Fractal Fract., Volume 8, Issue 6 (June 2024) – 62 articles

The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and employed a meshless technique for spatial derivatives utilizing Lucas and Fibonacci polynomials. The rising demand for meshless methods stems from their inherent mesh-free nature and suitability for higher dimensions. Moreover, this approach demonstrates the effective approximation of solutions across both regular and irregular domains. Error norms were used to assess the accuracy of the methodology across both regular and irregular domains. A comparative analysis was conducted between the exact solution and alternative numerical methods found in the contemporary literature. The findings demonstrate that our proposed approach exhibited better performance while demanding fewer computational resources. Full article

The present study presents a combination of two famous analytical techniques for the analytical solutions of linear and nonlinear time-fractional Emden–Fowler models. We combine the Elzaki transform (ET) and the homotopy perturbation method (HPM) for the development of the Elzaki transform homotopy perturbation method (ET-HPM). In this paper, we demonstrate that the Elzaki transform (ET) simplifies fractional differential problems by transforming them into algebraic formulas within the transform space. On the other hand, the HPM has the ability to discretize the nonlinear terms in fractional problems. The fractional orders are considered in the Caputo sense. The main purpose of this strategy is to use an alternative approach that has never been employed in the time-fractional Emden–Fowler model. This strategy does not require any variable or hypothesis constraints that ruin the physical nature of the actual problem. The derived series yields a convergent series using the Taylor series formula. The analytical data and visual illustrations for several kinds of fractional orders validate the effectiveness of the suggested scheme. The significant results demonstrate that our recommended strategy is quick and simple to use on fractional problems. Full article

To accurately investigate the nonlinear dynamic characteristics of a forward converter, a fractional-order state-space averaged model of a forward converter in continuous conduction mode (CCM) is established based on the fractional calculus theory. And nonlinear dynamical bifurcation maps which use PI controller parameters and a reference current as bifurcation parameters are obtained. The nonlinear dynamic behavior is analyzed and compared with that of an integral-order forward converter. The results show that under certain operating conditions, the fractional-order forward converter exhibits bifurcations characterized by low-frequency oscillations and period-doubling as certain circuit and control parameters change. Under the same circuit conditions, there is a difference in the stable parameter region between the fractional and integral-order models of the forward converter. The stable zone of the fractional-order forward converter is larger than that of the integral-order one. Therefore, the circuit struggles to enter states of bifurcation and chaos. The stability domain for low-frequency oscillations and period-doubling bifurcations can be accurately predicted by using a small signal model and a predictive correction model of the fractional-order forward converter, respectively. Finally, by performing circuit simulations and hardware-in-the-loop experiments, the rationality and correctness of the theoretical analysis are verified. Full article

Economic growth is resulting in serious environmental problems. Effectively establishing an economic growth model that considers environmental pollution is an important topic. To analyze the interplay between economic growth and environmental pollution, a fractional-order time-delayed economic growth model with environmental purification is proposed in this paper. The established model considers not only the environment and economic production but also the labor force population and total factor productivity. Furthermore, the asymptotic stability conditions and parameter stability interval are provided. Finally, in numerical experiments, the correctness of the theory is verified. Full article

Nano silica (NS) has been found to have a positive impact on enhancing the microporous structure of Ultra-High-Performance Concrete (UHPC). However, there is a lack of effective methods to accurately characterize the regulatory improvement mechanism of NS on the pore structure of UHPC. In this study, our objective is to investigate the influence of NS on various characteristic parameters of the pore structure in UHPC, including porosity, average pore size, box fractal dimension, and multifractal spectral parameters. To analyze these effects, we employ a combination of X- CT image processing techniques and fractal theory. Furthermore, we conducted regression analysis using linear functions to explore the correlation between these parameters and the 28d compressive strength of UHPC. The experimental results demonstrate that NS promotes the refinement of matrix pore size, leading to a denser microstructure of the matrix. Fractal analysis revealed that the pore structure of NS-modified UHPC exhibited favorable fractal characteristics. The fractal dimension and multiple fractal parameters provided complementary insights into the pore structure of NS-modified UHPC from different perspectives. The fractal dimension described the global information, indicating that NS improved matrix defects and reduced the complexity of the pore structure. On the other hand, the multiple fractal parameters supplemented local information, highlighting how the increase in micropores contributed to the heterogeneity of the pore structure. The results of the correlation analysis indicate that the developed mathematical model has a good fit with the 28d compressive strength of UHPC. Full article

Studying the temporal characteristics of earthquake activity contributes to enhancing earthquake prediction capabilities. The seismic b-value is a key indicator describing the relationship between seismic frequency and magnitude. This study investigates the correlation between the occurrence of major earthquakes and seismic b-values using earthquake activity records in Japan from 1990 to 2023. Local singularity analysis and wavelet analysis of earthquake frequency and b-value time series reveal significant 5-year periodic features in seismic activity in Japan. Furthermore, our research identifies that this periodicity is also prominent in major earthquakes with magnitudes of 7 and above. Additionally, through a detailed analysis of the cross-correlation between seismic b-values and the occurrence time of major earthquakes, we uncover a notable pattern: major earthquakes often occur approximately two years after the peak of seismic b-values. This discovery offers a new perspective on earthquake prediction and may play a crucial role in future earthquake early warning systems. Full article

Space plasma turbulence is inherently characterized by anisotropic fluctuations. The generalized k-th order correlation tensor of magnetic field increments allow us to separate the mixed isotropic and anisotropic structure functions from the purely anisotropic ones. In this work, we quantified the relative importance of anisotropic fluctuations in solar wind turbulence using two Alfvénic data samples gathered by the Solar Orbiter at 0.8 astronomical units. The results based on the joined statistics suggest that the anisotropic fluctuations are ubiquitous in solar wind turbulence and persist at kinetic scales. Using the $RTN$ coordinate system, we show that their presence depends on the anisotropic sector under consideration, e.g., the $RN$ and $RT$ sectors exhibit enhanced anisotropy toward kinetic scales, in contrast with the $TN$. We then study magnetic field fluctuations parallel and perpendicular to the local mean magnetic field separately. We find that perpendicular fluctuations are representative of the global statistics, resembling the typical picture of magnetohydrodynamic turbulence, whereas parallel fluctuations exhibit a scaling law with slope ∼1 for all the joined isotropic and anisotropic components. These results are in agreement with predictions based on the critical balance phenomenology. This topic is potentially of interest for future space missions measuring kinetic and MHD scales simultaneously in a multi-spacecraft configuration. Full article

The accurate recognition of a brain tumor (BT) is crucial for accurate diagnosis, intervention planning, and the evaluation of post-intervention outcomes. Conventional methods of manually identifying and delineating BTs are inefficient, prone to error, and time-consuming. Subjective methods for BT recognition are biased because of the diffuse and irregular nature of BTs, along with varying enhancement patterns and the coexistence of different tumor components. Hence, the development of an automated diagnostic system for BTs is vital for mitigating subjective bias and achieving speedy and effective BT segmentation. Recently developed deep learning (DL)-based methods have replaced subjective methods; however, these DL-based methods still have a low performance, showing room for improvement, and are limited to heterogeneous dataset analysis. Herein, we propose a DL-based parallel features aggregation network (PFA-Net) for the robust segmentation of three different regions in a BT scan, and we perform a heterogeneous dataset analysis to validate its generality. The parallel features aggregation (PFA) module exploits the local radiomic contextual spatial features of BTs at low, intermediate, and high levels for different types of tumors and aggregates them in a parallel fashion. To enhance the diagnostic capabilities of the proposed segmentation framework, we introduced the fractal dimension estimation into our system, seamlessly combined as an end-to-end task to gain insights into the complexity and irregularity of structures, thereby characterizing the intricate morphology of BTs. The proposed PFA-Net achieves the Dice scores (DSs) of 87.54%, 93.42%, and 91.02%, for the enhancing tumor region, whole tumor region, and tumor core region, respectively, with the multimodal brain tumor segmentation (BraTS)-2020 open database, surpassing the performance of existing state-of-the-art methods. Additionally, PFA-Net is validated with another open database of brain tumor progression and achieves a DS of 64.58% for heterogeneous dataset analysis, surpassing the performance of existing state-of-the-art methods. Full article

The pore-throat structure is a critical factor in the study of unconventional oil and gas reservoirs, drawing particular attention from petroleum geologists, and it is of paramount significance to analyze to enhance oil and gas production. In tight sandstone, which serves as a significant hydrocarbon reservoir, the internal pore-throat structure plays a decisive role in the storage and migration of fluids such as water, gases, and hydrocarbons. This paper employs casting thin section (CTS), field emission scanning electron microscope (FE-SEM), high-pressure mercury injection (HPMI), and low-temperature nitrogen gas adsorption (LT−N₂−GA) experimental tests to qualitatively and quantitatively analyze the characteristics of the pore-throat structure in tight sandstone. The results indicate that the pore types in tight sandstone include intergranular residual pores, dissolution pores, intercrystalline pores, and microfractures, while the throat types encompass sheet-shaped, curved-sheet-shaped, and tubular throats. Analysis of the physical and structural parameters from 13 HPMI and 5 LT−N₂−GA samples reveals a bimodal distribution of pore-throat radii. The complexity of the pore-throat structure is identified as a primary controlling factor for reservoir permeability. The fractal dimension (D) exhibits an average value of 2.45, displaying a negative correlation with porosity (R² = 0.22), permeability (R² = 0.65), the pore-throat diameter (R² = 0.58), and maximum mercury saturation (R² = 0.86) and a positive correlation with threshold pressure (R² = 0.56), median saturation pressure (R² = 0.49), BET specific surface area (R² = 0.51), and BJH total pore volume (R² = 0.14). As D increases, reservoir pores tend to decrease in size, leading to reduced flow and deteriorated physical properties, indicative of a more complex pore-throat structure. Full article

This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating the ccmKdVEs are proposed. Then, based on the msEs, a solvable RH problem related to the ccmKdVEs is constructed. By using the constructed RH problem with mixed spectrum, scattering data for the recovery of potential formulae are further determined. In the case of reflectionless coefficients, explicit N-soliton solutions of the ccmKdVEs are ultimately obtained. Taking N equal to 1 and 2 as examples, this paper reveals that the spatiotemporal solution structures with time-varying nonlinear dynamic characteristics localized in the ccmKdVEs is attributed to the multiple selectivity of mixed spectrum and time-varying coefficients. In addition, to further highlight the application of our work in fractional calculus, by appropriately selecting these time-varying coefficients, the ccmKdVEs are transformed into a conformable time-fractional order system of three-component coupled complex mKdV equations. Based on the obtained one-soliton solutions, a set of initial values are assigned to the transformed fractional order system, and the N-th iteration formulae of approximate solutions for this fractional order system are derived through the variational iteration method (VIM). Full article

In this paper, a fractional active disturbance rejection control (FADRC) scheme is proposed for remotely operated vehicles (ROVs) to enhance high-precision positioning and docking control in the presence of ocean current disturbances and model uncertainties. The scheme comprises a double closed-loop fractional-order ${\mathrm{PI}}^{\lambda }{\mathrm{D}}^{\mu }$ controller (DFOPID) and a model-assisted finite-time sliding-mode extended state observer (MFSESO). Among them, DFOPID effectively compensates for non-matching disturbances, while its fractional-order term enhances the dynamic performance and steady-state accuracy of the system. MFSESO contributes to enhancing the estimation accuracy through the integration of sliding-mode technology and model information, ensuring the finite-time convergence of observation errors. Numerical simulations and pool experiments have shown that the proposed control scheme can effectively resist disturbances and successfully complete high-precision tasks in the absence of an accurate model. This underscores the independence of this control scheme on accurate model data of an operational ROV. Meanwhile, it also has the advantages of a simple structure and easy parameter tuning. The FADRC scheme presented in this paper holds practical significance and can serve as a valuable reference for applications involving ROVs. Full article

We consider fractional integral operators ${(I-T)}^d,d\in (-1,1)$ acting on functions $g:{\mathbb{Z}}^{\nu }\to \mathbb{R},\nu \ge 1$, where T is the transition operator of a random walk on ${\mathbb{Z}}^{\nu }$. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels $\tau (\mathit{s};d),\mathit{s}\in {\mathbb{Z}}^{\nu }$ of ${(I-T)}^d$. The asymptotic behavior of $\tau (\mathit{s};d)$ as $\left|\mathit{s}\right|\to \infty$ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on ${\mathbb{Z}}^{\nu }$ solving the difference equation ${(I-T)}^{d}X=\epsilon$ with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail. Full article

The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the $\left(\frac{{\mathsf{\Omega }}^{\prime }}{\mathsf{\Omega }}, \frac{1}{\mathsf{\Omega }}\right)$ method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields. Full article

A novel analytical model based on the generalized ubiquitiformal Sierpinski carpet is proposed which can more accurately obtain the normal contact stiffness of the grinding joint surface. Firstly, the profile and the distribution of asperities on the grinding surface are characterized. Then, based on the generalized ubiquitiformal Sierpinski carpet, the contact characterization of the grinding joint surface is realized. Secondly, a contact mechanics analysis of the asperities on the grinding surface is carried out. The analytical expressions for contact stiffness in various deformation stages are derived, culminating in the establishment of a comprehensive analytical model for the grinding joint surface. Subsequently, a comparative analysis is conducted between the outcomes of the presented model, the KE model, and experimental data. The findings reveal that, under identical contact pressure conditions, the results obtained from the presented model exhibit a closer alignment with experimental observations compared to the KE model. With an increase in contact pressure, the relative error of the presented model shows a trend of first increasing and then decreasing, while the KE model has a trend of increasing. For the relative error values of the four surfaces under different contact pressures, the maximum relative error of the presented model is 5.44%, while the KE model is 22.99%. The presented model can lay a solid theoretical foundation for the optimization design of high-precision machine tools and provide a scientific theoretical basis for the performance analysis of machine tool systems. Full article

This work presents a model for solving the Economic-Environmental Dispatch (EED) challenge, which addresses the integration of thermal, renewable energy schemes, and natural gas (NG) units, that consider both toxin emission and fuel costs as its primary objectives. Three cases are examined using the IEEE 30-bus system, where thermal units (TUs) are replaced with NGs to minimize toxin emissions and fuel costs. The system constraints include equality and inequality conditions. A detailed modeling of NGs is performed, which also incorporates the pressure pipelines and the flow velocity of gas as procedure limitations. To obtain Pareto optimal solutions for fuel costs and emissions, three optimization algorithms, namely Fractional-Order Fish Migration Optimization (FOFMO), Coati Optimization Algorithm (COA), and Non-Dominated Sorting Genetic Algorithm (NSGA-II) are employed. Three cases are investigated to validate the effectiveness of the proposed model when applied to the IEEE 30-bus system with the integration of renewable energy sources (RESs) and natural gas units. The results from Case III, where NGs are installed in place of two thermal units (TUs), demonstrate that the economic dispatching approach presented in this study significantly reduces emission levels to 0.4232 t/h and achieves a lower fuel cost of 796.478 USD/MWh. Furthermore, the findings indicate that FOFMO outperforms COA and NSGA-II in effectively addressing the EED problem. Full article

The epidemic norovirus causes vomiting and diarrhea and is a highly contagious infection. The disease is affecting human lives in terms of deaths and medical expenses. This study examines the governing dynamics of norovirus by incorporating Lévy noise into a stochastic $SIRWF$ (susceptible, infected, recovered, water contamination, and food contamination) model. The existence of a non-negative solution and its uniqueness are proved after model formulation. Subsequently, the threshold parameter is calculated, and this number is used to explore the conditions under which disease tends to exist in the population. Likewise, additional conditions are derived that ensure the elimination of the disease from the community. It is proved that the norovirus is extinct whenever the threshold parameter is less than one and it persists for ${\mathbb{R}}_s>1$. The work assumes two working examples to numerically explain the theoretical findings. Simulations of the study are visually presented, and comparisons are made. The results of this study suggest a robust approach for handling complex biological and epidemic phenomena. Full article

The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation is applied to the M-truncated fractional paraxial wave equation. Moreover, a two-dimensional dynamical system and its disturbance system are obtained. The phase portraits of the two-dimensional dynamic system and Poincaré sections and a bifurcation portrait of its perturbation system are drawn. The obtained three-dimensional graphs of soliton solutions, two-dimensional graphs of soliton solutions, and contour graphs of the M-truncated fractional paraxial wave equation arising in a liquid crystal model are drawn. Full article

For a class of fractional-order singular multi-agent systems (FOSMASs) with local Lipschitz nonlinearity, this paper proposes a closed-loop ${\mathcal{D}}^{\alpha }$-type iterative learning formation control law via input sharing to achieve the stable formation of FOSMASs in a finite time. Firstly, the formation control issue of FOSMASs with local Lipschitz nonlinearity under the fixed communication topology (FCT) is transformed into the consensus tracking control scenario. Secondly, by virtue of utilizing the characteristics of fractional calculus and the generalized Gronwall inequality, sufficient conditions for the convergence of formation error are given. Then, drawing upon the FCT, the iteration-varying switching communication topology is considered and examined. Ultimately, the validity of the ${\mathcal{D}}^{\alpha }$-type learning method is showcased through two numerical cases. Full article

In this paper, we study the generalized F-iterated function system in G-metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F-Hutchinson operators are also established. We prove that the triplet of F-Hutchinson operators defined for a finite number of general contractive mappings on a complete G-metric space is itself a generalized F-contraction mapping on a space of compact sets. We also present several examples in 2-D and 3-D for our results. Full article

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order $\beta$ approaches 1, in addition to the fractional integrals we examined. Full article

Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks. Full article

This study addresses the issue of nonfragile state estimation for memristor-based fractional-order neural networks with hybrid randomly occurring delays. Considering the finite bandwidth of the signal transmission channel, quantitative processing is introduced to reduce network burden and prevent signal blocking and packet loss. In a real-world setting, the designed estimator may experience potential gain variations. To address this issue, a fractional-order nonfragile estimator is developed by incorporating a logarithmic quantizer, which ultimately improves the reliability of the state estimator. In addition, by combining the generalized fractional-order Lyapunov direct method with novel Caputo–Wirtinger integral inequalities, a lower conservative criterion is derived to guarantee the asymptotic stability of the augmented system. At last, the accuracy and practicality of the desired estimation scheme are demonstrated through two simulation examples. Full article

Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings. Full article

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software. Full article

The foreign exchange rate market is one of the most liquid and efficient. In this study, we address the efficient analysis of this market by verifying the day-of-the-week effect with fractal analysis. The presence of fractality was evident in the return series of each day and when analyzing an upward trend and a downward trend. The econometric models showed that the day-of-the-week effect in the studied currencies did not align with previous studies. However, analyzing the Hurst exponent of each day revealed that there a weekday effect in the fractal dimension. Thirty main world currencies from all continents were analyzed, showing weekday effects according to their fractal behavior. These results show a form of market inefficiency, as the returns or price variations of each day for the analyzed currencies should have behaved similarly and tended towards random walks. This fractal day-of-the-week effect in world currencies allows us to generate investment strategies and to better complement or support buying and selling decisions on certain days. Full article

In this paper, a new command filter-based adaptive NN control strategy is developed to address the prescribed tracking performance issue for a class of nonstrict-feedback nonlinear systems. Compared with the existing performance functions, a new performance function, the fixed-time performance function, which does not depend on the accurate initial value of the error signal and has the ability of fixed-time convergence, is proposed for the first time. A radial basis function neural network is introduced to identify unknown nonlinear functions, and the characteristic of Gaussian basis functions is utilized to overcome the difficulties of the nonstrict-feedback structure. Moreover, in contrast to the traditional Backstepping technique, a command filter-based adaptive control algorithm is constructed, which solves the “explosion of complexity” problem and relaxes the assumption on the reference signal. Additionally, it is guaranteed that the tracking error falls within a prescribed small neighborhood by the designed performance functions in fixed time, and the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB). The effectiveness of the proposed control scheme is verified by numerical simulation. Full article

This study presents a comparative analysis of classical model reference adaptive control (IO-DMRAC) and its fractional-order counterpart (FO-DMRAC), which are applied to the pitch-rate control of an F-16 aircraft longitudinal model. The research demonstrates a significant enhancement in control performance with fractional-order adaptive control. Notably, the FO-DMRAC achieves lower performance indices such as the Integral Square-Error criterion (ISE) and Integral Square-Input criterion (ISU) and eliminates system output oscillations during transient periods. This study marks the pioneering application of FO-DMRAC in aircraft pitch-rate control within the literature. Through simulations on an F-16 short-period model with a relative degree of 1, the FO-DMRAC design is assessed under specific flight conditions and compared with its IO-DMRAC counterpart. Furthermore, the study ensures the boundedness of all signals, including internal ones such as ω(t). Full article

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. Full article

The segmentation analysis of the Golding–Cox mRNA dataset clarifies the description of these trajectories as a Fractional Lévy Stable Motion (FLSM). The FLSM method has several important advantages. Using only a few parameters, it allows for the detection of jumps in segmented trajectories with non-Gaussian confined parts. The value of each parameter indicates the contribution of confined segments. Non-Gaussian features in mRNA trajectories are attributed to trajectory segmentation. Each segment can be in one of the following diffusion modes: free diffusion, confined motion, and immobility. When free diffusion segments alternate with confined or immobile segments, the mean square displacement of the segmented trajectory resembles subdiffusion. Confined segments have both Gaussian (normal) and non-Gaussian statistics. If random trajectories are estimated as FLSM, they can exhibit either subdiffusion or Lévy diffusion. This approach can be useful for analyzing empirical data with non-Gaussian behavior, and statistical classification of diffusion trajectories helps reveal anomalous dynamics. Full article

Distinguishing itself from marine shale formations, alkaline lake shale, as a significant hydrocarbon source rock and petroleum reservoir, exhibits distinct multifractal characteristics and evolutionary patterns. This study employs a combination of hydrous pyrolysis experimentation, nitrogen adsorption analysis, and multifractal theory to investigate the factors influencing pore heterogeneity and multifractal dimension during the maturation process of shale with abundant rich alkaline minerals. Utilizing partial least squares (PLS) analysis, a comparative examination is conducted, elucidating the disparate influence of mineralogical composition on their respective multifractal dimensions. The findings reveal a dynamic evolution of pore characteristics throughout the maturation process of alkaline lake shale, delineated into three distinct stages. Initially, in Stage 1 (200 °C to 300 °C), both ΔD and H demonstrate an incremental trend, rising from 1.2699 to 1.3 and from 0.8615 to 0.8636, respectively. Subsequently, in Stages 2 and 3, fluctuations are observed in the values of ΔD and D, while the H value undergoes a pronounced decline to 0.85. Additionally, the parameter D₁ exhibits a diminishing trajectory across all stages, decreasing from 0.859 to 0.829, indicative of evolving pore structure characteristics throughout the maturation process. The distinct alkaline environment and mineral composition of alkaline lake shale engender disparate diagenetic effects during its maturation process compared with other shale varieties. Consequently, this disparity results in contrasting evolutionary trajectories in pore heterogeneity and multifractal characteristics. Specifically, multifractal characteristics of alkaline lake shale are primarily influenced by quartz, potassium feldspar, clay minerals, and alkaline minerals. Full article