Fractal and Fractional

25 pages, 4931 KB

Open AccessArticle

Optical Multi-Peakon Dynamics in the Fractional Cubic–Quintic Nonlinear Pulse Propagation Model Using a Novel Integral Approach

by Ejaz Hussain, Aljethi Reem Abdullah, Khizar Farooq and Usman Younas

Fractal Fract. 2025, 9(10), 631; https://doi.org/10.3390/fractalfract9100631 (registering DOI) - 28 Sep 2025

Abstract

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, [...] Read more.

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, and self-focusing, arising from the balance between cubic and quintic nonlinearities. By employing the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method, we derive an extensive catalog of analytic solutions, multi-peakon structures, lump solitons, kinks, and bright and dark solitary waves, while periodic and singular solutions emerge as special cases. These outcomes are systematically constructed within a single framework and visualized through 2D, 3D, and contour plots under both anomalous and normal dispersion regimes. The analysis also addresses modulation instability (MI), interpreted as a sideband amplification of continuous-wave backgrounds that generates pulse trains and breather-type structures. Our results demonstrate that cubic–quintic contributions substantially affect MI gain spectrum, broadening instability bands and permitting MI beyond the anomalous-dispersion regime. These findings directly connect the obtained solution classes to experimentally observed routes for solitary wave shaping, pulse propagation, and instability and instability-driven waveform formation in optical communication devices, photonic platforms, and laser technologies. Full article

(This article belongs to the Special Issue Recent Computational Methods for Fractal and Fractional Nonlinear Partial Differential Equations)

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17 pages, 16129 KB

Open AccessArticle

Analytical Study of Soliton Solutions and Modulation Instability Analysis in the M-Truncated Fractional Coupled Ivancevic Option-Pricing Model

by Muhammad Bilal, Aljethi Reem Abdullah, Shafqat Ur Rehman and Usman Younas

Fractal Fract. 2025, 9(10), 630; https://doi.org/10.3390/fractalfract9100630 (registering DOI) - 27 Sep 2025

Abstract

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of [...] Read more.

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of novel exact soliton solutions, including bright, singular, dark, combined dark–bright, singular-periodic, complex solitons, exponential and Jacobi elliptic functions. A detailed analysis of option price wave functions and modulation instability analysis is conducted, with the conditions for valid solutions outlined. Additionally, a mathematical framework is established to capture market price fluctuations. Numerical simulations, illustrated through 2D, 3D and contour graphs, highlight the effects of parameter variations. Our findings demonstrate the effectiveness of the coupled Ivancevic model as a fractional nonlinear wave system, providing valuable insights into stock volatility and returns. This study contributes to creating new option-pricing models, which affect financial market analysis and risk management. Full article

16 pages, 11267 KB

Open AccessArticle

Seepage Characteristics and Critical Scale in Gas-Bearing Coal Pores Under Water Injection: A Multifractal Approach

by Qifeng Jia, Xiaoming Ni, Jingshuo Zhang, Bo Li, Lang Liu and Jingyu Wang

Fractal Fract. 2025, 9(10), 629; https://doi.org/10.3390/fractalfract9100629 (registering DOI) - 27 Sep 2025

Abstract

To investigate the flow characteristics of movable water in coal under the influence of micro-nano pore fractures with multiple fractal structures, this study employed nuclear magnetic resonance (NMR) and multifractal theory to analyze gas–water seepage under different injection pressures. Then, the scale threshold [...] Read more.

To investigate the flow characteristics of movable water in coal under the influence of micro-nano pore fractures with multiple fractal structures, this study employed nuclear magnetic resonance (NMR) and multifractal theory to analyze gas–water seepage under different injection pressures. Then, the scale threshold for mobile water entering coal pores and fractures was determined by clarifying the relationship among “injection pressure-T₂ dynamic multiple fractal parameter seepage resistance-critical pore scale”. The results indicate that coal samples from Yiwu (YW) and Wuxiang (WX) enter the nanoscale pore size range at an injection pressure of 8 MPa, while the coal sample from Malan (ML) enters the nanoscale pore size range at an injection pressure of 9 MPa. During the water injection process, there is a significant linear relationship between the multiple fractal parameters log X(q, ε) and log(ε) of the sample. The generalized fractal dimension D(q) decreases monotonically with increasing q in an inverse S-shape. This decrease occurs in two distinct stages: D(q) decreases rapidly in the low probability interval q < 0; D(q) decreases slowly in the high probability interval q > 0. The multiple fractal singularity spectrum function f(α) has an asymmetric upward parabolic convex function relationship with α, which is divided into a rapidly increasing left branch curve and a slowly decreasing right branch curve with α₀ as the boundary. Supporting evidence indicates the feasibility of a methodology for identifying the variation in multiple fractal parameters of gas–water NMR seepage and the critical scale transition conditions. This investigation establishes a methodological foundation for analyzing gas–water transport pathways within porous media materials. Full article

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16 pages, 319 KB

Open AccessArticle

A φ-Contractivity and Associated Fractal Dimensions

by Nifeen H. Altaweel, Olayan Albalawi and Razan Albalawi

Fractal Fract. 2025, 9(10), 628; https://doi.org/10.3390/fractalfract9100628 - 26 Sep 2025

Abstract

In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function

φ

, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power [...] Read more.

In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function

φ

, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power relative to the associated measure power, and where the gauge is a set-valued function or a measure in the majority of cases, we no longer assume this hypothesis. The introduced variant generalizes many existing cases, such as Haudorff, packing, Carathéodory, and Billingsley original variants. Many characteristics of the dimensions are investigated, such as bijectivity, convexity, monotony, asymptotic behavior, and fixed points. Full article

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

29 pages, 666 KB

Open AccessArticle

Super-Quadratic Stochastic Processes with Fractional Inequalities and Their Applications

by Yuanheng Wang, Usama Asif, Muhammad Zakria Javed, Muhammad Uzair Awan, Artion Kashuri and Omar Mutab Alsalami

Fractal Fract. 2025, 9(10), 627; https://doi.org/10.3390/fractalfract9100627 (registering DOI) - 26 Sep 2025

Abstract

The theory of stochastic processes is the prominent part of advanced probability theory and very influential in various mathematical models having randomness. One of the potential aspects is to investigate the stochastic convex processes. Working in the following direction, this study explores the [...] Read more.

The theory of stochastic processes is the prominent part of advanced probability theory and very influential in various mathematical models having randomness. One of the potential aspects is to investigate the stochastic convex processes. Working in the following direction, this study explores the set-valued super-quadratic processes through a unified approach under the centre-radius order relation, which is a totally ordered relation. First, we discuss some captivating properties and important results, which serve as a criterion. Relying on the newly proposed class of super-quadratic processes, we develop several fundamental inequalities within the fractional framework. Moreover, we present some novel deductions to complement the theoretical results with the existing literature. Also, we have provided the graphical breakdown, applications to the means, information theory, and divergence measures of the main inequalities. Full article

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

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18 pages, 2622 KB

Open AccessArticle

Phase-Based Fractional-Order Repetitive Control for Single-Phase Grid-Tied Inverters

by Qiangsong Zhao, Hao Dong, Guohui Zhou and Yongqiang Ye

Fractal Fract. 2025, 9(10), 626; https://doi.org/10.3390/fractalfract9100626 - 26 Sep 2025

Abstract

A novel fractional-order repetitive control based on phase angle information interpolation is proposed for single-phase LCL-type inverters in this paper. Conventional fractional-order repetitive control typically relies on inaccurate grid frequency information detected by a phase-locked loop or the frequency-locked loop, which may result [...] Read more.

A novel fractional-order repetitive control based on phase angle information interpolation is proposed for single-phase LCL-type inverters in this paper. Conventional fractional-order repetitive control typically relies on inaccurate grid frequency information detected by a phase-locked loop or the frequency-locked loop, which may result in a potential degradation in harmonics suppression capability. To address this issue, phase information is investigated to implement the fractional order of the repetitive controller through the linear interpolation method. A major advantage of the proposed scheme lies in that it avoids explicit frequency calculation and reduces sensitivity to frequency estimation fluctuations compared with conventional fractional-order repetitive control, enhancing its frequency adaptability. The stability analysis and the design process for the proposed scheme based on a plug-in-type repetitive control are given. Experimental results support the efficacy and advantages of the proposed control strategy. Full article

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15 pages, 514 KB

Open AccessArticle

Representation of Solutions and Ulam–Hyers Stability of the Two-Sided Fractional Matrix Delay Differential Equations

by Taoyu Yang and Mengmeng Li

Fractal Fract. 2025, 9(10), 625; https://doi.org/10.3390/fractalfract9100625 - 25 Sep 2025

Abstract

This paper investigates linear two-sided fractional matrix delay differential equations (TSFMDDE). Firstly, the two-sided fractional delayed Mittag-Leffler matrix functions (TSFDMLMF) are constructed. Further, the representation of solutions of two-sided homogeneous and nonhomogeneous problems are studied, and Ulam–Hyers (UH) stability of a two-sided nonhomogeneous [...] Read more.

This paper investigates linear two-sided fractional matrix delay differential equations (TSFMDDE). Firstly, the two-sided fractional delayed Mittag-Leffler matrix functions (TSFDMLMF) are constructed. Further, the representation of solutions of two-sided homogeneous and nonhomogeneous problems are studied, and Ulam–Hyers (UH) stability of a two-sided nonhomogeneous problem is discussed. Lastly, we provide a numerical example to demonstrate our results. In the numerical example, the fractional order

β = 0.6

, delay

ϱ = 2

, UH constant

u_{h} \approx 5.92479

,

n = 2

, and

s \in [- 2, 4]

. Full article

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17 pages, 5183 KB

Open AccessArticle

Multi-Scale Damage Evolution of Soil-Rock Mixtures Under Freeze–Thaw Cycles: Revealed by Electrochemical Impedance Spectroscopy Testing and Fractal Theory

by Junren Deng, Lei Wang, Guanglin Tian and Hongwei Deng

Fractal Fract. 2025, 9(10), 624; https://doi.org/10.3390/fractalfract9100624 - 25 Sep 2025

Abstract

The response of the microscopic structure and macroscopic mechanical parameters of SRM under F–T cycles is a key factor affecting the safety and stability of engineering projects in cold regions. In this study, F–T tests, EIS, and uniaxial compression tests were conducted on [...] Read more.

The response of the microscopic structure and macroscopic mechanical parameters of SRM under F–T cycles is a key factor affecting the safety and stability of engineering projects in cold regions. In this study, F–T tests, EIS, and uniaxial compression tests were conducted on SRM. The construct equivalent model of different conductive paths based on EIS was constructed. A peak strength prediction model was developed using characteristic parameters derived from the equivalent models, thereby revealing the mechanism by which F–T cycles influenced both microscopic structure and macroscopic strength. The results showed that with increasing cycles, both

R_{C P}

and

R_{C P P}

exhibited an exponential decreasing trend, whereas C_DSRP and D_f increased exponentially. Peak strength and peak secant modulus decreased exponentially, but peak strain increased exponentially. The expansion and interconnection of pores with different radii within

C P P

and

C P

caused smaller pores to evolve into larger ones while generating new pores, which led to a decline in

R_{C P P}

and

R_{C P}

. Moreover, this expansion enlarged the soil–rock contact area by connecting adjacent gas-phase pores and promoted the transformation of

C S R P P

into

D S R P P

, enhancing the parallel-plate capacitance effect and resulting in an increase in

C_{D S R P}

. Moreover, the interconnection increased the roughness of soil–soil and soil–rock contact surfaces, leading to a rising trend in

D_{f}

. The combined influence of

C_{D S R P}

and

D_{f}

yielded a strength prediction model with higher correlation than a single factor, providing more accurate predictions of UCS. However, the increases in

C_{D S R P}

and

D_{f}

induced by F–T cycles also contributed to microscopic structure damage and strength deterioration, reducing the load-bearing capacity and ultimately causing a decline in UCS. Full article

(This article belongs to the Special Issue Applications of Fractal Analysis in Structural Geology)

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22 pages, 6052 KB

Open AccessArticle

Dynamics of Complex Systems and Their Associated Attractors in a Multifractal Paradigm of Motion

by Vlad Ghizdovat, Monica Molcalut, Florin Nedeff, Valentin Nedeff, Diana Carmen Mirila, Mirela Panainte-Lehăduș, Dragos-Ioan Rusu, Maricel Agop and Decebal Vasincu

Fractal Fract. 2025, 9(10), 623; https://doi.org/10.3390/fractalfract9100623 - 25 Sep 2025

Abstract

In this paper we analyze complex systems dynamics using a multifractal framework derived from Scale Relativity Theory (SRT). By extending classical differential geometry to accommodate non-differentiable, scale-dependent behaviors, we formulate Schrödinger-type equations that describe multifractal geodesics. These equations reveal deep analogies between quantum [...] Read more.

In this paper we analyze complex systems dynamics using a multifractal framework derived from Scale Relativity Theory (SRT). By extending classical differential geometry to accommodate non-differentiable, scale-dependent behaviors, we formulate Schrödinger-type equations that describe multifractal geodesics. These equations reveal deep analogies between quantum mechanics and macroscopic complex dynamics. A key feature of this approach is the identification of hidden symmetries governed by multifractal analogs of classical groups, particularly the SL(2ℝ) group. These symmetries help explain universal dynamic behaviors such as double period dynamics, damped dynamics, modulated dynamics, or chaotic dynamics. The resulting framework offers a unified geometric and algebraic perspective on the emergence of order within complex systems, highlighting the fundamental role of fractality and scale covariance in nature. Full article

(This article belongs to the Section Complexity)

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29 pages, 2409 KB

Open AccessArticle

Mathematical Perspectives of a Coupled System of Nonlinear Hybrid Stochastic Fractional Differential Equations

by Rabeb Sidaoui, Alnadhief H. A. Alfedeel, Jalil Ahmad, Khaled Aldwoah, Amjad Ali, Osman Osman and Ali H. Tedjani

Fractal Fract. 2025, 9(10), 622; https://doi.org/10.3390/fractalfract9100622 - 24 Sep 2025

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Abstract

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of [...] Read more.

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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18 pages, 653 KB

Open AccessArticle

Zeroing Operators for Differential Initial Values Applied to Fractional Operators in the Self-Congruent Physical Space

by Gang Peng, Zhimo Jian, Meilin Li, Yu Wu, Meiling Yang and Yajun Yin

Fractal Fract. 2025, 9(10), 621; https://doi.org/10.3390/fractalfract9100621 - 24 Sep 2025

Viewed by 39

Abstract

Non-zero differential initial values hinder the application of fractional operator theory in practical systems. This paper proposes a differential initial values zeroing method, decomposing functions with non-zero differential initial values into a compensation function (Taylor polynomial) and a zeroing function (with all differential [...] Read more.

Non-zero differential initial values hinder the application of fractional operator theory in practical systems. This paper proposes a differential initial values zeroing method, decomposing functions with non-zero differential initial values into a compensation function (Taylor polynomial) and a zeroing function (with all differential initial values being zero). A differential initial values “zeroing operator” is defined, with properties such as initial value annihilation and linearity, and operational rules compatible with unilateral Laplace transforms and Mikusinski calculus operators. Based on the zeroing operator, the “zeroing differential operator” is defined to extract the zero-initial-value differential intrinsic properties of the functions with non-zero differential initial values. Using the zeroing operator, fractional constitutive equations are reconstructed in both time and complex Laplace domains in the self-congruent physical space, introducing complex fractional operators and generalized fractional operators. Validated by the complex fractional constitutive model of bone, this method breaks the bottleneck of zero-initial-value assumption in fractional operator theory in the self-congruent physical space, providing a rigorous mathematical foundation and a standardized tool for modeling sophisticated fractional systems with non-zero differential initial values. Full article

(This article belongs to the Special Issue Advances in Fractional Operators: Mathematical Perspectives and Multidisciplinary Applications)

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27 pages, 11163 KB

Open AccessArticle

Analysis of Vehicle Vibration Considering Fractional Damping in Suspensions and Tires

by Xianglong Su, Shuangning Xie and Jipeng Li

Fractal Fract. 2025, 9(10), 620; https://doi.org/10.3390/fractalfract9100620 - 24 Sep 2025

Viewed by 123

Abstract

Vehicle dynamics play a crucial role in assessing vehicle performance, comfort, and safety. To precisely depict the dynamic behaviors of a vehicle, fractional damping is employed to substitute the conventional damping in suspensions and tires. Taking the fractional damping into account, a four-degrees-of-freedom [...] Read more.

Vehicle dynamics play a crucial role in assessing vehicle performance, comfort, and safety. To precisely depict the dynamic behaviors of a vehicle, fractional damping is employed to substitute the conventional damping in suspensions and tires. Taking the fractional damping into account, a four-degrees-of-freedom vehicle model is developed, which encompasses the vertical vibration and pitch motion of the vehicle body, as well as the vertical motions of the front and rear axles. The vibration equations are solved in the Laplace domain using the transfer function method. The validity of the transfer function method is verified through comparison with a benchmark case. The vibrations of the vehicle are analyzed under the effects of suspension and tire properties, pavement excitation, and vehicle speed. The assessment methods employed include the time-domain vibration response, amplitude–frequency curves, phase diagrams, the frequency response function matrix, and weighted root mean square acceleration. The results show that the larger fractional order results in higher energy dissipation. Elevated values of the fractional order α, suspension stiffness, and the damping coefficient contribute to greater stable vibration amplitudes in vehicles and a consequent degradation in ride comfort. Higher tire stiffness reduces vehicle vibration amplitude, while the fractional order β and tire damping have a negligible effect. Moreover, increased vehicle speed and a greater pavement input amplitude adversely affect ride comfort. Full article

(This article belongs to the Special Issue Modeling, Simulation and Applications of Fractal/Fractional Calculus to Engineering Materials)

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33 pages, 12439 KB

Open AccessArticle

Fractional-Order PID Control of Two-Wheeled Self-Balancing Robots via Multi-Strategy Beluga Whale Optimization

by Huaqiang Zhang and Norzalilah Mohamad Nor

Fractal Fract. 2025, 9(10), 619; https://doi.org/10.3390/fractalfract9100619 - 23 Sep 2025

Viewed by 229

Abstract

In recent years, fractional-order controllers have garnered increasing attention due to their enhanced flexibility and superior dynamic performance in control system design. Among them, the fractional-order Proportional–Integral–Derivative (FOPID) controller offers two additional tunable parameters,

λ

and

μ

, expanding the design space and [...] Read more.

In recent years, fractional-order controllers have garnered increasing attention due to their enhanced flexibility and superior dynamic performance in control system design. Among them, the fractional-order Proportional–Integral–Derivative (FOPID) controller offers two additional tunable parameters,

λ

and

μ

, expanding the design space and allowing for finer performance tuning. However, the increased parameter dimensionality poses significant challenges for optimisation. To address this, the present study investigates the application of FOPID controllers to a two-wheeled self-balancing robot (TWSBR), with controller parameters optimised using intelligent algorithms. A novel Multi-Strategy Improved Beluga Whale Optimization (MSBWO) algorithm is proposed, integrating chaotic mapping, elite pooling, adaptive Lévy flight, and a golden sine search mechanism to enhance global convergence and local search capability. Comparative experiments are conducted using several widely known algorithms to evaluate performance. Results demonstrate that the FOPID controller optimised via the proposed MSBWO algorithm significantly outperforms both traditional PID and FOPID controllers tuned by other optimisation strategies, achieving faster convergence, reduced overshoot, and improved stability. Full article

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

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Fractal Fract., Volume 9, Issue 10 (October 2025) – 13 articles

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