Search Results (29)

In this paper a new generalized fractal equation for studying the behaviour of self-similar beams using the Timoshenko beam theory is introduced. This equation is established in fractal dimensions by applying the concept of fractal continuum calculus $F^{\alpha }$-CC introduced recently by Balankin and Elizarraraz in order to study engineering phenomena in complex bodies. Ultimately, the achieved formulation is a fourth-order fractal single equation generated by superposing a shear deformation on an Euler–Bernoulli beam. A mapping of the Timoshenko principle onto self-similar beams in the integer space into a corresponding principle for fractal continuum space is formulated employing local fractional differential operators. Consequently, the single equation that describes the stress/strain of a fractal Timoshenko beam is solved, which is simple, exact, and algorithmic as an alternative description of the fractal bending of beams. Therefore, the elastic curve function and rotation function can be described. Illustrative examples of classical beams are presented and show both the benefits and the efficiency of the suggested model. Full article

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order ($\alpha$, where $0<\alpha \le 1$). This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of $F^{\alpha }$ calculus. The Fisher–KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter ($\alpha$) on quasiparticle behavior. Full article

Background: The beat-by-beat fluctuation of heart rate (HR) in its temporal sequence (HR dynamics) provides information on HR regulation by the autonomic nervous system (ANS) and its dysregulation in pathological states. Commonly, linear analyses of HR and its variability (HRV) are used to draw conclusions about pathological states despite clear statistical and translational limitations. Objective: The main aim of this study was to compare linear and nonlinear HR measures, including detrended fluctuation analysis (DFA), based on ECG recordings by radiotelemetry in C57BL/6N mice to identify pathological HR dynamics. Methods: We investigated different behavioral and a wide range of pharmacological interventions which alter ANS regulation through various peripheral and/or central mechanisms including receptors implicated in psychiatric disorders. This spectrum of interventions served as a reference system for comparison of linear and nonlinear HR measures to identify pathological states. Results: Physiological HR dynamics constitute a self-similar, scale-invariant, fractal process with persistent intrinsic long-range correlations resulting in physiological DFA scaling coefficients of α~1. Strongly altered DFA scaling coefficients (α ≠ 1) indicate pathological states of HR dynamics as elicited by (1) parasympathetic blockade, (2) parasympathetic overactivation and (3) sympathetic overactivation but not inhibition. The DFA scaling coefficients are identical in mice and humans under physiological conditions with identical pathological states by defined pharmacological interventions. Conclusions: Here, we show the importance of tonic vagal function for physiological HR dynamics in mice, as reported in humans. Unlike linear measures, DFA provides an important translational measure that reliably identifies pathological HR dynamics based on altered ANS control by pharmacological interventions. Central ANS dysregulation represents a likely mechanism of increased cardiac mortality in psychiatric disorders. Full article

This paper addresses the extension of the Martinez–Kaabar fractal–fractional calculus (simply expressed as MK calculus) to the context of reduced differential transformation, with applications to the solution of some partial differential equations. Since this differential transformation is derived from the Taylor series expansion of real-valued functions of several variables, it is necessary to develop this theory in the context of such functions. Firstly, classical elements of the analysis of functions of several real variables are introduced, such as the concept of partial derivative and Clairaut’s theorem, in terms of the MK partial $\alpha ,\gamma$-derivative. Next, we establish the fractal–fractional (FrFr) Taylor formula with Lagrange residue and discuss a sufficient condition for a function of class $C_{\alpha ,\gamma }^{\infty }$ on an open and bounded set $D\subset R^2$ to be expanded into a convergent infinite series, the so-called FrFr Taylor series. The theoretical study is completed by defining the FrFr reduced differential transformation and establishing its fundamental properties, which will allow the construction of the FrFr reduced Pukhov differential transformation method (FrFrRPDTM). Based on the previous results, this new technique is applied to solve interesting non-integer order linear and non-linear partial differential equations that incorporate the fractal effect. Finally, the results show that the FrFrRPDTM represents a simple instrument that provides a direct, efficient, and effective solution to problems involving this class of partial differential equations. Full article

In this manuscript, we present the classical Hutchinson–Barnsley theory on the product neutrosophic fractal spaces by utilizing an iterated function system, which is enclosed by neutrosophic Edelstein contractions and a finite number of neutrosophic b-contractions. Further, we provide a sequence of sets that, under appropriate conditions and in terms of the Hausdorff neutrosophic metric, converge to the attractor set of specific neutrosophic iterated function systems. Furthermore, we present a fuzzy variant of α-dense curves that can accurately approximate the attractor set of certain iterated function systems with barely noticeable and controlled errors. In the end, we make a connection between the above-discussed concepts of neutrosophic theory and α-density theory. Full article

This article presents the process of creating a virtual reality (VR) game designed to assess the impact of stress on heart rate variability (HRV). The game features dynamic and challenging scenarios to induce stress responses, incorporating advanced 3D modelling and 3D animation techniques. A study involving 20 volunteers was conducted, with electrocardiographic (ECG) data collected before and during game play. HRV analysis focused on fractal and multifractal characteristics, utilizing detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MFDFA) methods. DFA results revealed decreased values of α₁, α₂, and α_all, indicating alterations in short-term and long-term correlations under stress. MFDFA further analyzed changes in fluctuation function Fq(s), generalized Hurst exponent Hq, multifractal scaling exponent τ(q), and multifractal spectrum f(α), showing significant differences in these parameters under stress. These findings validate the game’s effectiveness in simulating stress and its impact on HRV. The present study not only demonstrates the relationship between stress and the fractal characteristics of HRV but also offers a new foundation for future applications in psychology, physiology, and the development of VR technologies for stress management. Full article

This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function $sin\left(u^m\right)-\alpha u+\beta$, where $u,\alpha ,$$\beta \in \mathbb{C}$ and $m\ge 2$. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure. Full article

In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ and $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$ (which are analytic except at $z=0$) where $n\ge 2,$ $m,n\in \mathbb{N},$ $\alpha ,\beta ,\gamma \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry. Full article

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the $\alpha$-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter $\alpha$ ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with $(2-\alpha )$-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework. 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

In scenarios such as nearshore and inland waterways, the ship spots in a marine radar are easily confused with reefs and shorelines, leading to difficulties in ship identification. In such settings, the conventional ARPA method based on fractal detection and filter tracking performs relatively poorly. To accurately identify radar targets in such scenarios, a novel algorithm, namely YOSMR, based on the deep convolutional network, is proposed. The YOSMR uses the MobileNetV3(Large) network to extract ship imaging data of diverse depths and acquire feature data of various ships. Meanwhile, taking into account the issue of feature suppression for small-scale targets in algorithms composed of deep convolutional networks, the feature fusion module known as PANet has been subject to a lightweight reconstruction leveraging depthwise separable convolutions to enhance the extraction of salient features for small-scale ships while reducing model parameters and computational complexity to mitigate overfitting problems. To enhance the scale invariance of convolutional features, the feature extraction backbone is followed by an SPP module, which employs a design of four max-pooling constructs to preserve the prominent ship features within the feature representations. In the prediction head, the Cluster-NMS method and α-DIoU function are used to optimize non-maximum suppression (NMS) and positioning loss of prediction boxes, improving the accuracy and convergence speed of the algorithm. The experiments showed that the recall, accuracy, and precision of YOSMR reached 0.9308, 0.9204, and 0.9215, respectively. The identification efficacy of this algorithm exceeds that of various YOLO algorithms and other lightweight algorithms. In addition, the parameter size and calculational consumption were controlled to only 12.4 M and 8.63 G, respectively, exhibiting an 80.18% and 86.9% decrease compared to the standard YOLO model. As a result, the YOSMR displays a substantial advantage in terms of convolutional computation. Hence, the algorithm achieves an accurate identification of ships with different trail features and various scenes in marine radar images, especially in different interference and extreme scenarios, showing good robustness and applicability. Full article

Self-similarity is a common feature among mathematical fractals and various objects of our natural environment. Therefore, escape criteria are used to determine the dynamics of fractal patterns through various iterative techniques. Taking motivation from this fact, we generate and analyze fractals as an application of the proposed Mann iterative technique with h-convexity. By doing so, we develop an escape criterion for it. Using this established criterion, we set the algorithm for fractal generation. We use the complex function $f\left(x\right)=x^n+c^t$, with $n\ge 2,c\in \mathbb{C}$ and $t\in R$ to generate and compare fractals using both the Mann iteration and Mann iteration with h-convexity. We generalize the Mann iterative scheme using the convexity parameter $h\left(\alpha \right)={\alpha }^2$ and provide the detailed representations of quadratic and cubic fractals. Our comparative analysis consistently proved that the Mann iteration with h-convexity significantly outperforms the standard Mann iteration scheme regarding speed and efficiency. It is noticeable that the average number of iterations required to perform the task using Mann iteration with h-convexity is significantly less than the classical Mann iteration scheme. Moreover, the relationship between the fractal patterns and the input parameters of the proposed iteration is extremely intricate. Full article

Picard iteration is on the basis of a great number of numerical methods and applications of mathematics. However, it has been known since the 1950s that this method of fixed-point approximation may not converge in the case of nonexpansive mappings. In this paper, an extension of the concept of nonexpansiveness is presented in the first place. Unlike the classical case, the new maps may be discontinuous, adding an element of generality to the model. Some properties of the set of fixed points of the new maps are studied. Afterwards, two iterative methods of fixed-point approximation are analyzed, in the frameworks of b-metric and Hilbert spaces. In the latter case, it is proved that the symmetrically averaged iterative procedures perform well in the sense of convergence with the least number of operations at each step. As an application, the second part of the article is devoted to the study of fractal mappings on Hilbert spaces defined by means of nonexpansive operators. The paper considers fractal mappings coming from $\phi$-contractions as well. In particular, the new operators are useful for the definition of an extension of the concept of $\alpha$-fractal function, enlarging its scope to more abstract spaces and procedures. The fractal maps studied here have quasi-symmetry, in the sense that their graphs are composed of transformed copies of itself. Full article

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: $\sqrt{p+{\alpha }^2}$-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators. Full article

The time dynamics of the instrumental seismicity recorded in the area of the Lai Chau reservoir (Vietnam) between 2015 and 2021 were analyzed in this study. The Gutenberg–Richter analysis of the frequency–magnitude distribution has revealed that the seismic catalog is complete for events with magnitudes larger or equal to 0.6. The fractal method of the Allan Factor applied to the series of the occurrence times suggests that the seismic series is characterized by time-clustering behavior with rather large degrees of clustering, as indicated by the value of the fractal exponent $\alpha \approx 0.55$. The time-clustering of the time distribution of the earthquakes is also confirmed by a global coefficient of variation value of 1.9 for the interevent times. The application of the correlogram-based periodogram, which is a robust method used to estimate the power spectrum of short series, has revealed three main cycles with a significance level of $p<0.01$ (of 10 months, 1 year, and 2 years) in the monthly variation of the mean water level of the reservoir, and two main periodicities with a significance level of $p<0.01$ (at 6 months and 2 years) in the monthly number of earthquakes. By decomposing the monthly earthquake counts into intrinsic mode functions (IMFs) using the empirical decomposition method (EMD), we identified two IMFs characterized by cycles of 10 months and 2 years, significant at the 1% level, and one cycle of 1 year, significant at the 5% level. The cycles identified in these two IMFs are consistent with those detected in the water level, showing that, in a rigorously statistical manner, the seismic process occurring in the Lai Chau area might be triggered by the loading–unloading operational cycles of the reservoir. Full article