Fractal Fract., Volume 8, Issue 10 (October 2024) – 63 articles

Industrial mobile robots easily experience actuator loss of some effectiveness and additive bias faults due to the working scenarios, resulting in unexpected performance degradation. This article proposes a novel adaptive fault-tolerant control (FTC) strategy for nonholonomic mobile robot systems subject to simultaneous actuator lock-in-place (LIP) and partial loss-of-effectiveness (LOE) faults. First, a nominal fractional-order sliding mode controller based on the designed exponential super-twisting reaching law is investigated to reduce the reaching phase time and eliminate the chattering. To address the time-varying LIP faults and uncertainties, a novel barrier function (BF)-based gain is explored to assist the super-twisting law. An estimator is designed to estimate the lower bound of the time-varying partial LOE fault coefficients, thus without requiring the boundary information of faults that is commonly requested in traditional FTC schemes. Combined with the nominal controller clubbed with BF and estimator-based LOE fault compensation term, the fault-tolerant controller is finally constructed. The proposed FTC scheme achieves fast convergence and the sliding variables can be confined in a predetermined neighborhood of the sliding manifold under actuator faults. The results show that the proposed controller has superior tracking performance under faulty conditions compared with other state-of-the-art adaptive FTC approaches. Full article

Contact interface is essential for the dynamic response of the bolted structures. To accurately predict the dynamic characteristics of bolted joint structures, a fractal extension of the segmented scale model, i.e., the JK model, is proposed in this paper to comprehensively analyze the dynamic contact performance of engineering surfaces and revisit the multi-scale model based on the concept of asperities. The influence of asperity geometry, dimensionless material properties, and the elastic, elastoplastic, and full plastic mechanical models of a single asperity is established considering the asperity–substrate interaction. Then, a segmented scale contact model of rough surfaces is proposed based on the island distribution function in a strict sense. The mechanical contact process of determining rough surfaces is divided into small-scale, medium-scale, and large-scale stages. Moreover, cross-scale boundary conditions, i.e., a_l1′, a_l2′, and a_l3′, are provided through strict mathematical deduction. The results show that the real contact area and contact stiffness are positively correlated with fractal dimension and negatively correlated with fractal roughness. On a small scale, the contact damping decreases with an increase in load. In meso-scale and large-scale stages, the contact damping increases with the load. Finally, the reliability of the proposed model is verified by setting up three groups of modal vibration experiments. Full article

Aiming at the problem of system controller performance failure caused by improperly setting the value of each weighting coefficient of the model predictive control (MPC), a fractional-order MPC strategy with Takagi–Sugeno fuzzy optimization (T–SFO MPC) is proposed for a vehicle active suspension system. Firstly, the fractional-order model predictive control framework for active suspension systems is designed based on a 1/4 vehicle model. Then, we analyze the influence of different weighting coefficients on the suspension performance and introduce the Takagi–Sugeno fuzzy optimization theory to adaptively adjust the weighting coefficients of the fractional-order MPC controller. Finally, the system responses of the T–SFO MPC, traditional MPC, linear quadratic regulator (LQR), and passive suspension control are numerically analyzed under various road conditions. Simulation results show that suspension response with the T–SFO MPC is significantly improved compared with passive suspension control, traditional MPC control, and LQR control, and the weight coefficients of the T–SFO MPC can be adaptively adjusted according to the dynamic changes of suspension response. Compared with passive suspension, the root mean square (RMS) value of the vertical acceleration of the T–SFO MPC under various roads decreased by a maximum of 37.97%, and the RMS value of suspension dynamic deflection and tire dynamic load decreased by a maximum of 32.94% and 37.8%, respectively. These results validate that the proposed control method can achieve coordinated optimization of vehicle comfort and handling stability. Full article

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems. Full article

In this paper, we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of the first derivative for the numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper. Full article

This research introduces a fractional high-order sliding mode control (FHOSMC) method that utilises an inverse integral fractional order, $0<\beta <1$, as the high order on the FHOSMC reaching law, exhibiting a novel contribution in the related field of study. The application of the proposed approach into a bioreactor system via diffeomorphism operations demonstrates a notable improvement in the management of the bioreactor dynamics versus classic controllers. The numerical findings highlight an improved precision in tracking reference signals and an enhanced plant stability compared to proportional–integral–derivative (PID) controller implementations within challenging disturbance scenarios. The FHOSMC effectively maintains the biomass concentration at desired levels, reducing the wear of the system as well as implementation expenses. Furthermore, the theoretical analysis of the convergence within time indicates substantial potential for further enhancements. Subsequent studies might focus on extending this control approach to bioreactor systems that integrate sensor technologies and the formulation of adaptive algorithms for real-time adjustments of $\beta$-type fractional-orders. Full article

This paper presents an extensive investigation into the state feedback stabilization, observer design, and observer-based controller design for a specific category of nonlinear Hadamard fractional-order systems. The research extends the conventional integer-order derivative to the Hadamard fractional-order derivative, offering a more universally applicable method for system analysis. Furthermore, the traditional Lipschitz condition is adapted to a one-sided Lipschitz condition, broadening the range of systems amenable to analysis using these techniques. The efficacy of the proposed theoretical findings is illustrated through several numerical examples. For instance, in Example 1, we select an order of derivative r = 0.8; in Example 2, r is set to 0.9; and in Example 3, r = 0.95. This study enhances the comprehension and regulation of nonlinear Hadamard fractional-order systems, setting the stage for future explorations in this domain. Full article

This paper addresses critical gaps in the digital implementations of fractional-order memelement emulators, particularly given the challenges associated with the development of solid-state devices using nanomaterials. Despite the potentials of these devices for industrial applications, the digital implementation of fractional-order models has received limited attention. This research contributes to bridging this knowledge gap by presenting the FPGA realization of the memelements based on a universal voltage-controlled circuit topology. The digital emulators successfully exhibit the pinched hysteresis behaviors of memristors, memcapacitors, and meminductors, showing the retention of historical states of their constitutive electronic variables. Additionally, we analyze the impact of the fractional-order parameters and excitation frequencies on the behaviors of the memelements. The design methodology involves using Xilinx System Generator for DSP blocks to lay out the architectures of the emulators, with synthesis and gate-level implementation performed on the Xilinx Artix-7 AC701 Evaluation kit, where resource utilization on hardware accounts for about $1\%$ of available hardware resources. Further hardware analysis shows successful timing validation and low power consumption across all designs, with an average on-chip power of 0.23 Watts and average worst negative slack of 0.6 ns against a 5 ns constraint. We validate these results with Matlab 2020b simulations, which aligns with the hardware models. Full article

This study examines the effectiveness of Convolutional Autoencoder (CAE) and Variational Autoencoder (VAE) models in detecting anomalies within occupational accident data from the Mining of Coal and Lignite (NACE05), Manufacture of Other Transport Equipment (NACE30), and Manufacture of Basic Metals (NACE24) sectors. By applying fractional dimension methods—Box Counting, Hall–Wood, Genton, and Wavelet—we aim to uncover hidden risks and complex patterns that traditional time series analyses often overlook. The results demonstrate that the VAE model consistently detects a broader range of anomalies, particularly in sectors with complex operational processes like NACE05 and NACE30. In contrast, the CAE model tends to focus on more specific, moderate anomalies. Among the fractional dimension methods, Genton and Hall–Wood reveal the most significant differences in anomaly detection performance between the models, while Box Counting and Wavelet yield more consistent outcomes across sectors. These findings suggest that integrating VAE models with appropriate fractional dimension methods can significantly enhance proactive risk management in high-risk industries by identifying a wider spectrum of safety-related anomalies. This approach offers practical insights for improving safety monitoring systems and contributes to the advancement of data-driven occupational safety practices. By enabling earlier detection of potential hazards, the study supports the development of more effective safety policies, and could lead to substantial improvements in workplace safety outcomes. Full article

This paper explores the fractal properties of token embedding spaces in GPT-2 language models by analyzing the stability of the correlation dimension, a measure of geometric complexity. Token embeddings represent words or subwords as vectors in a high-dimensional space. We hypothesize that the correlation dimension $D_2$ remains consistent across different vocabulary subsets, revealing fundamental structural characteristics of language representation in GPT-2. Our main objective is to quantify and analyze the stability of $D_2$ in these embedding subspaces, addressing the challenges posed by their high dimensionality. We introduce a new theorem formalizing this stability, stating that for any two sufficiently large random subsets $S_1,S_2\subset E$, the difference in their correlation dimensions is less than a small constant $\epsilon$. We validate this theorem using the Grassberger–Procaccia algorithm for estimating $D_2$, coupled with bootstrap sampling for statistical consistency. Our experiments on GPT-2 models of varying sizes demonstrate remarkable stability in $D_2$ across different subsets, with consistent mean values and small standard errors. We further investigate how the model size, embedding dimension, and network depth impact $D_2$. Our findings reveal distinct patterns of $D_2$ progression through the network layers, contributing to a deeper understanding of the geometric properties of language model representations and informing new approaches in natural language processing. Full article

This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations. Full article

The aim of this paper is to identify a time-independent source term in the Rayleigh–Stokes equation with a fractional derivative where additional data are considered at a fixed time point. This inverse problem is proved to be ill-posed in the sense of Hadamard. By using a fractional Tikhonov regularization method, we construct a regularized solution. Then, according to a priori and a posteriori regularization parameter selection rules, we prove the convergence estimates of the regularization method. Finally, we provide some numerical examples to prove the effectiveness of the proposed method. Full article

The global incidence of cutaneous squamous cell carcinoma (cSCC), a prevalent and aggressive skin cancer, has risen significantly, posing a substantial public health challenge. This study investigates the tumor microenvironment (TME) of cSCC by focusing on the spatial distribution patterns of immune and vascular markers (CD31, CD20, CD4, and CD8) using fractal dimension (FD) analysis. Our analysis encompassed 141 cases, including 100 invasive cSCCs and 41 specimens with pre-invasive lesions exclusively, and the rest were peripheral pre-invasive lesions from the invasive cSCC class. The FD values for each marker were computed and compared between pre-invasive and invasive lesion classes. The results revealed significant differences in FD values between the two classes for CD20 and CD31 markers, suggesting distinct alterations in B cell distribution and angiogenic activity during cSCC progression. However, CD4 and CD8 markers did not exhibit significant changes individually. Still, the CD4/CD8 ratio showed a significant difference, suggesting a potential shift in the balance between T helper and cytotoxic T cell responses, impacting the immune landscape as lesions progressed from pre-invasive to invasive stages. These findings underscore the complexity and heterogeneity of the TME in cSCC and highlight the potential of FD analysis as a quantitative tool for characterizing tumor progression. Further research is needed to elucidate the implications of these differences in the clinical management of cSCC. Full article

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, $G^{\prime }/G$-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations. Full article

In the present work, we study some existing results related to a new class of Steklov $p\left(x\right)$-Kirchhoff problems with critical exponents. More precisely, we propose and prove some properties of the associated energy functional. In the first existence result, we use the mountain pass theorem to prove that the energy functional admits a critical point, which is a weak solution for such a problem. In the second main result, we use a symmetric version of the mountain pass theorem to prove that the investigated problem has an infinite number of solutions. Finally, in the third existence result, we use a critical point theorem proposed by Kajikiya to prove the existence of a sequence of solutions that tend to zero. Full article

The emission of carbon dioxide is the main reason for many global warming problems. Although China has made tremendous efforts to reduce carbon emission, the space–time dynamics of the carbon emission trend is still imbalanced. To forecast CDED in China, the Dagum Gini coefficient was applied to measure regional CDED. Then, a grey correlation model was used to select potential influence factors and a wrapping method for selecting the optimal subset. DGMC is proposed to forecast CDED. The research results showed that the DGMC generalization performance is significantly superior to other models. The MAPE of DGMC in six cases are 1.18%, 1.11%, 0.66%, 1.13%, 1.27% and 0.51%, respectively. The RMSPEPR of DGMC in six cases are 1.08%, 1.21%, 0.97%, 1.36%, 1.41% and 0.57%, respectively. The RMSPEPO of DGMC in six cases are 1.29%, 0.69%, 0.02%, 0.58%, 0.78% and 0.32%, respectively. In future trends, the eastern carbon dioxide emission intraregional differences will decrease. Additionally, the intraregional differences in western and middle-region carbon dioxide emissions will expand. Interregional carbon emission difference will display a narrowing trend. Compared with the traditional grey model and ANN model, integrating the influence factor information significantly improved forecasting accuracy. The proposed model will present better balanced historical information and accurately forecast future trends. Finally, policy recommendations are proposed based on the research results. Full article

Ebola virus disease (EVD) is a severe and often fatal illness posing significant public health challenges. This study investigates EVD transmission dynamics using a novel fractional mathematical model with five distinct compartments: individuals with low susceptibility (${\mathbb{S}}_1$), individuals with high susceptibility (${\mathbb{S}}_2$), infected individuals ($\mathbb{I}$), exposed individuals ($\mathbb{E}$), and recovered individuals ($\mathbb{R}$). To capture the complex dynamics of EVD, we employ a $\Phi$-piecewise hybrid fractional derivative approach. We investigate the crossover effect and its impact on disease dynamics by dividing the study interval into two subintervals and utilize the $\Phi$-Caputo derivative in the first interval and the $\Phi$-ABC derivative in the second interval. The study determines the basic reproduction number ${\mathit{R}}_0$, analyzes the stability of the disease-free equilibrium and investigates the sensitivity of the parameters to understand how variations affect the system’s behavior and outcomes. Numerical simulations support the model and demonstrate consistent results with the theoretical analysis, highlighting the importance of fractional calculus in modeling infectious diseases. This research provides valuable information for developing effective control strategies to combat EVD. Full article

In this paper, the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise is investigated with consideration of a new method for dealing with singular integrals. Firstly, the estimate on higher moments for the solution is given. Secondly, under some suitable assumptions, we prove the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise by using the Hölder inequality. Finally, a simulation example is given to verify the theoretical results. Full article

In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and coupled wave packets in nonlinear optical fibers and describes transverse effects in nonlinear fiber optics. Initially, a complex transformation is used to convert the model into a single Nonlinear Ordinary Differential Equation (NODE), from which hyperbolic, exponential, rational, trigonometric, and rational hyperbolic solutions are produced. In order to better understand the physical dynamics, we offer several 3D, contour, and 2D illustrations for the independent selections of physical parameter values. These illustrations highlight the graphic behaviour of some optical solitons and demonstrate that, under certain constraint conditions, acquired optical solitons lose their stability when they approach an axis and display periodic-axial perturbations, which lead to the generation of optical fractals. As a framework, the generated optical solitons have several useful applications in the field of telecommunications. Furthermore, our suggested RMESEM demonstrates its use by broadening the spectrum of optical soliton solutions, offering important insights into the dynamics of the CNHEs, and suggesting possible applications in the management of nonlinear models. Full article

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions. Full article

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. Full article

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results. Full article

In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms. Full article

At present, the integration of microgrids into power systems presents significant power quality challenges in terms of the rising adoption of nonlinear loads and electric vehicles. Ensuring the stability and efficiency of the electrical network in this evolving landscape is crucial. This paper explores the implementation of cascading Proportional–Integral (PI-PI) and cascading Fractional-Order PI (FOPI-FOPI) controllers for a Distribution Static Compensator (DSTATCOM) in hybrid microgrids that include photovoltaic (PV) systems and fuel cells. A novel hybrid optimization algorithm, WSO-WOA, is introduced to enhance power quality. This algorithm leverages the strengths of the White Shark Optimization (WSO) algorithm and the Whale Optimization Algorithm (WOA), with WSO generating new candidate solutions and WOA exploring alternative search areas when WSO does not converge on optimal results. The proposed approach was rigorously tested through multiple case studies and compared with established metaheuristic algorithms. The findings demonstrate that the WSO-WOA hybrid algorithm significantly outperforms others in optimizing the PI-PI and FOPI-FOPI controllers. The WSO-WOA algorithm showed an improvement in accuracy, surpassing the other algorithms by approximately 7.29% to 14.1% in the tuning of the PI-PI controller and about 8.5% to 21.2% in the tuning of the FOPI-FOPI controller. Additionally, the results confirm the superior performance of the FOPI-FOPI controller over the PI-PI controller in enhancing the effectiveness of the DSTATCOM across various scenarios. The FOPI-FOPI provided controller a reduced settling time by at least 30.5–56.1%, resulting in marked improvements in voltage regulation and overall power quality within the microgrid. Full article

The pore structure and mineral characteristics affect the accumulation and migration of hydrocarbons in shale, which determines the production capacity of shale oil. In this study, shale samples from the Chang 7 member of the Ordos Basin in China were selected to investigate the pore space characteristics, the effect of hydrocarbon accumulation on pore heterogeneity, and the hydrocarbon migration changes based on fractal theory, and a series of experiments were conducted involving X-ray diffraction (XRD), total organic carbon (TOC), Soxhlet extraction, and low-temperature nitrogen (N₂) and carbon dioxide (CO₂) adsorption. Then, the factors affecting extraction efficiency in shale pores were discussed. The interparticle pores contributed most to the accumulation of shale oil, and the organic matter (OM) pores contributed positively to the adsorption of hydrocarbons. The accumulation of hydrocarbons in the pore space did not increase the heterogeneity of the shale pore structure. The contents, states, and positions of hydrocarbons changed during the extraction process. Hydrocarbons were redistributed on the pore surface after Soxhlet extraction, and the heterogeneity of hydrocarbon adsorption and pore surface roughness were improved. Some heavy hydrocarbons and adsorbed components were pyrolyzed, resulting in the gradual escape of the adsorbed layer in the large pores. However, the free oil in the small pores diffused to the large pores and reaggregated on the surface, restoring a stable adsorption layer. The extraction rate was closely related to the pore throat structure and the wettability of mineral surfaces. The configuration between pores and throats had a crucial influence on the extraction rate. A high proportion of meso-pores, which effectively connect micro- and macro-pores, had a higher diffusion efficiency and a higher extraction rate. The OM pores with high energy adsorption were located in the micro-pores, and the shale oil existed in a dissolved state with high mobile capacity. The wettability of mineral surfaces affected the adsorption behavior during extraction, and strong oil wetting promoted hydrocarbon re-adsorption in clay minerals, so that the volume of micro-pores was smaller after extraction. Full article

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard ($\mathcal{H}$-$\mathcal{H}$)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and $\mathcal{I}.\mathcal{V}$-$(⋏,\hslash )$ convexity to derive new improvements of the $\mathcal{H}$-$\mathcal{H}$- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for $\mathcal{I}.\mathcal{V}$ R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results. Full article

Norovirus is a leading global cause of viral gastroenteritis, significantly affecting mortality, morbidity, and healthcare costs. This paper develops and analyzes a stochastic $\mathsf{S}\mathsf{E}\mathsf{I}\mathsf{Q}\mathsf{R}$ epidemic model for norovirus dynamics, incorporating temporal immunity and a generalized incidence rate. The model is proven to have a unique positive global solution, with extinction conditions explored. Using Khasminskii’s method, the model’s ergodicity and equilibrium distribution are investigated, demonstrating a unique ergodic stationary distribution when ${\widehat{\mathcal{R}}}_s>1$. Extinction occurs when ${\mathcal{R}}_0^E<1$. Computer simulations confirm that noise level significantly influences epidemic spread. Full article

This paper explores the exact solutions of the fractional Hirota–Satsuma coupled KdV (fHScKdV) equation in the Beta fractional derivative. The logistic method is first proposed to construct analytical solutions for the fHScKdV equation. In order to better comprehend the physical structure of the solutions, three-dimensional visualizations and line graphs of the exponent function solutions are depicted with the aid of Matlab. Furthermore, the phase portraits and bifurcation behaviors of the fHScKdV model under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These investigations enhance our understanding of practical phenomena governed by the fHScKdV model, and are crucial for examining the dynamic behaviors and phase portraits of the fHScKdV system. The strategies utilized here are more direct and effective, and can be applied effortlessly to other fractional order differential equations. Full article

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: ${\Gamma }_{\mathit{\ell }}(.)$). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete $\mathit{\ell }-$fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators. Full article