Fractal and Fractional

17 pages, 499 KiB

Open AccessArticle

Analysis of Non-Local Integro-Differential Equations with Hadamard Fractional Derivatives: Existence, Uniqueness, and Stability in the Context of RLC Models

by Manigandan Murugesan, Saravanan Shanmugam, Mohamed Rhaima and Ragul Ravi

Fractal Fract. 2024, 8(7), 409; https://doi.org/10.3390/fractalfract8070409 (registering DOI) - 12 Jul 2024

Abstract

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, [...] Read more.

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, we convert the problem into an equivalent integro-differential equation. Additionally, we explore several versions of Ulam’s stability findings. Two numerical examples are provided to illustrate the applications of our main results. We also observe that modifications to the Hadamard fractional derivative lead to asymmetric outcomes. The study concludes with an applied example demonstrating the existence results derived from Schaefer’s fixed point theorem. These findings represent novel contributions to the literature on this topic, significantly advancing our understanding. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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26 pages, 556 KiB

Open AccessArticle

Fractional Hermite–Hadamard–Mercer-Type Inequalities for Interval-Valued Convex Stochastic Processes with Center-Radius Order and Their Related Applications in Entropy and Information Theory

by Ahsan Fareed Shah, Serap Özcan, Miguel Vivas-Cortez, Muhammad Shoaib Saleem and Artion Kashuri

Fractal Fract. 2024, 8(7), 408; https://doi.org/10.3390/fractalfract8070408 - 11 Jul 2024

Viewed by 120

Abstract

We propose a new definition of the

γ

-convex stochastic processes

(C SP)

using center and radius

(CR)

order with the notion of interval valued functions (

_{C . R}^{I . V}

). By utilizing this definition and [...] Read more.

We propose a new definition of the

γ

-convex stochastic processes

(C SP)

using center and radius

(CR)

order with the notion of interval valued functions (

_{C . R}^{I . V}

). By utilizing this definition and Mean-Square Fractional Integrals, we generalize fractional Hermite–Hadamard–Mercer-type inclusions for generalized

_{C . R}^{I . V}

versions of convex, tgs-convex, P-convex, exponential-type convex, Godunova–Levin convex, s-convex, Godunova–Levin s-convex, h-convex, n-polynomial convex, and fractional n-polynomial

(C SP)

. Also, our work uses interesting examples of

_{C . R}^{I . V} (C SP)

with Python-programmed graphs to validate our findings using an extension of Mercer’s inclusions with applications related to entropy and information theory. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)

21 pages, 62211 KiB

Open AccessArticle

Damage Law and Reasonable Width of Coal Pillar under Gully Area: Linking Fractal Characteristics of Coal Pillar Fractures to Their Stability

by Zhaopeng Wu, Yunpei Liang, Kaijun Miao, Qigang Li, Sichen Liu, Qican Ran, Wanjie Sun, Hualong Yin and Yun Ma

Fractal Fract. 2024, 8(7), 407; https://doi.org/10.3390/fractalfract8070407 - 11 Jul 2024

Viewed by 102

Abstract

The coal pillar is an important structure to control the stability of the roadway surrounding rock and maintain the safety of underground mining activities. An unreasonable design of the coal pillar size can result in the failure of the surrounding rock structure or [...] Read more.

The coal pillar is an important structure to control the stability of the roadway surrounding rock and maintain the safety of underground mining activities. An unreasonable design of the coal pillar size can result in the failure of the surrounding rock structure or waste of coal resources. The northern Shaanxi mining area of China belongs to the shallow buried coal seam mining in the gully area, and the gully topography makes the bearing law of the coal pillar and the development law of the internal fracture more complicated. In this study, based on the geological conditions of the Longhua Mine 20202 working face, a PFC^2D numerical model was established to study the damage characteristics of coal pillars under the different overlying strata base load ratios in the gentle terrain area and the different gully slope sections in the gully terrain area, and the coal pillar design strategy based on the fractal characteristics of the fractures was proposed to provide a reference for determining the width of the coal pillars in mines under similar geological conditions. The results show that the reliability of the mathematical equation between the overlying strata base load ratio and the fractal dimension of the fractures in the coal pillar is high, the smaller the overlying strata base load ratio is, the greater the damage degree of the coal pillar is, and the width of the coal pillar of 15 m under the condition of the actual overlying strata base load ratio (1.19) is more reasonable. Compared with the gentle terrain area, the damage degree of the coal pillar in the gully terrain area is larger, in which the fractal dimension of the fracture in the coal pillar located below the gully bottom is the smallest, and the coal pillar in the gully terrain should be set as far as possible to make the coal pillar located below the gully bottom, so as to ensure the stability of the coal pillar. Full article

(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)

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31 pages, 450 KiB

Open AccessArticle

Derivation of Closed-Form Expressions in Apéry-like Series Using Fractional Calculus and Applications

by Ampol Duangpan, Ratinan Boonklurb, Udomsak Rakwongwan and Phiraphat Sutthimat

Fractal Fract. 2024, 8(7), 406; https://doi.org/10.3390/fractalfract8070406 - 11 Jul 2024

Viewed by 140

Abstract

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their [...] Read more.

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their properties across various domains. The study focuses on establishing rigorous mathematical frameworks that unveil new insights into the behaviors of these series, contributing to a deeper understanding of number theory and mathematical analysis. Key results include proofs of convergence and divergence within specified intervals and the derivation of closed-form solutions through fractional integration and differentiation. This paper also introduces a method aimed at conjecturing mathematical constants through continued fractions as an application of our results. Finally, we provide the proof of validation for three unproven conjectures of continued fractions obtained from the Ramanujan Machine. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

12 pages, 326 KiB

Open AccessArticle

Fuzzy Differential Subordination for Classes of Admissible Functions Defined by a Class of Operators

by Ekram E. Ali, Miguel Vivas-Cortez and Rabha M. El-Ashwah

Fractal Fract. 2024, 8(7), 405; https://doi.org/10.3390/fractalfract8070405 - 11 Jul 2024

Viewed by 167

Abstract

This paper’s findings are related to geometric function theory (GFT). We employ one of the most recent methods in this area, the fuzzy admissible functions methodology, which is based on fuzzy differential subordination, to produce them. To do this, the relevant fuzzy admissible [...] Read more.

This paper’s findings are related to geometric function theory (GFT). We employ one of the most recent methods in this area, the fuzzy admissible functions methodology, which is based on fuzzy differential subordination, to produce them. To do this, the relevant fuzzy admissible function classes must first be defined. This work deals with fuzzy differential subordinations, ideas borrowed from fuzzy set theory and applied to complex analysis. This work examines the characteristics of analytic functions and presents a class of operators in the open unit disk

J_{η, ς}^{κ} (a, e, x)

for

ς > - 1, η > 0,

such that

a, e \in R, (e - a) \geq 0, a > - x

. The fuzzy differential subordination results are obtained using (GFT) concepts outside the field of complex analysis because of the operator’s compositional structure, and some relevant classes of admissible functions are studied by utilizing fuzzy differential subordination. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)

31 pages, 794 KiB

Open AccessArticle

Performance Analysis of Fully Intuitionistic Fuzzy Multi-Objective Multi-Item Solid Fractional Transportation Model

by Sultan Almotairi, Elsayed Badr, M. A. Elsisy, F. A. Farahat and M. A. El Sayed

Fractal Fract. 2024, 8(7), 404; https://doi.org/10.3390/fractalfract8070404 - 9 Jul 2024

Viewed by 263

Abstract

An investigation is conducted in this paper into a performance analysis of fully intuitionistic fuzzy multi-objective multi-item solid fractional transport model (FIF-MMSFTM). It is to be anticipated that the parameters of the conveyance model will be imprecise by virtue of numerous uncontrollable factors. [...] Read more.

An investigation is conducted in this paper into a performance analysis of fully intuitionistic fuzzy multi-objective multi-item solid fractional transport model (FIF-MMSFTM). It is to be anticipated that the parameters of the conveyance model will be imprecise by virtue of numerous uncontrollable factors. The model under consideration incorporates intuitionistic fuzzy (IF) quantities of shipments, costs and profit coefficients, supplies, demands, and transport. The FIF-MMSFTM that has been devised is transformed into a linear form through a series of operations. The accuracy function and ordering relations of IF sets are then used to reduce the linearized model to a concise multi-objective multi-item solid transportation model (MMSTM). Furthermore, an examination is conducted on several theorems that illustrate the correlation between the FIF-MMSFTM and its corresponding crisp model, which is founded upon linear, hyperbolic, and parabolic membership functions. A numerical example was furnished to showcase the efficacy and feasibility of the suggested methodology. The numerical data acquired indicates that the linear, hyperbolic, and parabolic models require fewer computational resources to achieve the optimal solution. The parabolic model has the greatest number of iterations, in contrast to the hyperbolic model which has the fewest. Additionally, the elapsed run time for the three models is a negligible amount of time: 0.2, 0.15, and 1.37 s, respectively. In conclusion, suggestions for future research are provided. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

5 pages, 174 KiB

Open AccessEditorial

Applications of Fractional-Order Calculus in Robotics

by Abhaya Pal Singh and Kishore Bingi

Fractal Fract. 2024, 8(7), 403; https://doi.org/10.3390/fractalfract8070403 - 6 Jul 2024

Viewed by 380

Abstract

Fractional calculus, a branch of mathematical analysis, extends traditional calculus that encompasses integrals and derivatives of non-integer orders [...] Full article

(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)

25 pages, 3439 KiB

Open AccessArticle

Split-Step Galerkin FE Method for Two-Dimensional Space-Fractional CNLS

by Xiaogang Zhu, Yaping Zhang and Yufeng Nie

Fractal Fract. 2024, 8(7), 402; https://doi.org/10.3390/fractalfract8070402 - 5 Jul 2024

Viewed by 431

Abstract

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in [...] Read more.

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in space, which avoids iteration at each time layer. The analysis of mass conservative and convergent properties for this split-step FE scheme is performed. To test its capability, some numerical tests and the simulation of the double solitons intersection and plane wave are carried out. The results and comparisons with the algorithm combined with Newton’s iteration illustrate its effectiveness and advantages in computational efficiency. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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18 pages, 3448 KiB

Open AccessArticle

Employing the Laplace Residual Power Series Method to Solve (1+1)- and (2+1)-Dimensional Time-Fractional Nonlinear Differential Equations

by Adel R. Hadhoud, Abdulqawi A. M. Rageh and Taha Radwan

Fractal Fract. 2024, 8(7), 401; https://doi.org/10.3390/fractalfract8070401 - 4 Jul 2024

Viewed by 264

Abstract

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional [...] Read more.

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms

E_{2}

and

E_{\infty}

, and compared to existing methods. Full article

(This article belongs to the Section Numerical and Computational Methods)

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14 pages, 284 KiB

Open AccessArticle

Certain Geometric Study Involving the Barnes–Mittag-Leffler Function

by Abdulaziz Alenazi and Khaled Mehrez

Fractal Fract. 2024, 8(7), 400; https://doi.org/10.3390/fractalfract8070400 - 4 Jul 2024

Viewed by 452

Abstract

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, [...] Read more.

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, some consequences and examples are presented. Full article

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

16 pages, 1702 KiB

Open AccessArticle

Influence of Local Thermodynamic Non-Equilibrium to Photothermally Induced Acoustic Response of Complex Systems

by Slobodanka Galovic, Aleksa I. Djordjevic, Bojan Z. Kovacevic, Katarina Lj. Djordjevic and Dalibor Chevizovich

Fractal Fract. 2024, 8(7), 399; https://doi.org/10.3390/fractalfract8070399 - 3 Jul 2024

Viewed by 354

Abstract

In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account [...] Read more.

In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed. Full article

(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)

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16 pages, 301 KiB

Open AccessArticle

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

by Yujun Cui, Chunyu Liang and Yumei Zou

Fractal Fract. 2024, 8(7), 398; https://doi.org/10.3390/fractalfract8070398 - 2 Jul 2024

Viewed by 245

Abstract

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given [...] Read more.

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

20 pages, 2586 KiB

Open AccessArticle

Robust Consensus Analysis in Fractional-Order Nonlinear Leader-Following Systems with Delays: Incorporating Practical Controller Design and Nonlinear Dynamics

by Asad Khan, Muhammad Awais Javeed, Azmat Ullah Khan Niazi, Saadia Rehman and Yubin Zhong

Fractal Fract. 2024, 8(7), 397; https://doi.org/10.3390/fractalfract8070397 - 2 Jul 2024

Viewed by 309

Abstract

This article investigates the resilient-based consensus analysis of fractional-order nonlinear leader-following systems with distributed and input lags. To enhance the practicality of the controller design, an incorporation of a disturbance term is proposed. Our modeling framework provides a more precise and flexible approach [...] Read more.

This article investigates the resilient-based consensus analysis of fractional-order nonlinear leader-following systems with distributed and input lags. To enhance the practicality of the controller design, an incorporation of a disturbance term is proposed. Our modeling framework provides a more precise and flexible approach that considers the memory and heredity aspects of agent dynamics through the utilization of fractional calculus. Furthermore, the leader and follower equations of the system incorporate nonlinear functions to explore the resulting changes. The leader-following system is expressed by a weighted graph, which can be either undirected or directed. Analyzed using algebraic graph theory and the fractional-order Razumikhin technique, the case of leader-following consensus is presented algebraically. To increase robustness in multi-agent systems, input and distributive delays are used to accommodate communication delays and replicate real-time varying environments. This study lays the groundwork for developing control methods that are more robust and flexible in complex networked systems. It does so by advancing our understanding and practical application of fractional-order multi-agent systems. Additionally, experiments were conducted to show the effectiveness of the design in achieving consensus within the system. Full article

(This article belongs to the Special Issue New Insights and Trends in Fractional-Order Dynamical Networks, Their Control and Synchronization)

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15 pages, 384 KiB

Open AccessArticle

A New Fractional-Order Grey Prediction Model without a Parameter Estimation Process

by Yadong Wang and Chong Liu

Fractal Fract. 2024, 8(7), 396; https://doi.org/10.3390/fractalfract8070396 - 2 Jul 2024

Viewed by 304

Abstract

The fractional-order grey prediction model is widely recognized for its performance in time series prediction tasks with small sample characteristics. However, its parameter-estimation method, namely the least squares method, limits the predictive performance of the model and requires time to address the ill-conditioning [...] Read more.

The fractional-order grey prediction model is widely recognized for its performance in time series prediction tasks with small sample characteristics. However, its parameter-estimation method, namely the least squares method, limits the predictive performance of the model and requires time to address the ill-conditioning of the system. To address these issues, this paper proposes a novel parameter-acquisition method treating structural parameters as hyperparameters, obtained through the marine predators optimization algorithm. The experimental analysis on three datasets validate the effectiveness of the method proposed in this paper. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)

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15 pages, 5823 KiB

Open AccessArticle

Analytical Scheme for Time Fractional Kawahara and Modified Kawahara Problems in Shallow Water Waves

by Muhammad Nadeem, Asad Khan, Muhammad Awais Javeed and Zhong Yubin

Fractal Fract. 2024, 8(7), 395; https://doi.org/10.3390/fractalfract8070395 - 2 Jul 2024

Viewed by 373

Abstract

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) [...] Read more.

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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15 pages, 613 KiB

Open AccessArticle

The Stability of a Tumor–Macrophages Model with Caputo Fractional Operator

by Kaushik Dehingia and Salah Boulaaras

Fractal Fract. 2024, 8(7), 394; https://doi.org/10.3390/fractalfract8070394 - 30 Jun 2024

Viewed by 311

Abstract

This study proposes a fractional-order model in the Caputo sense to describe the interaction between tumor and immune macrophages by assuming that the pro-tumor macrophages induce a Holling type-II response to the tumor. Then, the basic properties of the solutions to the model [...] Read more.

This study proposes a fractional-order model in the Caputo sense to describe the interaction between tumor and immune macrophages by assuming that the pro-tumor macrophages induce a Holling type-II response to the tumor. Then, the basic properties of the solutions to the model are studied. Local stability analysis is conducted at each of the equilibria in the model, and a numerical study is performed with varying activation rates of type-II or pro-tumor macrophages and the order of the fractional operator. The numerical findings suggest that type-I or anti-tumor macrophages can stabilize the system if the activation rate of type-II or pro-tumor macrophages is low. Still, for a higher value of the activation rate for type-II or pro-tumor macrophages, the proliferation of tumor cells is uncontrollable and the system becomes unstable. Furthermore, the stability of the system decreases as the order of the fractional operator increases. Full article

(This article belongs to the Section Engineering)

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21 pages, 9483 KiB

Open AccessArticle

Strength and Fractal Characteristics of Artificial Frozen–Thawed Sandy Soft Soil

by Bowen Kong, Yuntian Yan, Huan He, Jing Yu, Baoping Zou and Qizhi Chen

Fractal Fract. 2024, 8(7), 393; https://doi.org/10.3390/fractalfract8070393 - 29 Jun 2024

Viewed by 411

Abstract

In regions with sandy soft soil strata, the subway foundation commonly undergoes freeze–thaw cycles during construction. This study focuses on analyzing the microstructural and fractal characteristics of frozen–thawed sandy soft soil to improve our understanding of its strength behavior and stability. Pore size [...] Read more.

In regions with sandy soft soil strata, the subway foundation commonly undergoes freeze–thaw cycles during construction. This study focuses on analyzing the microstructural and fractal characteristics of frozen–thawed sandy soft soil to improve our understanding of its strength behavior and stability. Pore size distribution curves before and after freeze–thaw cycles were examined using nuclear magnetic resonance technology. Additionally, fractal theory was applied to illustrate the soil’s fractal properties. The strength properties of frozen remolded clay under varying freezing temperatures and sand contents were investigated through uniaxial compression tests, indicating that soil strength is significantly influenced by fractal dimensions. The findings suggest that lower freezing temperatures lead to a more dispersed soil skeleton, resulting in a higher fractal dimension for the frozen–thawed soil. Likewise, an increase in sand content enlarges the soil pores and the fractal dimension of the frozen–thawed soil. Furthermore, an increase in fractal dimension caused by freezing temperatures results in increased soil strength, while an increase in fractal dimension due to changes in sand content leads to a decrease in soil strength. Full article

(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)

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21 pages, 7253 KiB

Open AccessArticle

Modeling and Control Research of Fractional-Order Cascaded H-Bridge Multilevel STATCOM

by Junhua Xu, Songqin Tang, Guopeng He, Zheng Gong, Guangqing Lin and Jiayu Liu

Fractal Fract. 2024, 8(7), 392; https://doi.org/10.3390/fractalfract8070392 - 29 Jun 2024

Viewed by 266

Abstract

This paper introduces fractional-order capacitors and fractional-order inductors into the conventional integer-order cascaded H-bridge multilevel static compensator (ICHM-STATCOM), thereby constructing the main circuit of the fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM). Mechanism-based modeling is employed to establish switching function models and low-frequency [...] Read more.

This paper introduces fractional-order capacitors and fractional-order inductors into the conventional integer-order cascaded H-bridge multilevel static compensator (ICHM-STATCOM), thereby constructing the main circuit of the fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM). Mechanism-based modeling is employed to establish switching function models and low-frequency dynamic models for the FCHM-STATCOM in the three-phase stationary coordinate system (a-b-c). Subsequently, fractional-order rotating coordinate transformation is introduced to establish the mathematical model of the FCHM-STATCOM in the synchronous rotating coordinate system (d-q). Additionally, a fractional-order proportional-integral (FOPI)-based fractional-order dual closed-loop current decoupling control strategy is proposed. Finally, this paper validates the correctness of the established mathematical models through digital simulation. Moreover, the simulation results demonstrate that by appropriately selecting the order of fractional-order capacitors and fractional-order inductors, the FCHM-STATCOM exhibits superior dynamic and static characteristics compared to the conventional ICHM-STATCOM, and the FCHM-STATCOM provides a more flexible reactive power compensation solution for power systems. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing, 2nd Edition)

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18 pages, 324 KiB

Open AccessArticle

Multiplicity of Normalized Solutions to a Fractional Logarithmic Schrödinger Equation

by Yan-Cheng Lv and Gui-Dong Li

Fractal Fract. 2024, 8(7), 391; https://doi.org/10.3390/fractalfract8070391 (registering DOI) - 29 Jun 2024

Viewed by 250

Abstract

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation

{(- Δ)}^{s} u + V (ϵ x) u = λ u + u log u^{2} in R^{N},

under the mass [...] Read more.

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation

{(- Δ)}^{s} u + V (ϵ x) u = λ u + u log u^{2} in R^{N},

under the mass constraint

\int_{R^{N}} {| u |}^{2} d x = a .

Here,

N \geq 2

,

a, ϵ > 0

,

λ \in R

is an unknown parameter,

{(- Δ)}^{s}

is the fractional Laplacian and

s \in (0, 1)

. We introduce a function space where the energy functional associated with the problem is of class

C^{1}

. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum. Full article

(This article belongs to the Special Issue Variational Problems and Fractional Differential Equations)

10 pages, 275 KiB

Open AccessArticle

Time-Stepping Error Estimates of Linearized Grünwald–Letnikov Difference Schemes for Strongly Nonlinear Time-Fractional Parabolic Problems

by Hongyu Qin, Lili Li, Yuanyuan Li and Xiaoli Chen

Fractal Fract. 2024, 8(7), 390; https://doi.org/10.3390/fractalfract8070390 - 29 Jun 2024

Viewed by 278

Abstract

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G-L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through [...] Read more.

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G-L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

44 pages, 24273 KiB

Open AccessArticle

A Piecewise Linear Approach for Implementing Fractional-Order Multi-Scroll Chaotic Systems on ARMs and FPGAs

by Daniel Clemente-López, Jesus M. Munoz-Pacheco, Ernesto Zambrano-Serrano, Olga G. Félix Beltrán and Jose de Jesus Rangel-Magdaleno

Fractal Fract. 2024, 8(7), 389; https://doi.org/10.3390/fractalfract8070389 - 29 Jun 2024

Viewed by 287

Abstract

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators [...] Read more.

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and

4 \times 4

-grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

32 pages, 2592 KiB

Open AccessArticle

Kink, Dark, Bright, and Singular Optical Solitons to the Space–Time Nonlinear Fractional (4+1)-Dimensional Davey–Stewartson–Kadomtsev–Petviashvili Model

by Abdulaziz Khalid Alsharidi and Moin-ud-Din Junjua

Fractal Fract. 2024, 8(7), 388; https://doi.org/10.3390/fractalfract8070388 - 29 Jun 2024

Viewed by 364

Abstract

The new types of exact solitons of the space–time fractional nonlinear (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) model are achieved by applying the unified technique and modified extended tanh-expansion function technique. A novel definition of the fractional derivative known as the truncated M-fractional derivative is also [...] Read more.

The new types of exact solitons of the space–time fractional nonlinear (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) model are achieved by applying the unified technique and modified extended tanh-expansion function technique. A novel definition of the fractional derivative known as the truncated M-fractional derivative is also used. This model describes both the non-elastic and elastic interactions between internal waves. This model is used to represent intricate nonlinear phenomena like shallow-water waves, plasma physics, and others. The obtained results are in the form of kink, singular, bright, periodic, and dark solitons. The observed results are verified and represented by 2D and 3D graphs. The observed results are not present in the literature due to the use of fractional derivatives. The impact of the truncated M-fractional derivative on the observed results is also represented by graphs. Hence, our observed results are fruitful for the future study of these models. The applied techniques are simple, fruitful, and reliable in solving the other models in applied mathematics. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

20 pages, 42171 KiB

Open AccessArticle

Fractal Characteristics and Energy Evolution Analysis of Rocks under True Triaxial Unloading Conditions

by Cheng Pan, Chongyan Liu, Guangming Zhao, Wei Yuan, Xiao Wang and Xiangrui Meng

Fractal Fract. 2024, 8(7), 387; https://doi.org/10.3390/fractalfract8070387 - 28 Jun 2024

Viewed by 294

Abstract

To investigate the mechanical properties and energy evolution laws of rocks under true triaxial unloading conditions, a study was conducted using a true triaxial rock testing system on three different types of rocks: coal, sandy mudstone, and siltstone. The study examined the mechanical [...] Read more.

To investigate the mechanical properties and energy evolution laws of rocks under true triaxial unloading conditions, a study was conducted using a true triaxial rock testing system on three different types of rocks: coal, sandy mudstone, and siltstone. The study examined the mechanical behavior, failure patterns, and fractal dimensions of these rocks under true triaxial unloading conditions. The tests revealed significant variations in stress–strain curves and failure patterns among the different rock types. Observation indicated that rocks with lower peak strength exhibited higher fractal dimensions and increased fragmentation upon failure. Subsequently, based on the experimental data of siltstone, the impact of the unloading rate and particle size distribution on the energy evolution under true triaxial single-sided unloading paths was further investigated using the three-dimensional particle flow software PFC3D 6.0, revealing the micro-mechanisms of rock energy evolution. The study revealed that when the initial stress unloading level was low, the total energy and strain energy at the peak strength exhibited a strong linear relationship with the unloading rate. Before the stress peak, the dissipative energy was mainly composed of frictional energy. After the stress peak, the dissipative energy consisted of frictional energy, damping energy, and kinetic energy. The heterogeneity of rock significantly affected the distribution of dissipative energy, with an increase in rock heterogeneity leading to a decrease in frictional energy and an increase in kinetic energy. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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27 pages, 1651 KiB

Open AccessArticle

PQMLE and Generalized F-Test of Random Effects Semiparametric Model with Serially and Spatially Correlated Nonseparable Error

by Shuangshuang Li, Jianbao Chen and Danqing Chen

Fractal Fract. 2024, 8(7), 386; https://doi.org/10.3390/fractalfract8070386 - 28 Jun 2024

Viewed by 199

Abstract

Semiparametric panel data models are powerful tools for analyzing data with complex characteristics such as linearity and nonlinearity of covariates. This study aims to investigate the estimation and testing of a random effects semiparametric model (RESPM) with serially and spatially correlated nonseparable error, [...] Read more.

Semiparametric panel data models are powerful tools for analyzing data with complex characteristics such as linearity and nonlinearity of covariates. This study aims to investigate the estimation and testing of a random effects semiparametric model (RESPM) with serially and spatially correlated nonseparable error, utilizing a combination of profile quasi-maximum likelihood estimation and local linear approximation. Profile quasi-maximum likelihood estimators (PQMLEs) for unknowns and a generalized F-test statistic

F_{N T}

are built to determine the beingness of nonlinear relationships. The asymptotic properties of PQMLEs and

F_{N T}

are proven under regular assumptions. The Monte Carlo results imply that the PQMLEs and

F_{N T}

performances are excellent on finite samples; however, missing the spatially and serially correlated error leads to estimator inefficiency and bias. Indonesian rice-farming data is used to illustrate the proposed approach, and indicates that

l a n d a r e a

exhibits a significant nonlinear relationship with

r i c e y i e l d

, in addition,

h i g h

-

y i e l d v a r i e t i e s

,

m i x e d

-

y i e l d

v a r i e t i e s

, and

s e e d w e i g h t

have significant positive impacts on rice yield. Full article

(This article belongs to the Special Issue Fractional Models and Statistical Applications)

16 pages, 4634 KiB

Open AccessArticle

Actor-Critic Neural-Network-Based Fractional-Order Sliding Mode Control for Attitude Tracking of Spacecraft with Uncertainties and Actuator Faults

by Chenghu Jing, Xiaole Ma, Kun Zhang, Yanfeng Wang, Bingsheng Yan and Yanbo Hui

Fractal Fract. 2024, 8(7), 385; https://doi.org/10.3390/fractalfract8070385 - 28 Jun 2024

Viewed by 251

Abstract

This paper investigates the attitude control of rigid spacecraft in the presence of uncertainties, disturbances, and actuator faults. In order to effectively address these challenges and improve the performance of the system, a novel actor-critic neural-network-based fractional-order sliding mode control (ACNNFOSMC) has been [...] Read more.

This paper investigates the attitude control of rigid spacecraft in the presence of uncertainties, disturbances, and actuator faults. In order to effectively address these challenges and improve the performance of the system, a novel actor-critic neural-network-based fractional-order sliding mode control (ACNNFOSMC) has been developed for spacecraft. The integration of actor-critic neural network, fractional-order theory, and sliding mode control enables dual functionality: the actor-critic neural network serves to approximate the aggregate of uncertain parameters, disturbances, and actuator faults, thereby facilitating their compensation, while the fractional-order sliding mode control mechanism significantly improves the system’s tracking precision and overall robustness against uncertainties. Theoretical analyses are presented to analyze the stability of the proposed control framework. Thorough examination via simulation experiments affirms the effectiveness and control precision of attitude of our proposed control strategy, even in complex operational scenarios. Full article

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24 pages, 349 KiB

Open AccessArticle

Solving Fractional Random Differential Equations by Using Fixed Point Methodologies under Mild Boundary Conditions

by Hasanen A. Hammad and Saleh Fahad Aljurbua

Fractal Fract. 2024, 8(7), 384; https://doi.org/10.3390/fractalfract8070384 - 28 Jun 2024

Viewed by 258

Abstract

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we [...] Read more.

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results. Full article

(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)

15 pages, 3651 KiB

Open AccessArticle

Dynamical Behavior of the Fractional BBMB Equation on Unbounded Domain

by Wei Zhang, Haijing Wang, Haolu Zhang, Zhiyuan Li and Xiaoyu Li

Fractal Fract. 2024, 8(7), 383; https://doi.org/10.3390/fractalfract8070383 - 28 Jun 2024

Viewed by 225

Abstract

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order [...] Read more.

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

15 pages, 310 KiB

Open AccessArticle

Certain Properties and Characterizations of Two-Iterated Two-Dimensional Appell and Related Polynomials via Fractional Operators

by Mohra Zayed and Shahid Ahmad Wani

Fractal Fract. 2024, 8(7), 382; https://doi.org/10.3390/fractalfract8070382 - 28 Jun 2024

Viewed by 248

Abstract

This paper introduces the operational rule for 2-iterated 2D Appell polynomials and derives its generalized form using fractional operators. It also presents the generating relation and explicit forms that characterize the generalized 2-iterated 2D Appell polynomials. Additionally, it establishes the monomiality principle for [...] Read more.

This paper introduces the operational rule for 2-iterated 2D Appell polynomials and derives its generalized form using fractional operators. It also presents the generating relation and explicit forms that characterize the generalized 2-iterated 2D Appell polynomials. Additionally, it establishes the monomiality principle for these polynomials and obtains their recurrence relations. The paper also establishes corresponding results for the generalized 2-iterated 2D Bernoulli, 2-iterated 2D Euler, and 2-iterated 2D Genocchi polynomials. Full article

(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)

14 pages, 297 KiB

Open AccessArticle

Existence of Positive Solutions for Non-Local Magnetic Fractional Systems

by Tahar Bouali, Rafik Guefaifia, Salah Boulaaras and Taha Radwan

Fractal Fract. 2024, 8(7), 381; https://doi.org/10.3390/fractalfract8070381 - 27 Jun 2024

Viewed by 296

Abstract

In this paper, the existence of a weak positive solution for non-local magnetic fractional systems is studied in the fractional magnetic Sobolev space through a sub-supersolution method combined with iterative techniques. Full article

14 pages, 317 KiB

Open AccessArticle

A Fractional Magnetic System with Critical Nonlinearities

by Libo Yang, Shapour Heidarkhani and Jiabin Zuo

Fractal Fract. 2024, 8(7), 380; https://doi.org/10.3390/fractalfract8070380 - 27 Jun 2024

Viewed by 291

Abstract

In the present paper, we investigate a fractional magnetic system involving critical concave–convex nonlinearities with Laplace operators. Specifically,

{(- Δ)}_{A}^{s} u_{1} = λ_{1} {| u_{1} |}^{q - 2} u_{1}

+ [...] Read more.

In the present paper, we investigate a fractional magnetic system involving critical concave–convex nonlinearities with Laplace operators. Specifically,

{(- Δ)}_{A}^{s} u_{1} = λ_{1} {| u_{1} |}^{q - 2} u_{1}

+

\frac{2 α_{1}}{α_{1} + β_{1}} | u_{1} |^{α_{1} - 2} u_{1} {| u_{2} |}^{β_{1}}

in

Ω

,

{(- Δ)}_{A}^{s} u_{2} = λ_{2} | u_{2} |^{q - 2} u_{2} + \frac{2 β_{1}}{α_{1} + β_{1}} | u_{2} |^{β_{1} - 2} u_{2} {| u_{1} |}^{α_{1}}

in

Ω

,

u_{1} = u_{2} = 0

in

R^{n} ∖ Ω

, where

Ω

is a bounded set with Lipschitz boundary

\partial Ω

in

R^{n}

,

1 < q < 2 < \frac{n}{s}

with

s \in (0, 1)

,

λ_{1}, λ_{2}

are two real positive parameters,

α_{1} > 1, β_{1} > 1

,

α_{1} + β_{1} = 2_{s}^{*} = \frac{2 n}{n - 2 s}

,

2_{s}^{*}

is the fractional critical Sobolev exponent, and

{(- Δ)}_{A}^{s}

is a fractional magnetic Laplace operator. By using Lusternik–Schnirelmann’s theory, we prove the existence result of infinitely many solutions for the magnetic fractional system. Full article

(This article belongs to the Special Issue Nonlinear Equations Driven by Fractional Laplacian Operators)

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Fractal Fract., Volume 8, Issue 7 (July 2024) – 45 articles

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