Fractal and Fractional

Semiparametric panel data models are powerful tools for analyzing data with complex characteristics such as linearity and nonlinearity of covariates. This study aims to investigate the estimation and testing of a random effects semiparametric model (RESPM) with serially and spatially correlated nonseparable error, utilizing a combination of profile quasi-maximum likelihood estimation and local linear approximation. Profile quasi-maximum likelihood estimators (PQMLEs) for unknowns and a generalized F-test statistic $F_{NT}$ are built to determine the beingness of nonlinear relationships. The asymptotic properties of PQMLEs and $F_{NT}$ are proven under regular assumptions. The Monte Carlo results imply that the PQMLEs and $F_{NT}$ performances are excellent on finite samples; however, missing the spatially and serially correlated error leads to estimator inefficiency and bias. Indonesian rice-farming data is used to illustrate the proposed approach, and indicates that $landarea$ exhibits a significant nonlinear relationship with $riceyield$, in addition, $high$-$yieldvarieties$, $mixed$-$yield$$varieties$, and $seedweight$ have significant positive impacts on rice yield. Full article

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

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

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

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

In the present paper, we investigate a fractional magnetic system involving critical concave–convex nonlinearities with Laplace operators. Specifically, ${(-\Delta )}_A^s{u}_1={\lambda }_1{|u_1|}^{q-2}{u}_1$ + $\frac{2{\alpha }_1}{{\alpha }_1+{\beta }_1}|u_1{|}^{{\alpha }_1-2}{u}_1{|u_2|}^{{\beta }_1}$ in $\Omega$, ${(-\Delta )}_A^s{u}_2={\lambda }_2|u_2{|}^{q-2}{u}_2+\frac{2{\beta }_1}{{\alpha }_1+{\beta }_1}|u_2{|}^{{\beta }_1-2}{u}_2{|u_1|}^{{\alpha }_1}$ in $\Omega$, $u_1=u_2=0$ in ${\mathbb{R}}^n\setminus \Omega$, where $\Omega$ is a bounded set with Lipschitz boundary $\partial \Omega$ in ${\mathbb{R}}^n$, $1<q<2<\frac{n}{s}$ with $s\in (0,1)$, ${\lambda }_1, {\lambda }_2$ are two real positive parameters, ${\alpha }_1>1,{\beta }_1>1$, ${\alpha }_1+{\beta }_1=2_s^{\ast }=\frac{2n}{n-2s}$, $2_s^{\ast }$ is the fractional critical Sobolev exponent, and ${(-\Delta )}_A^s$ is a fractional magnetic Laplace operator. By using Lusternik–Schnirelmann’s theory, we prove the existence result of infinitely many solutions for the magnetic fractional system. Full article

This article introduces a novel optimization approach known as fractional order whale optimization algorithm (FWOA). The proposed optimizer incorporates the idea of fractional calculus (FC) into the mathematical structure of the conventional whale optimization algorithm (WOA). To validate the efficiency of the proposed FWOA, it is applied to address the challenges associated with the economic load dispatch (ELD) problem, which is a nonconvex, nonlinear, and non-smooth optimization problem. The objectives associated with ELD such as fuel cost and wind power generation cost minimization are achieved by taking into consideration different practical constraints like valve point loading effect (VPLE), transmission line losses, generator constraints, and stochastically variation of renewable energy sources (RES) integration. RES, particularly wind energy, has garnered more attention in recent times due to a range of environmental and economic factors. Stochastic wind (SW) power is also included in the ELD problem formulation. The incomplete gamma function (IGF) quantifies the influence of wind power. To assess its efficacy, the suggested approach is applied to a range of power systems including 3 generating units, 13 generating units and 40 generating units, consisting of 37 thermal units and 3 wind power units. To further strengthen the performance of the optimizer, the FWOA is hybridized with the interior point algorithm (IPA) to further refine the outcomes of the FWOA. The FWOA and IPA are used to address the problem of ELD while including the unpredictable nature of wind power. The simulation results of the suggested technique are compared with the most advanced heuristic optimization methods available, and it has been observed that the proposed optimizer obtained a superior and refined solution when compared to other state of the art optimization techniques. Furthermore, the efficacy of the suggested strategy in enhancing the solution of the ELD issue is validated through statistical analysis in terms of minimum fitness value. Full article

An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, instead of the Cauchy condition, the non-local in time condition $u\left(0\right)=u\left(T\right)$ is taken. The right-hand side of the equation has the form $g\left(t\right)f$ with a given function $g\left(t\right)$ and an unknown element $f\in H$. If the function $g\left(t\right)$ preserves its sign, then under a over-determined condition $u\left(t_0\right)=\psi$, $t_0\in (0,T)$, it is proved that the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution for some sign-changing functions $g\left(t\right)$. For such functions $g\left(t\right)$, under certain conditions on this function, one can achieve the well-posedness of the problem by choosing properly $t_0$. Moreover, we show that for some $g\left(t\right)$, for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied, but in this case there is no uniqueness. To the best knowledge of authors, the inverse problem with the non-local condition $u\left(0\right)=u\left(T\right)$ has been considered for the first time. Moreover, all the results obtained are new not only for the subdiffusion equation, but also for the classical diffusion equation. Full article

Based on the spectral analysis method, Gelfand’s formula, and the cones fixed point theorem, some positive solutions with their existence of a nonlinear infinite-point Hadamard fractional-order differential equation is achieved on the interval [a, b] under some conditions, and particularly the nonlinear term allows singularities for time and spatial parameters in the present study. Finally, an analysis case is carried out to reveal the principal results. Full article

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, $D_{0,t}^{\sigma ,v,\varrho ,\omega }$, of order $\sigma \in (1,2),$ in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator $D_{0,t}^{\sigma ,v,\varrho ,\omega }$, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions. Full article

In magnetic data processing, a fractional derivative can enhance details without excessively amplifying high-frequency noise. To obtain a fractional derivative numerically, a large number of survey points are required. This article demonstrates how a few survey points can be used to obtain the fractional derivative of a two-dimensional magnetic field through the application of Taylor–Riemann series. First, we derive the measurement method for the fractional gradient. This method is achieved by measuring the magnetic field at several survey points on a circle, then constructing analytical functions and finally calculating the fractional derivative. Next, an experiment is designed and simulated to demonstrate the impact of the fractional derivative start point and the ability to suppress Gaussian noise. Finally, the experiment is performed, which verifies the feasibility of the proposed method in a two-dimensional magnetic field. Full article

The reservoir quality of tight sandstone is usually affected by pore throat structures, and understanding pore throat structures and their fractal characteristics is crucial for the exploration and development of tight sandstone gas. In this study, fractal dimensions of pore throat structures and the effect of diagenesis on the fractal dimension of tight sandstone sweet spot in Huagang Formation, Jiaxing area, East China Sea Basin were studied by means of thin sections, scanning electron microscopes, X-ray diffraction analysis, scanning electron microscope quantitative mineral evaluation, and high pressure mercury injection experiments. The results show that the total fractal dimension ranges of type I, type II, and type III sweet spots were 2.62–2.87, 2.22–2.56, and 2.71–2.77, respectively. The negative correlation between total fractal dimensions, porosity, and permeability of type I sweet spots was different from those of type II and type III sweet spots. The negative correlation between total fractal dimensions of type II and type III sweet spots and maximum mercury saturation, average pore throat radius, and skewness were significant, whereas the correlation between total fractal dimensions of type I sweet spots, and maximum mercury saturation, average pore throat radius and skewness were not significant. The positive correlation between the total fractal dimensions of type II and type III sweet spots and the relative sorting coefficient, displacement pressure, and efficiency of mercury withdrawal were significant, whereas the correlation between the total fractal dimension of type I sweet spots and relative sorting coefficients, displacement pressures and efficiency of mercury withdrawal were not significant. The effect of diagenesis on fractal dimensions was investigated. Compaction reduced the pore space of tight sandstone and increased fractal dimensions. Quartz cementation and calcite cementation blocked pores and throats, reduced pore space, and increased fractal dimensions. Chlorite coat can inhibit compaction, protect pore throat structures, and maintain fractal dimensions. Most clay minerals filled primary pores and secondary pores and increased fractal dimensions. Dissolution increased the pore space of tight sandstone and decreased the fractal dimensions of the pore throat structures. The pore throat structures of type I sweet spots were mainly composed of macropores, mesopores, transitional pores, and micropores, and the fractal dimension of type I sweet spots was chiefly controlled by chlorite coat formation, dissolution, and a small amount of compaction. This study provides a reference for pore throat structure and fractal dimension analysis of tight sandstone sweet spots. Full article

In this paper, we first present multiple numerical simulations of the anti-symmetric matrix in the stability criteria for fractional order systems (FOSs). Subsequently, this paper is devoted to the study of the admissibility criteria for descriptor fractional order systems (DFOSs) whose order belongs to (0, 2). The admissibility criteria are provided for DFOSs without eigenvalues on the boundary axes. In addition, a unified admissibility criterion for DFOSs involving the minimal linear matrix inequality (LMI) variable is provided. The results of this paper are all based on LMIs. Finally, numerical examples were provided to validate the accuracy and effectiveness of the conclusions. Full article

This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions by fractional integrals. Some fractional Euler–Maclaurin-type inequalities are established for Lipschitzian functions. Finally, several Euler–Maclaurin-type inequalities are constructed by fractional integrals of bounded variation. Full article

Pore structure characterization and fractal analysis have great significance for understanding and evaluating tight limestone reservoirs. In this work, the pore structure of tight limestone, low-temperature nitrogen adsorption (LTNA), and low-field nuclear magnetic resonance (NMR) are characterized, and the fractal dimension of the pore structure of tight limestone is discussed based on LTNA and NMR data. The results indicate that the pores of tight limestone have H3 and H4 types, the pore size distribution (PSD) of the H3 type is a wave distribution ranging from 2 to 10 nm, and the PSD of the H4 type is a unimodal distribution ranging from 2 to 10 nm. The transverse relaxation time (T₂) spectrum of tight limestone shows a single peak (DF), double peak (SF), and triple peak (TF), and the ranges for the T₂ spectra for micropores, mesopores, and macropores are 0.1 to 10 ms, 10 to 100 ms, and greater than 100 ms, respectively. The LTNA fractal dimension of tight limestone (D_L) ranges between 2.4446 and 2.7688, with an average of 2.5729, and the NMR fractal dimensions of micropores (D_NMR1), mesopores (D_NMR2), and macropores (D_NMR3) are distributed between 0.3744 and 1.1293, 2.4263 and 2.9395, and 2.6582 and 2.9989, respectively. Moreover, there is a negative correlation between D_L and average pore radius, a positive correlation between D_L and specific surface area, and a positive correlation between D_NMR2 and D_NMR3 and micropore content, while D_NMR2 and D_NMR3 are negatively correlated with the content of mesopores and macropores. Full article

The traditional Langmuir equation displays drawback in accurately characterizing the methane adsorption behavior in coal, due to it assuming the uniform surface of coal pores. Additionally, the decay law of gas adsorption capacity with an increasing coal reservoir temperature remains unknown. In this study, the fractal adsorption model is proposed based on the fractal dimension (D_f) of coal pores and the attenuation coefficient (n) of the adsorption capacity. The principles and methods of this fractal adsorption model are deduced and summarized in detail. The results show that the pore structures of the two coal samples exhibit obvious fractal characteristics, with the values of fractal dimensions (D_f) being 2.6279 and 2.93. The values of adsorption capacity attenuation coefficients (n) are estimated as −0.006 and −0.004 by the adsorption experiments with different temperatures. The proposed fractal adsorption model presents a greater theoretical significance and higher accuracy than that of the Langmuir equation. The accuracy of the fractal adsorption model with temperature effect dependence is verified, establishing a prediction method for methane adsorption capacity in deep coal reservoirs. This study can serve as a theoretical foundation for coalbed methane exploration and development, as well as provide valuable insights for unconventional natural gas exploitation. Full article

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems. Full article

This paper proposes a novel nonlinear speed control method for permanent magnet synchronous motors that enhances their robustness and tracking performance. This technique integrates a sliding-mode disturbance observer and variable-gain fractional-order super-twisting sliding-mode control within a vector-control framework. The proposed control scheme employs a sliding-mode control method to mitigate chattering and improve dynamics by implementing fractional-order theory with a variable-gain super-twisting sliding manifold design while regulating the speed of the considered motor system. The aforementioned observer is suggested to enhance the control accuracy by estimating and compensating for the lumped disturbances. The proposed methodology demonstrates its superiority over other control schemes such as traditional sliding-mode control, super-twisting sliding-mode control, and the proposed technique. MATLAB/Simulink simulations and real-time implementation validate its performance, showing its potential as a reliable and efficient control approach for the system under study in practical applications. Full article