-L) fractional derivative. According to the property of fractional calculus and the Lyapunov direct method, an original controller is proposed to achieve the asymptotic synchronization of FCDNDP. Our controller is more adaptable and effective than those in other literature. Secondly, a sufficient condition is given for the asymptotic synchronization of FCDNDP based on the asymptotic stability theorem and the matrix inequality technique. Finally, the numerical simulations verify the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)

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17 pages, 362 KiB

Open AccessArticle

Approximate Controllability of Non-Instantaneous Impulsive Stochastic Evolution Systems Driven by Fractional Brownian Motion with Hurst Parameter

H \in (0, \frac{1}{2})

by Jiankang Liu, Wei Wei and Wei Xu

Fractal Fract. 2022, 6(8), 440; https://doi.org/10.3390/fractalfract6080440 - 13 Aug 2022

Cited by 4 | Viewed by 1173

Abstract

This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index

0 < H < 1 / 2

. First, to overcome the irregular [...] Read more.

This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index

0 < H < 1 / 2

. First, to overcome the irregular or singular properties of fBm with Hurst parameter

0 < H < 1 / 2

, we define a new type of control function. Then, by virtue of the stochastic analysis theory, inequality technique, the semigroup approach, Krasnoselskii’s fixed-point theorem and Schaefer’s fixed-point theorem, we derive two new sets of sufficient conditions for the existence and approximate controllability of the concerned system. In the end, a concrete example is worked out to demonstrate the applicability of our obtained results. Full article

17 pages, 499 KiB

Open AccessArticle

Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation

by Ying Ma and Lizhen Chen

Fractal Fract. 2022, 6(8), 439; https://doi.org/10.3390/fractalfract6080439 - 11 Aug 2022

Cited by 2 | Viewed by 1421

Abstract

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is [...] Read more.

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have

\min {2 - α, 2 - β}

-order convergence in time and spectral accuracy in space for smooth solutions, where

α, β

are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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15 pages, 376 KiB

Open AccessArticle

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

by Lijuan Nong, Qian Yi, Jianxiong Cao and An Chen

Fractal Fract. 2022, 6(8), 438; https://doi.org/10.3390/fractalfract6080438 - 11 Aug 2022

Cited by 4 | Viewed by 1330

Abstract

The time-fractional Cattaneo equation is an equation where the fractional order

α \in (1, 2)

has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation [...] Read more.

The time-fractional Cattaneo equation is an equation where the fractional order

α \in (1, 2)

has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1

_{σ}

formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1

_{σ}

difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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15 pages, 342 KiB

Open AccessArticle

On Sharp Estimate of Third Hankel Determinant for a Subclass of Starlike Functions

by Lei Shi, Muhammad Arif, Khalil Ullah, Naseer Alreshidi and Meshal Shutaywi

Fractal Fract. 2022, 6(8), 437; https://doi.org/10.3390/fractalfract6080437 - 11 Aug 2022

Cited by 4 | Viewed by 1108

Abstract

In our present investigation, a subclass of starlike function

S_{n - 1, L}^{*}

connected with a domain bounded by an epicycloid with

n - 1

cusps was considered. The main work is to investigate some coefficient inequalities, and second and [...] Read more.

In our present investigation, a subclass of starlike function

S_{n - 1, L}^{*}

connected with a domain bounded by an epicycloid with

n - 1

cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for

f \in S_{4 L}^{*}

with

\frac{z f^{'} (z)}{f (z)}

bounded by a four-leaf shaped domain under the unit disk

D

. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

16 pages, 392 KiB

Open AccessArticle

Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators

by Christopher N. Angstmann, Stuart-James M. Burney, Bruce I. Henry and Byron A. Jacobs

Fractal Fract. 2022, 6(8), 436; https://doi.org/10.3390/fractalfract6080436 - 11 Aug 2022

Viewed by 1414

Abstract

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to [...] Read more.

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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18 pages, 384 KiB

Open AccessArticle

(q₁,q₂)-Trapezium-Like Inequalities Involving Twice Differentiable Generalized m-Convex Functions and Applications

by Muhammad Uzair Awan, Muhammad Zakria Javed, Ibrahim Slimane, Artion Kashuri, Clemente Cesarano and Kamsing Nonlaopon

Fractal Fract. 2022, 6(8), 435; https://doi.org/10.3390/fractalfract6080435 - 10 Aug 2022

Cited by 2 | Viewed by 1033

Abstract

A new auxiliary result pertaining to twice (

q_{1}, q_{2}

)-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard’s inequality involving the class of generalized

m

-convex functions are obtained. Finally, to demonstrate the significance [...] Read more.

A new auxiliary result pertaining to twice (

q_{1}, q_{2}

)-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard’s inequality involving the class of generalized

m

-convex functions are obtained. Finally, to demonstrate the significance of the main outcomes, some applications are discussed for hypergeometric functions, Mittag–Leffler functions, and bounded functions. Full article

(This article belongs to the Section General Mathematics, Analysis)

11 pages, 301 KiB

Open AccessArticle

Study of a Fractional Creep Problem with Multiple Delays in Terms of Boltzmann’s Superposition Principle

by Amar Chidouh, Rahima Atmania and Delfim F. M. Torres

Fractal Fract. 2022, 6(8), 434; https://doi.org/10.3390/fractalfract6080434 - 10 Aug 2022

Viewed by 1073

Abstract

We study a class of nonlinear fractional differential equations with multiple delays, which is represented by the Voigt creep fractional model of viscoelasticity. We discuss two Voigt models, the first being linear and the second being nonlinear. The linear Voigt model give us [...] Read more.

We study a class of nonlinear fractional differential equations with multiple delays, which is represented by the Voigt creep fractional model of viscoelasticity. We discuss two Voigt models, the first being linear and the second being nonlinear. The linear Voigt model give us the physical interpretation and is associated with important results since the creep function characterizes the viscoelastic behavior of stress and strain. For the nonlinear model of Voigt, our theoretical study and analysis provides existence and stability, where time delays are expressed in terms of Boltzmann’s superposition principle. By means of the Banach contraction principle, we prove existence of a unique solution and investigate its continuous dependence upon the initial data as well as Ulam stability. The results are illustrated with an example. Full article

(This article belongs to the Section Mathematical Physics)

13 pages, 1184 KiB

Open AccessArticle

Parameter Estimation for Several Types of Linear Partial Differential Equations Based on Gaussian Processes

by Wenbo Zhang and Wei Gu

Fractal Fract. 2022, 6(8), 433; https://doi.org/10.3390/fractalfract6080433 - 08 Aug 2022

Cited by 2 | Viewed by 1292

Abstract

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to [...] Read more.

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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17 pages, 3050 KiB

Open AccessArticle

CORDIC-Based FPGA Realization of a Spatially Rotating Translational Fractional-Order Multi-Scroll Grid Chaotic System

by Wafaa S. Sayed, Merna Roshdy, Lobna A. Said, Norbert Herencsar and Ahmed G. Radwan

Fractal Fract. 2022, 6(8), 432; https://doi.org/10.3390/fractalfract6080432 - 07 Aug 2022

Cited by 4 | Viewed by 1628

Abstract

This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational [...] Read more.

This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational parameters. The rotated system is successfully utilized as a Pseudo-Random Number Generator (PRNG) in an image encryption scheme. It preserves the chaotic dynamics and exhibits continuous chaotic behavior for all values of the rotation angle. The Coordinate Rotation Digital Computer (CORDIC) algorithm is used to implement rotation and the Grünwald–Letnikov (GL) technique is used for solving the fractional-order system. CORDIC enables complete control and dynamic spatial rotation by providing real-time computation of the sine and cosine functions. The proposed hardware architectures are realized on a Field-Programmable Gate Array (FPGA) using the Xilinx ISE 14.7 on Artix 7 XC7A100T kit. The Intellectual-Property (IP)-core-based implementation generates sine and cosine functions with a one-clock-cycle latency and provides a generic framework for rotating any chaotic system given its system of differential equations. The achieved throughputs are

821.92

Mbits/s and

520.768

Mbits/s for two- and three-dimensional rotating chaotic systems, respectively. Because it is amenable to digital realization, the proposed spatially rotating translational fractional-order multi-scroll grid chaotic system can fit various secure communication and motion control applications. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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Fractal Fract., Volume 6, Issue 8 (August 2022) – 58 articles

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