Fractal and Fractional

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

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

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation ${(-\Delta )}^{s}u+V\left(ϵx\right)u=\lambda u+ulog{u}^{2}in{\mathbb{R}}^N,$ under the mass constraint ${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{2}dx=a.$ Here, $N\ge 2$, $a,ϵ>0$, $\lambda \in \mathbb{R}$ is an unknown parameter, ${(-\Delta )}^s$ is the fractional Laplacian and $s\in (0,1)$. We introduce a function space where the energy functional associated with the problem is of class ${\mathcal{C}}^1$. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum. Full article

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

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and $4\times 4$-grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms. Full article

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

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

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