Fractal Fract., Volume 8, Issue 7 (July 2024) – 13 articles

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

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions. Full article

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications. Full article

Intelligent control methodologies and artificial intelligence (AI) are essential components for the efficient management of energy storage modern systems, specifically those utilizing superconducting magnetic energy storage (SMES). Through the implementation of AI algorithms, SMES units are able to optimize their operations in real time, thereby maximizing energy efficiency. To have a more advanced understanding of this issue, DynamoMan is presented in this paper. For use with SMES systems, DynamoMan, an Artificial Neural Network (ANN)-tuned Fractional Order PID Brain Emotional Learning-Based Intelligent Controller (ANN-FOPID-BELBIC), has been developed. ANN tuning is employed to optimize the key settings of the reward/penalty generator of a BELBIC, which are important for its overall efficacy. Following this, DynamoMan is integrated into the SMES control system and compared to scenarios in which a BELBIC, PID, PI, and P are utilized. The findings indicate that DynamoMan performs considerably better than other models, demonstrating robust and control attributes alongside a considerably reduced period of settling time, especially when incorporated with the power grid. Full article