Fractal and Fractional

In this paper, a fractional-order model for African swine fever with limited medical resources is proposed and analyzed. First, the existence and uniqueness of a positive solution is proven. Second, the basic reproduction number and the conditions sufficient for the existence of two equilibriums are obtained. Third, the local stability of the two equilibriums is studied. Next, some numerical simulations are performed to verify the theoretical results. The mathematical and simulation results show that the values of some parameters, such as fractional order and medical resources, are critical for the stability of the equilibriums. Full article

In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allows for the efficient computation of the solution in the form of a power series expansion. The method is based on the Laplace transform and the residual power series, and is applied to a system of coupled FSPDEs. The method is validated using several test problems, and the results show that the LRPS method is a reliable and efficient method for solving FSPDEs. Full article

After multi-stage volume hydraulic fracturing in a shale oil reservoir, massive amounts of water can be imbibed into the matrix pores. One of the key imbibition characteristics of a shale reservoir is the imbibition water and its height distribution. Based on high pressure mercury injection (HPMI) experiments and nuclear magnetic resonance (NMR) analyses, this study quantitatively evaluated the pore-size distribution of Chang 7 continental shale oil reservoirs in Yanchang Formation, Ordos Basin. The pores could be divided into three types as micropores (≤0.1 μm), mesopores (0.1–1.0 μm), and macropores (>1.0 μm), while the volume of micropores and mesopores accounted for more than 90%. This demonstrated that there were strong heterogeneity and micro–nano characteristics. According to the spontaneous imbibition (SI) experiments, the cumulative proportion of imbibition water content was the largest in micropores, exceeding 43%, followed by mesopores around 30%, and that of macropores was the lowest, and basically less than 20%. The negative values of stage water content in the macropore or mesopore indicated that these pores became a water supply channel for other dominant imbibition pores. Additionally, combining the fractal theory with the NMR T₂ spectrum, the relative imbibition water and actual height were calculated in different pores, while the height distribution varied with cores and shale oil. The shorter the core, the higher was the relative height, while the radius of macropores filled with imbibition water was reduced. This indicates that the height distribution was affected by the pore structure, oil viscosity, and core length. Full article

The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved via the methods of reducing and topological degree in cone, which extend and enrich some previous results. Full article

In the field of nonlinear optics, quantum mechanics, condensed matter physics, and wave propagation in rigid and other nonlinear instability phenomena, the nonlinear Schrödinger equation has significant applications. In this study, the soliton solutions of the space-time fractional cubic nonlinear Schrödinger equation with Kerr law nonlinearity are investigated using an extended direct algebraic method. The solutions are found in the form of hyperbolic, trigonometric, and rational functions. Among the established solutions, some exhibit wide spectral and typical characteristics, while others are standard. Various types of well-known solitons, including kink-shape, periodic, V-shape, and singular kink-shape solitons, have been extracted here. To gain insight into the internal formation of these phenomena, the obtained solutions have been depicted in two- and three-dimensional graphs with different parameter values. The obtained solitons can be employed to explain many complicated phenomena associated with this model. Full article

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order $1<\alpha \le 2$. In order to create novel, strongly continuous associated operators, the infinitesimal generator of the sine and cosine families is examined. Additionally, two approaches are used to discuss the solution’s total controllability. A compact strategy based on the non-linear Leray–Schauder alternative theorem is one of them. In contrast, a measure of a non-compactness technique is implemented using the Sadovskii fixed point theorem with the Kuratowski measure of non-compactness. These conclusions are applied using simulation findings for the non-homogeneous fractional wave equation. Full article

This study aimed to uncover the impact of COVID-19 on the leading cryptocurrency (Bitcoin) and on sustainable finance with specific attention to their potential long memory properties. In this article, the application of the selected methodologies is based on a fractal and entropy analysis of the econometric model in the financial market. To detect the regularity/irregularity property of a time series, approximate entropy is introduced to measure deterministic chaos. Using daily data for Bitcoin and sustainable finance, namely DJSW, Green Bond, Carbon, and Clean Energy, we examine long memory behaviour by employing a rescaled range statistic (R/S) methodology. The results of the research present that the returns of Bitcoin, the Dow Jones Sustainability World Index (DJSW), Green Bond, Carbon, and Clean Energy have a significant long memory. Contrastingly, an interdisciplinary approach, namely wavelet analysis, is also used to obtain complementary results. Wavelet analysis can provide warning information about turmoil phenomena and offer insights into co-movements in the time–frequency space. Our findings reveal that approximate entropy shows crisis (turmoil) conditions in the Bitcoin market, despite the nature of the pandemic’s origin. Crucially, compared to Bitcoin assets, sustainable financial assets may play a better safe haven role during a pandemic turmoil period. The policy implications of this study could improve trading strategies for the sake of portfolio managers and investors during crisis and non-crisis periods. Full article

The permeability of crushed coal bodies plays a bottom neck role in seepage processes, which significantly limits the coal resource utilisation. To study the permeability of crushed coal bodies under pressure, the particle size distribution of crushed coal body grains is quantitatively considered by fractal theory. In addition, the parameters of the percolation characteristics of crushed coal body grains are calculated. Moreover, the permeability of the crushed coal body during recrushing is determined by the fractal dimension and porosity. A lateral limit compression test with the crushed coal bodies was carried out to illustrate the effect of the porosity on the permeability, In addition, a compressive crushed coal body size fractal–permeability model was proposed by combination of the fractal dimension and the non-Darcy equivalent permeability. The results show (1) the migration and loss of fine particles lead to a rapid increase in the porosity of the crushed coal body. (2) Increases in the effective stress cause the porosity and permeability to decrease. When the porosity decreases to approximately 0.375, its effect is undermined. (3) The migration and loss of fine particles change the pore structure and enhance the permeability properties of the skeleton, causing sudden seepage changes. (4) At low porosity, the permeability k is slightly larger than the non-Darcy equivalent permeability ke. Thus, the experimental data show an acceptable agreement with the present model. A particle size fractal–percolation model for crushed coal bodies under pressure provides a solution for effectively determining the grain permeability of the crushed coal bodies. The research results can contribute to the formation of more fractal-seepage theoretical models in fractured lithosphere, karst column pillars and coal goaf, and provide theoretical guidance for mine water disaster prevention. Full article

To accurately predict the time series of energy data, an optimized Hausdorff fractional grey seasonal model was proposed based on the complex characteristics of seasonal fluctuations and local random oscillations of seasonal energy data. This paper used a new seasonal index to eliminate the seasonal variation of the data and weaken the local random fluctuations. Furthermore, the Hausdorff fractional accumulation operator was introduced into the traditional grey prediction model to improve the weight of new information, and the particle swarm optimization algorithm was used to find the nonlinear parameters of the model. In order to verify the reliability of the new model in energy forecasting, the new model was applied to two different energy types, hydropower and wind power. The experimental results indicated that the model can effectively predict quarterly time series of energy data. Based on this, we used China’s quarterly natural gas production data from 2015 to 2021 as samples to forecast those for 2022–2024. In addition, we also compared the proposed model with the traditional statistical models and the grey seasonal models. The comparison results showed that the new model had obvious advantages in predicting quarterly data of natural gas production, and the accurate prediction results can provide a reference for natural gas resource allocation. Full article

In this article, the problems of the fractional calculus of variations are discussed based on generalized fractional operators, and the corresponding Lagrange equations are established. Then, the Noether symmetry method and the perturbation to Noether symmetry are analyzed in order to find the integrals of the equations. As a result, the conserved quantities and the adiabatic invariants are obtained. Due to the universality of the generalized fractional operators, the results achieved here can be used to solve other specific problems. Several examples are given to illustrate the universality of the methods and results. Full article

Edge detection is a highly researched topic in the field of image processing, with numerous methods proposed by previous scholars. Among these, ant colony algorithms have emerged as a promising approach for detecting image edges. These algorithms have demonstrated high efficacy in accurately identifying edges within images. For this paper, due to the long-term memory, nonlocality, and weak singularity of fractional calculus, fractional-order ant colony algorithm combined with fractional differential mask and coefficient of variation (FACAFCV) for image edge detection is proposed. If we set the order of the fractional-order ant colony algorithm and fractional differential mask to $v=0$, the edge detection method we propose becomes an integer-order edge detection method. We conduct experiments on images that are corrupted by multiplicative noise, as well as on an edge detection dataset. Our experimental results demonstrate that our method is able to detect image edges, while also mitigating the impact of multiplicative noise. These results indicate that our method has the potential to be a valuable tool for edge detection in practical applications. Full article

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in $\psi$-fractional space ${\mathbb{H}}_p^{\alpha ,\beta ;\psi }\left(\mathsf{\Omega }\right)$. The results obtained in this paper are the first to make use of the theory of $\psi$-Hilfer fractional operators with p-Laplacian. Full article

We have developed a Jungck version of the DK iterative scheme called the Jungck–DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck–DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck–DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration. Full article

A Schrödinger equation with a time-fractional derivative, posed in $(0,\infty )\times \mathbb{I}$, where $\mathbb{I}=]a,b]$, is investigated in this paper. The equation involves a singular Hardy potential of the form $\frac{\lambda }{{(x-a)}^2}$, where the parameter $\lambda$ belongs to a certain range, and a nonlinearity of the form $\mu {(x-a)}^{-\rho }{\left|u\right|}^p$, where $\rho \ge 0$. Using some a priori estimates, necessary conditions for the existence of weak solutions are obtained. Full article

This study aims to improve the performance of a pneumatic positioning system by designing a control system based on Fuzzy Fractional Order Proportional Integral Derivative (Fuzzy FOPID) controllers. The pneumatic system’s mathematical model was obtained using a system identification approach, and the Fuzzy FOPID controller was optimized using a PSO algorithm to achieve a balance between performance and robustness. The control system’s performance was compared to that of a Fuzzy PID controller through real-time experimental results, which showed that the former provided better rapidity, stability, and precision. The proposed control system was applied to a pneumatically actuated ball and beam (PABB) system, where a Fuzzy FOPID controller was used for the inner loop and another Fuzzy FOPID controller was used for the outer loop. The results demonstrated that the intelligent pneumatic actuator, when coupled with a Fuzzy FOPID controller, can accurately and robustly control the positioning of the ball and beam system. Full article

We consider a general metric Steiner problem, which involves finding a set $\mathcal{S}$ with the minimal length, such that $\mathcal{S}\cup A$ is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties). Full article

In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat–Toma transform. The solutions of Cauchy problems and the diffusion equations modeled with the HP fractional derivative are computed in the form of the generalized Mittag–Leffler function. Full article