Fractal and Fractional

The study investigates the nonlinear contagion, tail dependence, and Granger causality relations with TAR-TR-GARCH–copula causality methods for daily Bitcoin, Fintech, energy consumption, and CO₂ emissions in addition to examining these series for entropy, long-range dependence, fractionality, complexity, chaos, and nonlinearity with a dataset spanning from 25 June 2012 to 22 June 2024. Empirical results from Shannon, Rényi, and Tsallis entropy measures; Kolmogorov–Sinai complexity; Hurst–Mandelbrot and Lo’s R/S tests; and Phillips’ and Geweke and Porter-Hudak’s fractionality tests confirm the presence of entropy, complexity, fractionality, and long-range dependence. Further, the largest Lyapunov exponents and Hurst exponents confirm chaos across all series. The BDS test confirms nonlinearity, and ARCH-type heteroskedasticity test results support the basis for the use of novel TAR-TR-GARCH–copula causality. The model estimation results indicate moderate to strong levels of positive and asymmetric tail dependence and contagion under distinct regimes. The novel method captures nonlinear causality dynamics from Bitcoin and Fintech to energy consumption and CO₂ emissions as well as causality from energy consumption to CO₂ emissions and bidirectional feedback between Bitcoin and Fintech. These findings underscore the need to take the chaotic and complex dynamics seriously in policy and decision formulation and the necessity of eco-friendly technologies for Bitcoin and Fintech. Full article

Little research has been carried out in terms of modeling and control of analgesia. However, emerging new technology and recent prototypes paved the way for several ideas on pain modeling for control. Recently, such an idea has been proposed for measuring the Depth of Analgesia (DoA). In this paper, that solution is further exploited towards obtaining a novel fractional-order model and dedicated controller for DoA. First, clinical data from patients undergoing general anesthesia are used to determine a commensurate fractional-order model of the skin impedance at each sampling period. Second, we provide a proof of concept indicating that fractional order changes due to variations in the infused opioid drug (Remifentanil). Third, a fractional-order model for DoA is developed correlating the changes in the pain index (as the output signal) and the Remifentanil infusion rate (as the input signal). Standard optimization routines are used to estimate the parameters. A database of 19 real patients is used. Lastly, a preliminary fractional-order controller is designed and tested in simulation for the 19 patients. The closed-loop simulation results correspond to the expected clinical outcomes. Full article

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. Full article

In this article, a diffusion component in an SIR model is introduced, and its impact is analyzed using fractional calculus. We have included the diffusion component in the SIR model. in order to illustrate the variations. Here, we have applied the general fractional derivative to analyze the impact. The Laplace decomposition technique is employed to obtain the numerical outcomes of the model. In order to observe the effect of the diffusion component in the SIR model, graphical solutions are also displayed. Full article

This study aims to predict the carbon sequestration capacity of Chinese grasslands to address climate change and achieve carbon neutrality goals. Grassland carbon sequestration is a crucial part of the global carbon cycle. However, its capacity is significantly impacted by climate change and human activities, making its dynamic changes complex and challenging to predict. This study adopts a fractional-order accumulation grey model, using 11 provinces in China as samples, to analyze and forecast grassland carbon sequestration. The study finds significant differences in grassland carbon sequestration trends across the sample regions. The carbon sequestration capacity of the grasslands in Xizang (Tibet) and Heilongjiang province is increasing, while it is decreasing in other provinces. The varying prediction results are influenced not only by regional climatic and natural conditions, but also by human interventions such as overgrazing, irrational reclamation, excessive mineral resource exploitation, and increased tourism development. Therefore, more region-specific grassland management and protection strategies should be formulated to enhance the carbon sequestration capacity of grasslands and promote the sustainable development of ecosystems. The significance of this study lies not only in providing scientific guidance for the protection and sustainable management of Chinese grasslands, but also in contributing theoretical and practical insights into global carbon sequestration strategies. Full article

In this work, some properties of the general convolutional operators of general fractional calculus (GFC), which satisfy analogues of the fundamental theorems of calculus, are described. Two types of general fractional (GF) operators on a finite interval exist in GFC that are conventionally called the L-type and T-type operators. The main difference between these operators is that the additivity property holds for T-type operators and is violated for L-type operators. This property is very important for the application of GFC in physics and other sciences. The presence or violation of the additivity property can be associated with qualitative differences in the behavior of physical processes and systems. In this paper, we define L-type line GF integrals and L-type line GF gradients. For these L-type operators, the gradient theorem is proved in this paper. In general, the L-type line GF integral over a simple line is not equal to the sum of the L-type line GF integrals over lines that make up the entire line. In this work, it is shown that there exist two cases when the additivity property holds for the L-type line GF integrals. In the first case, the L-type line GF integral along the line is equal to the sum of the L-type line GF integrals along parts of this line only if the processes, which are described by these lines, are independent. Processes are called independent if the history of changes in the subsequent process does not depend on the history of the previous process. In the second case, we prove the additivity property holds for the L-type line GF integrals, if the conditions of the GF gradient theorems are satisfied. Full article

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition for bounded variation functions is examined, as well as expansions to bounded and Lipschitzian derivatives. Our comprehension is improved by comparison with current findings, and recommendations for future study areas are given. Full article

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

Reactive power dispatch (RPD) in electric power systems, integrated with renewable energy sources, is gaining popularity among power engineers because of its vital importance in the planning, designing, and operation of advanced power systems. The goal of RPD is to upgrade the power system performance by minimizing the transmission line losses, enhancing voltage profiles, and reducing the total operating costs by tuning the decision variables such as transformer tap setting, generator’s terminal voltages, and capacitor size. But the complex, non-linear, and dynamic characteristics of the power networks, as well as the presence of power demand uncertainties and non-stationary behavior of wind generation, pose a challenging problem that cannot be solved efficiently with traditional numerical techniques. In this study, a new fractional computing strategy, namely, fractional hybrid particle swarm optimization (FHPSO), is proposed to handle RPD issues in electric networks integrated with wind power plants (WPPs) while incorporating the power demand uncertainties. To improve the convergence characteristics of the Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA), the proposed FHPSO incorporates the concepts of Shannon entropy inside the mathematical model of traditional PSOGSA. Extensive experimentation validates FHPSO effectiveness by computing the best value of objective functions, namely, voltage deviation index and line loss minimization in standard power systems. The proposed FHPSO shows an improvement in percentage of 61.62%, 85.44%, 86.51%, 93.15%, 84.37%, 67.31%, 61.64%, 61.13%, 8.44%, and 1.899%, respectively, over ALC_PSO, FAHLCPSO, OGSA, ABC, SGA, CKHA, NGBWCA, KHA, PSOGSA, and FPSOGSA in case of traditional optimal reactive power dispatch(ORPD) for IEEE 30 bus system. Furthermore, the stability, robustness, and precision of the designed FHPSO are determined using statistical interpretations such as cumulative distribution function graphs, quantile-quantile plots, boxplot illustrations, and histograms. Full article

This paper investigates the problem of observer-based control for a class of nonlinear systems described by the Caputo–Hadamard fractional-order derivative. Given the growing interest in fractional-order systems for their ability to capture complex dynamics, ensuring their practical stability remains a significant challenge. We propose a novel concept of practical stability tailored to nonlinear Hadamard fractional-order systems, which guarantees that the system solutions converge to a small ball containing the origin, thereby enhancing their robustness against perturbations. Furthermore, we introduce a practical observer design that extends the classical observer framework to fractional-order systems under an enhanced One-Sided Lipschitz (OSL) condition. This extended OSL condition ensures the convergence of the proposed practical observer, even in the presence of significant nonlinearities and disturbances. Notably, the novelty of our approach lies in the extension of both the practical observer and the stability criteria, which are innovative even in the integer-order case. Theoretical results are substantiated through numerical examples, demonstrating the feasibility of the proposed method in real-world control applications. Our contributions pave the way for the development of robust observers in fractional-order systems, with potential applications across various engineering domains. Full article

We utilize Lyapunov exponents to quantitatively assess the hyperchaos and categorize the limit sets of complex dynamical systems. While there are numerous methods for computing Lyapunov exponents in integer-order systems, these methods are not suitable for fractional-order systems because of the nonlocal characteristics of fractional-order derivatives. This paper introduces innovative eight-dimensional chaotic systems that investigate fractional-order dynamics. These systems exploit the memory effect inherent in the Grünwald–Letnikov (G-L) derivative. This approach enhances the system’s applicability and compatibility with traditional integer-order systems. An 8D Chen’s fractional-order system is utilized to showcase the effectiveness of the presented methodology for hyperchaotic systems. The simulation results demonstrate that the proposed algorithm outperforms existing algorithms in both accuracy and precision. Moreover, the study utilizes the 0–1 Test for Chaos, Kolmogorov–Sinai (KS) entropy, the Kaplan–Yorke dimension, and the Perron Effect to analyze the proposed eight-dimensional fractional-order system. These additional metrics offer a thorough insight into the system’s chaotic behavior and stability characteristics. Full article

Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique $\alpha$ value; instead, they can use a wide range of $\alpha$ values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known. Full article

In this paper, we consider a logarithmic fractional Schrödinger-Poisson system where the potential is a sign-changing function. When the potential is coercive, we get the existence of infinitely many solutions for the system. When the potential is bounded, we get the existence of a ground state solution for the system. Full article

Fractional cointegration in time series data has been explored by several authors, but panel data applications have been largely neglected. A previous study of ours discovered that the Chen and Hurvich fractional cointegration test for time series was fairly robust to a moderate degree of heterogeneity across sections of the six tests considered. Therefore, this paper advances a customized version of the Chen and Hurvich methodology to detect cointegrating connections in panels with unobserved fixed effects. Specifically, we develop a test statistic that accommodates variation in the long-term cointegrating vectors and fractional cointegration parameters across observational units. The behavior of our proposed test is examined through extensive Monte Carlo experiments under various data-generating processes and circumstances. The findings reveal that our modified test performs quite well comparatively and can successfully identify fractional cointegrating relationships in panels, even in the presence of idiosyncratic disturbances unique to each cross-sectional unit. Furthermore, the proposed modified test procedure established the presence of long-run equilibrium between the exchange rate and labor wage of 36 countries’ agricultural markets. Full article

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order $\theta \in [0,1]$. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. Full article

This paper studies synchronization behaviors of two sorts of non-linear fractional-order complex spatio-temporal networks modeled by hyperbolic space-varying PDEs (FCSNHSPDEs), respectively, with time-invariant delays and time-varying delays, including one delayed coupling. One distributed controller with space-varying control gains is firstly designed. For time-invariant delayed cases, sufficient conditions for synchronization of FCSNHSPDEs are presented via LMIs, which have no relation to time delays. For time-varying delayed cases, synchronization conditions of FCSNHSPDEs are presented via spatial algebraic LMIs (SALMIs), which are related to time delay varying speeds. Finally, two examples show the validity of the control approaches. Full article

This paper presents an application of a fractional-order system on modeling an industrial process system with large inertia and time delay. The traditional integer-order model of the process system is extended to a fractional-order one in this work. To identify the parameters of the proposed fractional-order model, an output-error identification algorithm is presented. Based on the experimental step response data of the selective catalytic reduction (SCR) denitrification process in a power plant, this proposed fractional-order model shows a better fitting result compared with the typical integer-order models. An integer-order proportional–integral (PI) controller is designed for the process plant using a simple scheme according to the identified fractional-order and integer-order models, respectively. Validation tests are performed based on the obtained fractional-order and integer-order models, demonstrating the advantages of the proposed fractional-order model with the corresponding system identification approach for industrial processes with large inertia and time delay. Full article

Due to system complexity, research on fuzzy fractional-order, singular perturbation, multi-agent systems (FOSPMASs) remains limited in control theory. This article focuses on the leader-following consensus of fuzzy FOSPMASs with orders in the range of $\left(0, 2\right)$. By employing the T-S fuzzy modeling approach, a fuzzy FOSPMAS is constructed. In order to achieve the consensus of a FOSPMAS with multiple time-scale characteristics, a fuzzy observer-based controller is designed, and the error system corresponding to each agent is derived. Through a series of equivalent transformations, the error system is decomposed into fuzzy singular fractional-order systems (SFOSs). The consensus conditions of the fuzzy FOSPMASs are obtained based on linear matrix inequalities (LMIs) without an equality constraint. The theorems provide a way to tackle the uncertainty and nonlinearity in FOSPMASs with orders in the range of $\left(0, 2\right)$. Finally, the effectiveness of the theorems is verified through an RLC circuit model and a numerical example. Full article

The occurrence of landslide hazards significantly induces changes in slope surface displacement. This study conducts an in-depth analysis of the multifractal characteristics and displacement prediction of highway slope surface displacement sequences. Utilizing automated monitoring devices, data are collected to analyze the deformation patterns of the slope surface layer. Specifically, the multifractal detrended fluctuation analysis (MF-DFA) method is employed to examine the multifractal features of the monitoring data for slope surface displacement. Additionally, the Mann–Kendall (M-K) method is combined to construct the α indicator and $f\left(\alpha \right)$ indicator criteria, which provide early warnings for slope stability. Furthermore, the long short-term memory (LSTM) model is optimized using the particle swarm optimization (PSO) algorithm to enhance the prediction of slope surface displacement. The results indicate that the slope displacement monitoring data exhibit a distinct fractal sequence characterized by $h\left(q\right)$, with values decreasing as the fluctuation function q decreases. Through this study, the slope landslide warning classification has been determined to be Level III. Moreover, the PSO-LSTM model demonstrates superior prediction accuracy and stability in slope displacement forecasting, achieving a root mean square error (RMSE) of 0.72 and a coefficient of determination (R²) of 91%. Finally, a joint response synthesis of the slope landslide warning levels and slope displacement predictions resulted in conclusions. Subsequent surface displacements of the slope are likely to stabilize, indicating the need for routine monitoring and inspection of the site. Full article