Fractal and Fractional

There are many techniques for the extension and generalization of fractional theories, one of which improves fractional operators by means of their kernels. This paper is devoted to the most general concept of interval-valued functions, studying fractional integral operators for interval-valued functions, along with the multi-variate extension of the Bessel–Maitland function, which acts as kernel. We discuss the behavior of Hermite–Hadamard Fejér (HHF)-type inequalities by using the convex fuzzy interval-valued function (C-FIVF) with generalized fuzzy fractional operators. Also, we obtain some refinements of Hermite–Hadamard(H-H)-type inequalities via convex fuzzy interval-valued functions (C-FIVFs). Our results extend and generalize existing findings from the literature. Full article

In the real world, the competition and cooperation relationship exists in numerous systems. For instance, the competition–cooperation structure of a biological neural network is determined by the excitatory and inhibitory effects of neurons. The dynamic behaviors of a neural network model with a competition–cooperation structure are studied in this article, focusing particularly on the bifurcation and control problems. By selecting time delay as the parameter, a new sufficient condition for Hopf bifurcation is given and the impact of the fractional order on bifurcation behavior is determined for the network. Furthermore, a time-delay feedback controller is introduced to manage Hopf bifurcation behaviors, and, meanwhile, the stability domain is expanded. Our findings indicate that both fractional order and time delay play a crucial role in controlling the stability and Hopf bifurcation of the given model. Lastly, the accuracy of our theoretical results is verified through several numerical simulations, and the impact of control parameters on the bifurcation behavior of the network model is discussed in detail. Full article

This paper introduces a novel approach for solving multi-term time-fractional convection–diffusion equations with the fractional derivatives in the Caputo sense. The proposed highly accurate numerical algorithm is based on the barycentric rational interpolation collocation method (BRICM) in conjunction with the Gauss–Legendre quadrature rule. The discrete scheme constructed in this paper can achieve high computational accuracy with very few interval partitioning points. To verify the effectiveness of the present discrete scheme, some numerical examples are presented and are compared with the other existing method. Numerical results demonstrate the effectiveness of the method and the correctness of the theoretical analysis. Full article

The current research endeavors to explore the mechanical properties of recycled concrete cubic specimens, predominantly concentrating on macroscopic attributes such as compressive strength and splitting tensile strength. However, at the mesoscopic scale, the internal structure of recycled concrete becomes increasingly intricate due to the adherence of substantial mortar on the surface of recycled coarse aggregates, ultimately influencing its macroscopic mechanical behavior and crack propagation trajectories. To address this complexity, Digital Image Correlation (DIC) technology is harnessed to quantitatively analyze crack tip displacement fields and crack propagation behavior in recycled concrete, considering variations in aggregate replacement ratios and particle sizes. This analysis encompasses quantitative assessments of pre-cracking damage, post-cracking crack morphology characteristics, and size. Utilizing the advanced 3D-DIC algorithmic technique, this study delves into the evolution of crack propagation patterns in recycled concrete during the loading process, evaluating the impact of both aggregate replacement ratios and particle sizes on crack propagation behavior. Ultimately, fractal theory is employed to provide a quantitative analysis of the crack morphology on the surface of recycled concrete. Full article

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

The study of the fractal characteristics of the pore throat radius (PTR) and throat radius of sweet spots is crucial for the exploration and development of tight gas sandstone. This study used conventional core analysis, X-ray diffraction analysis, scanning electron microscopy (SEM), and constant-rate mercury injection experiment (CRMI), high-pressure mercury injection experiment (HPMI), and nuclear magnetic resonance (NMR) techniques to investigate the fractal characteristics of the PTR and throat radius of the tight sandstone sweet spots of the Huagang Formation in the central uplift belt of the East China Sea Basin. Based on conventional core analysis and SEM, the main pore types of the tight sandstone samples in the Huagang Formation were determined to be intergranular dissolved pore, intragranular dissolved pore, intergranular pore, and moldic pore. HPMI and NMR techniques were used to evaluate the full-size PTR distribution of type I (TI), type II (TII), and type III (TIII) sweet spots. Based on fractal theory, CRMI was used to calculate the fractal dimension of the PTR and throat radius of three types of sweet spots, and the relationship between the fractal dimensions and pore throat structure parameters and mineral composition were investigated. The results showed that the full-size PTR distribution curve exhibited bimodal or unimodal characteristics. The peak values of the PTR distribution of the TI, TII, and TIII sweet spots were mainly concentrated at 0.002–22.5 μm, 0.001–2.5 μm, and 0.0004–0.9 μm, respectively. The fractal dimensions of the PTR and throat radius were calculated. The average throat radius fractal dimensions of the TI, TIII, and TIII sweet spots were 2.925, 2.875, and 2.786, respectively. The average PTR fractal dimensions of the TI, TII, and TIII sweet spots were 2.677, 2.684, and 2.702, respectively. The throat radius fractal dimension of the TI, TII, and TIII sweet spots was positively correlated with mercury saturation, average throat radius, feldspar content, and clay mineral content and negatively correlated with displacement pressure, quartz content, and carbonate cement content. The PTR fractal dimension of the TI, TII, and TIII sweet spots was positively correlated with displacement pressure, quartz content, and carbonate cement content and negatively correlated with feldspar content. The throat size of the TI sweet spot was large, and the heterogeneity of the throat was strong. The PTR heterogeneity of the TI sweet spot was lower than that of the TII and TIII sweet spots. The findings of this study can provide important guidance for the exploration and development of tight gas sandstone. Full article

Delay partial differential equations have significant applications in numerous fields, such as population dynamics, control systems, neuroscience, and epidemiology, where they are required to efficiently model the effects of past states on current system behavior. This work presents an RBF-based localized meshless method for the numerical solution of delay partial differential equations. In the suggested numerical scheme, the localized meshless method is combined with the Laplace transform. The main attractive features of the localized meshless method are its simplicity, adaptability, and ease of implementation for complex problems defined on complex shaped domains. In a localized meshless scheme, a linear system of equations is solved. The Laplace transform, which is one of the most powerful techniques for solving integer- and non-integer-order problems, is used to represent the desired solution as a contour integral in the complex plane, known as the Bromwich integral. However, the analytic inversion of contour integral becomes very laborious in many situations. Therefore, a contour integration method is utilized to numerically approximate the Bromwich integral. The aim of utilizing the Laplace transform is to handle the costly convolution integral associated with the Caputo derivative and to avoid the effects of time-stepping techniques on the stability and accuracy of the numerical solution. We also discuss the convergence and stability of the suggested scheme. Furthermore, the existence and uniqueness of the solution for the considered model are studied. The efficiency, efficacy, and accuracy of the proposed numerical scheme have been demonstrated through numerical experiments on various problems. Full article

The mathematical theories and methods of fractional calculus are relatively mature, which have been widely used in signal processing, control systems, nonlinear dynamics, financial models, etc. The studies of some basic theories of fractional differential equations can provide more understanding of mechanisms for the applications. In this paper, the expression of the Green function as well as its special properties are acquired and presented through theoretical analyses. Subsequently, on the basis of these properties of the Green function, the existence and uniqueness of positive solutions are achieved for a singular p-Laplacian fractional-order differential equation with nonlocal integral and infinite-point boundary value systems by using the method of a nonlinear alternative of Leray–Schauder-type Guo–Krasnoselskii’s fixed point theorem in cone, and the Banach fixed point theorem, respectively. Some existence results are obtained for the case in which the nonlinearity is allowed to be singular with regard to the time variable. Several examples are correspondingly provided to show the correctness and applicability of the obtained results, where nonlinear terms are controlled by the integrable functions ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{1}{2}}{(1-lnt)}^{\frac{1}{2}}}}$ and ${\displaystyle \frac{1}{\pi {(lnt)}^{\frac{3}{4}}{(1-lnt)}^{\frac{3}{4}}}}$ in Example 1, and by the integrable functions $\theta ,\overline{\theta }$ and $\phi \left(v\right),\psi \left(u\right)$ in Example 2, respectively. The present work may contribute to the improvement and application of the coupled p-Laplacian Hadamard fractional differential model and further promote the development of fractional differential equations and fractional differential calculus. Full article

The problems associated with the visible set have been explored by various scholars. In this paper, we investigate the Hausdorff dimension and the topological properties of the visible set in relation to the products of homogeneous Cantor sets. To address these issues and establish our results, we employ beta expansion theory, numerical calculations and several technical results from fractal geometry. Our research reveals that the case of the homogeneous Cantor set differs from those of the middle Cantor sets. Furthermore, we identify a critical number that is linked to both the Hausdorff dimension and the topological properties of the visible set. Full article

The generalization of strongly convex and strongly m-convex functions is presented in this paper. We began by proving the properties of a strongly modified $(h,m)$-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for the proposed class. Moreover, H-H inequalities are also proved in the context of Riemann–Liouville (R-L) integrals. Some examples and graphs are also presented in order to show the existence of this newly defined class. Full article

In this paper, fractional calculus is used to develop a generalized fractional dynamic model of an electrohydraulic system composed of a servo valve and a hydraulic cylinder, where a fractional position controller $P{I}^{\gamma }{D}^{\mu }$ is proposed for minimizing the performance index according to the integral of the time-weighted absolute error (ITAE). First, the general mathematical equations of the cylinder and servo valve are used to obtain the transfer functions in fractional order by applying Caputo’s definition and a Laplace transform. Then, through a block diagram of the closed-loop system without a controller, the fractional model is validated by comparing its performance concerning the integer-order electrohydraulic system model reported in the literature. Subsequently, a fractional PID controller is designed to control the cylinder position. This controller is included in the closed-loop system to determine the fractional exponents of the transfer functions of the servo valve, cylinder, and control, as well as to tune the controller gains, by using the ITAE objective function, with a comparison of the following: (1) the electrohydraulic system model in integer order and the controller in fractional order; (2) the electrohydraulic system model in fractional order and the controller in integer order; and (3) both the system model and the controller in fractional order. For each of the above alternatives, numerical simulations were carried out using MATLAB^®/Simulink^® R2023b and adding white noise as a perturbation. The results show that strategy (3), where electrohydraulic system and controller model are given in fractional order, develops the best performance because it generates the minimum value of ITAE. Full article

In engineering applications, it is crucial to consider the size dependence of a material’s mechanical properties and its overall behavior. One of the theories that quantifies this phenomenon in quasi-brittle materials is the cohesive fractal theory (CFT) introduced by Carpinteri and his collaborators. This theory describes the behavior of materials using fractal dimensions. To investigate whether the scale effect can be analyzed using the CFT, a version of the Lattice Discrete Element Method (LDEM) is employed. The accuracy of the LDEM in capturing the scale effect is evaluated through simulations of three primary tests. Specifically, rock specimens are subjected to tensile, compressive, and bending loads to determine their mechanical properties. The influence of material heterogeneity and boundary conditions is also examined. In scenarios involving tensile and bending loads, the localization of a significant crack leads to failure. According to the CFT, the sum of the fractal exponents is close to unity, with values of 1.0 (mean value) for tensile loading and 0.97 for bending loading. However, the compressive loading results do not exhibit this characteristic, as no single prominent crack leads to failure. Overall, the LDEM results are consistent with the CFT, effectively quantifying the scale effect without modifying the elementary constitutive law. Full article

The degree of rock mass discontinuity is crucial for evaluating surrounding rock quality, yet its accurate and rapid measurement at construction sites remains challenging. This study utilizes fractal dimension to characterize the geometric characteristics of rock mass discontinuity and develops a data-driven surrounding rock classification (SRC) model integrating machine learning algorithms. Initially, the box-counting method was introduced to calculate the fractal dimension of discontinuity from the excavation face image. Subsequently, crucial parameters affecting surrounding rock quality were analyzed and selected, including rock strength, the fractal dimension of discontinuity, the discontinuity condition, the in-situ stress condition, the groundwater condition, and excavation orientation. This study compiled a database containing 246 railway and highway tunnel cases based on these parameters. Then, four SRC models were constructed, integrating Bayesian optimization (BO) with support vector machine (SVM), random forest (RF), adaptive boosting (AdaBoost), and gradient boosting decision tree (GBDT) algorithms. Evaluation indicators, including 5-fold cross-validation, precision, recall, F1-score, micro-F1-score, macro-F1-score, accuracy, and the receiver operating characteristic curve, demonstrated the GBDT-BO model’s superior robustness in learning and generalization compared to other models. Furthermore, four additional excavation face cases validated the intelligent SRC approach’s practicality. Finally, the synthetic minority over-sampling technique was employed to balance the training set. Subsequent retraining and evaluation confirmed that the imbalanced dataset does not adversely affect SRC model performance. The proposed GBDT-BO model shows promise for predicting surrounding rock quality and guiding dynamic tunnel excavation and support. Full article

This article mainly delves into finite-time passivity and finite-time synchronization of fractional-order spatiotemporal directed networks with multiple weights via boundary control schemes. Firstly, several concepts of finite-time passivity for fractional spatiotemporal models are proposed by virtue of a fractional-order differential inequality. Next, a boundary controller is presented to realize the passivity, which only relies on the information of the state at the spatial boundary. Subsequently, by constructing the Lyapunov function and leveraging some inequality techniques, some sufficient conditions in the form of linear matrix inequalities are established to ensure the finite-time strict passivity, finite-time input strict passivity, and finite-time output strict passivity. As a consequence of the derived result of finite-time output strict passivity, several criteria are obtained for realizing finite-time synchronization of the addressed fractional spatiotemporal networks. A numerical example is eventually presented to verify the developed control protocol and the theoretical criteria. Full article

In this paper, an effective numerical approach that combines the fast L1 formula and barycentric Lagrange interpolation is proposed for solving a kind of time-fractional evolution equations. This type of equation contains a nonlocal term involving the time variable, resulting in extremely high computational complexity of numerical discrete formats in general. To reduce the computational burden, the fast L1 technique based on the L1 formula and sum-of-exponentials approximation is employed to evaluate the Caputo time-fractional derivative. Meanwhile, a fast and unconditionally stable time semi-discrete format is obtained. Subsequently, we utilize the barycentric Lagrange interpolation and its differential matrices to achieve spatial discretizations so as to deduce fully discrete formats. Then error estimates of related fully discrete formats are explored. Eventually, some numerical experiments are simulated to testify to the effective and fast behavior of the presented method. Full article

We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called ${\mathsf{\Phi }}_s$ functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ${\mathsf{\Phi }}_M$ functions and apply the Hölder continuous condition in the extended b-metric space. The obtained results are applied to prove the existence and uniqueness of solutions and positive solutions for nonlinear integral equations and fractional boundary value problems. Examples and numerical simulation are given to illustrate the applications. Full article

This study explores the novel dynamics of the (3+1)-dimensional generalized Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation. A Galilean transformation is employed to derive the associated system of equations. Perturbing this system allows us to investigate the presence and characteristics of chaotic behavior, including return maps, fractal dimension, power spectrum, recurrence plots, and strange attractors, supported by 2D and time-dependent phase portraits. A sensitivity analysis is demonstrated to show how the system behaves when there are small changes in initial values. Finally, the planar dynamical system method is used to derive anti-kink, dark soliton, and kink soliton solutions, advancing our understanding of the range of solutions admitted by the model. Full article

The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to derive new solutions for the governing model. Second, it analyzes the complex dynamical structure of the governing model using bifurcation, chaos, and sensitivity analyses. To provide a more accurate depiction of the underlying dynamics, they use quantum mechanics to explain the intricate behavior of the system. To illustrate the physical behavior of the obtained solutions, 2D and 3D plots, along with a phase plane analysis, are presented using appropriate parameter values. These results validate the effectiveness of the employed method, providing thorough and consistent solutions with significant computational efficiency. The investigated soliton solutions will be valuable in understanding complex physical structures in various scientific fields, including ferromagnetic dynamics, nonlinear optics, soliton wave theory, and fiber optics. This approach proves highly effective in handling the complexities inherent in engineering and mathematical problems, especially those involving fractional-order systems. Full article