Fractal Fract., Volume 7, Issue 10 (October 2023) – 74 articles

Fractal analysis (FA) is a contemporary computational technique that can assist in identifying and assessing nuanced structural alterations in cells and tissues after exposure to certain toxic chemical agents. Its application in toxicology may be particularly valuable for quantifying structural changes in cell nuclei during conventional microscopy assessments. In recent years, the fractal dimension and lacunarity of cell nuclei, considered among the most significant FA features, have been suggested as potentially important indicators of cell damage and death. In this study, we demonstrate the feasibility of developing a random forest machine learning model that employs fractal indicators as input data to identify yeast cells treated with oxidopamine (6-hydroxydopamine, 6-OHDA), a powerful toxin commonly applied in neuroscience research. The model achieves notable classification accuracy and discriminatory power, with an area under the receiver operating characteristics curve of more than 0.8. Moreover, it surpasses alternative decision tree models, such as the gradient-boosting classifier, in differentiating treated cells from their intact counterparts. Despite the methodological challenges associated with fractal analysis and random forest training, this approach offers a promising avenue for the continued exploration of machine learning applications in cellular physiology, pathology, and toxicology. Full article

The Robin boundary condition initial value problem for transient heat conduction with the time-fractional Caputo derivative in a semi-infinite domain with a convective heat transfer (Newton’s law) at the boundary has been solved and analyzed by two analytical approaches. The uniqueness and the stability of the solution on the half-axis have been analyzed. The problem solutions by application of the operational method (Laplace transform in the time domain) and the integral-balance method (double integration technique) have been developed analytically. Full article

This paper focuses on fractional-order modeling and the design of a robust speed controller for a nonlinear system. An induction motor (IM), widely used in Electrical Vehicles (EVs), is preferred in this study as a well-known nonlinear system. The major challenge in designing a robust speed controller for IM is the insufficiency of the machine model due to inherent machine dynamics. Fractional calculus is employed to model the IM using the small-signal method, accounting for model uncertainties. In this context, experimental data is approximated using a fractional-order small-signal transfer function. Consequently, a mixed sensitivity problem is formulated with fractional-order weighting functions. The primary advantage of these weighting functions is their greater flexibility in solving the mixed sensitivity problem by involving more coefficients. Hereby, three robust speed controllers are designed using the PID toolkit of the Matlab program and solving the H_∞ mixed sensitivity problem, respectively. The novelty and contribution of the proposed method lie in maintaining the closed-loop response within a secure margin determined by fractional weighting functions while addressing the controller design. After evaluating the robust speed controllers with Bode diagrams, it is proven that all the designed controllers meet the desired nominal performance and robustness criteria. Subsequently, real-time implementations of the designed controllers are performed using the dsPIC microcontroller unit. Experimental results confirm that the designed H_∞-based fractional-order proportional-integral-derivative (FOPID) controller performs well in terms of tracking dynamics, exhibits robustness against load disturbances, and effectively suppresses sensor noise compared to the robust PID and fixed-structured H_∞ controller. Full article

The majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine $T_c\left(z\right)=asin\left(z^r\right)+bz+c$ and complex exponential $T_c\left(z\right)=a{e}^{z^r}+bz+c$ functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters ‘a’ and ‘b’, and the parameters involved in the series expansion of the sine and exponential functions. Full article

This paper focuses on the equilibrium problem of an urban public transportation system with time delay. Time delay, multi-weights, and stochastic disturbances are considered in the urban public transportation system. Hence, one can regard the urban public transportation system as a stochastic multi-weighted delayed complex network. By combining graph theory and the Lyapunov method, the global Lyapunov function is constructed indirectly. Moreover, the response system can realize synchronization with the drive system under the adaptive controller. In other words, the urban public transportation system is balanced in the actual running traffic network. Finally, numerical examples about the Chua system and small-world network are presented to confirm the accuracy and validity of the theoretical results. Full article

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results. Full article

In the literature, different approaches that are employed in designing automatic voltage regulators (AVRs) usually model the AVR as a single-input-single-output system, where the input is the generator reference voltage, and the output is the generator voltage. Alternately, it could be thought of as a double-input, single-output system, with the excitation voltage change serving as the additional input. In this paper, unlike in the existing literature, we designed the AVR system as a sextuple-input single-output (6ISO) system. The inputs in the model include the generator reference voltage, regulator signal change, exciter signal change, amplifier signal change, generator output signal change, and the sensor signal change. We also compared the generator voltage responses for various structural configurations and regulator parameter choices reported in the literature. The effectiveness of numerous controllers is investigated; the proportional, integral and differential (PID) controller, the PID with second-order derivative (PIDD2) controller, and the fractional order PID (FOPID) controller are the most prevalent types of controllers. The findings reveal that the regulator signal change and the generator output signal change significantly impact the generator voltage. Based on these findings, we propose a new approach to design the regulator parameter to enhance the response to generator reference voltage changes. This approach takes into consideration changes in the generator reference voltage as well as the regulator signal. We calculate the regulator settings using a cutting-edge hybrid technique called the Particle Swarm Optimization African Vultures Optimization algorithm (PSO–AVOA). The effectiveness of the regulator design technique and the proposed optimization algorithm are demonstrated. Full article

In this paper, a hybrid cuckoo search technique is combined with a single-layer neural network (BHCS-ANN) to approximate the solution to a differential equation describing the curvature shape of the cornea of the human eye. The proposed problem is transformed into an optimization problem such that the $L_2–$error remains minimal. A single hidden layer is chosen to reduce the sink of the local minimum values. The weights in the neural network are trained with a hybrid cuckoo search algorithm to refine them so that we obtain a better approximate solution for the given problem. To show the efficacy of our method, we considered six different corneal models. For validation, the solution with Adam’s method is taken as a reference solution. The results are presented in the form of figures and tables. The obtained results are compared with the fractional order Darwinian particle swarm optimization (FO-DPSO). We determined that results obtained with BHCS-ANN outperformed the ones acquired with other numerical routines. Our findings suggest that BHCS-ANN is a better methodology for solving real-world problems. Full article

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article

This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions. Full article

In this article, we investigate the iterative properties of positive solutions for a tempered fractional equation under the case where the boundary conditions and nonlinearity all involve tempered fractional derivatives of unknown functions. By weakening a basic growth condition, some new and complete results on the iterative properties of the positive solutions to the equation are established, which include the uniqueness and existence of positive solutions, the iterative sequence converging to the unique solution, the error estimate of the solution and convergence rate as well as the asymptotic behavior of the solution. In particular, the iterative process is easy to implement as it can start from a known initial value function. Full article

This paper mainly studies fault-tolerant control for a class of semi-linear fractional-order multi-agent systems with diffusion characteristics, where the actuator fault is considered. The adaptive fault-tolerant control protocol based on the adjacency relationship of agents is firstly designed, which can adjust the coupling gain online through the adaptive mechanism. Using the Lyapunov stability theory, the adaptive fault-tolerant control protocol can drive the agents to achieve consensus for leader-following and leaderless cases. Finally, the simulation experiment is carried out, showing the effectiveness of the proposed theory. Full article

In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s fixed point theorem. A part of this study is dedicated to transforming this problem into an equivalent integral representation and then proving some existence results to achieve our aims. Furthermore, the continuous dependence of the unique solution on the control variable and on the set of selections will be discussed. Moreover, we provide an illustration to support our results. Full article

Transport of vesicles and organelles inside cells consists of constant-speed bidirectional movement along cytoskeletal filaments interspersed by periods of idling. This transport shows many features of anomalous diffusion. In this paper, we develop a non-Markovian persistent random walk model for intracellular transport that incorporates the removal rate of organelles. The model consists of two active states with different speeds and one resting state. The organelle transitions between states with switching rates that depend on the residence time the organelle spends in each state. The mesoscopic master equations that describe the average densities of intracellular transport in each of the three states are the main results of the paper. We also derive ordinary differential equations for the dynamics for the first and second moments of the organelles’ position along the cell. Furthermore, we analyse models with power-law distributed random times, which reveal the prevalence of the Mittag-Leffler resting state and its contribution to subdiffusive and superdiffusive behaviour. Finally, we demonstrate a non-Markovian non-additivity effect when the switching rates and transport characteristics depend on the rate of organelles removal. The analytical calculations are in good agreement with numerical Monte Carlo simulations. Our results shed light on the dynamics of intracellular transport and emphasise the effects of rest times on the persistence of random walks in complex biological systems. Full article

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable $q=a\left(x\right)u_x$. An H${}^1$-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babu$\check{\mathrm{s}}$ka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The $L^2\left(L^2\right)$ norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H${}^1$-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function. Full article

Image deblurring is a fundamental image processing task, and research for efficient image deblurring methods is still a great challenge. Most of the currently existing methods are focused on TV-based models and regularization term construction; little efforts are paid to model proposal and correlated algorithms for the fidelity term in fractional-order derivative space. In this paper, we propose a novel fractional-order variational model for image deblurring, which can efficiently address three different blur kernels. The objective functional contains a fractional-order gradient fidelity term and a total generalized variation (TGV) regularization term, and it highlights the ability to preserve details and eliminate the staircase effect. To solve the problem efficiently, we provide two numerical algorithms based on the Chambolle-Pock primal-dual method (PD) and the alternating direction method of multipliers (ADMM). A series of experiments show that the proposed method achieves a good balance between detail preservation and deblurring compared with several existing advanced models. Full article

In this study, we delve into the general theory of operator kernel functions (OKFs) in operational calculus (OC). We established the rigorous mapping relation between the kernel function and the corresponding operator through the primary translation operator $e^{-\mathit{p}t}$, which bears a striking resemblance to the Laplace transform. Our research demonstrates the uniqueness of the kernel function, determined by the rules of operational calculus and its integral representation. This discovery provides a novel perspective on how the operational calculus can be understood and applied, particularly through convolution with kernel functions. We substantiate the accuracy of the proposed method by demonstrating the consistency between the operator solution and the classical solution for the heat conduction problem. Subsequently, on the fractal tree, fractal loop, and fractal ladder structures, we illustrate the application of operational calculus in viscoelastic constitutive and hemodynamics confirming that the method proposed unifies the OKFs in the existing OC theory and can be extended to the operator field. These results underscore the practical significance of our results and open up new possibilities for future research. Full article

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d>2$. This solution family is equivalent to a fractal curve in complex space ${\mathbb{C}}^d$ with random steps parametrized by N Ising variables ${\sigma }_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number r, related by $\sum {\sigma }_i=qr$. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in ${\mathbb{R}}_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N\to \infty$ or its chemical potential $\mu \to 0$. The system with fixed N has different asymptotics at odd and even $N\to \infty$, but the limit $\mu \to 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu /{\mu }^2$ in the continuum limit $\mu \to 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu }\to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda }$, with a continuous spectrum of indexes $\lambda$ in the local limit $\mu \to 0$. The spectrum is determined by a resolvent, which is represented as an infinite product of $3\otimes 3$ matrices depending of the element of the Euler ensemble. Full article

This work investigates the complex dynamics of the stochastic fractional Kuramoto–Sivashinsky equation (SFKSE) with conformable fractional derivatives. The research begins with the creation of singular stochastic soliton solutions utilizing the modified extended direct algebraic method (mEDAM). Comprehensive contour, 3D, and 2D visual representations clearly depict the categorization of these stochastic soliton solutions as kink waves or shock waves, offering a clear description of these soliton behaviors within the context of the SFKSE framework. The paper also illustrates the flexibility of the transformation-based approach mEDAM for investigating soliton occurrence not only in SFKSE but also in a wide range of nonlinear fractional partial differential equations (FPDEs). Furthermore, the analysis considers the effect of noise, specifically Brownian motion, on soliton solutions and wave dynamics, revealing the significant influence of randomness on the propagation, generation, and stability of soliton in complex stochastic systems and advancing our understanding of extreme behaviors in scientific and engineering domains. Full article

From the very start of modelling with power-tail distributions, concerns were expressed about the actual applicability of distributions with infinite expectations to real-world distributions, which usually have bounded ranges. Here, we suggest resolving this issue by shifting the analysis from the true convergence in various CLTs to some kind of quasi convergence, where a stable approximation to, say, normalised sums of n i.i.d. random variables (or more generally, in a functional setting, to the processes of random walks), holds for large n, but not “too large” n. If the range of “large n” includes all imaginable applications, the approximation is practically indistinguishable from the true limit. This approach allows us to justify a stable approximation to random walks with bounded jumps and, moreover, it leads to some kind of cascading (quasi) asymptotics, where for different ranges of a small parameter, one can have different stable or light-tail approximations. The author believes that this development might be relevant to all applications of stable laws (and thus of fractional equations), say, in Earth systems, astrophysics, biological transport and finances. Full article

This study examines the dynamics of a stochastic prey–predator model using a functional response function driven by Lévy noise and a mixed Holling-II and Beddington–DeAngelis functional response. The proposed model presents a computational analysis between two prey and one predator population dynamics. First, we show that the suggested model admits a unique positive solution. Second, we prove the extinction of all the studied populations, the extinction of only the predator, and the persistence of all the considered populations under several sufficient conditions. Finally, a special Runge–Kutta method for the stochastic model is illustrated and implemented in order to show the behavior of the two prey and one predator subpopulations. Full article

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. Full article

This paper investigates the distributed optimization problem (DOP) for fractional high-order nonstrict-feedback multiagent systems (MASs) where each agent is multiple-input–multiple-output (MIMO) dynamic and contains uncertain dynamics. Based on the penalty-function method, the consensus constraint is eliminated and the global objective function is reconstructed. Different from the existing literatures, where the DOPs are addressed for linear MASs, this paper deals with the DOP through using radial basis function neural networks (RBFNNs) to approximate the unknown nonlinear functions for high-order MASs. To reduce transmitting and computational costs, event-triggered scheme and quantized control technology are combined to propose an adaptive backstepping neural network (NN) control protocol. By applying the Lyapunov stability theory, the optimal consensus error is proved to be bounded and all signals remain semi-global uniformly ultimately bounded. Simulation shows that all agents reach consensus and errors between agents’ outputs and the optimal solution is close to zero with low computational costs. Full article

Previous studies have shown that solar wind, a plasma medium with turbulent dynamics, exhibits anomalous scaling features, i.e., intermittency, in the inertial domain. This intermittent nature has primarily been investigated through the study of the scaling features of the structure functions of single quantities. We use a novel approach based on joint multifractal analysis in this study to simultaneously investigate the scaling characteristics of both the magnetic field and the plasma velocity in solar wind turbulence. Specifically, we focus on the joint multifractal behavior of magnetic and velocity field fluctuations in both fast and slow solar wind streams observed by the ESA-Ulysses satellite, with the goal of identifying any differences in their joint multifractal characteristics. Full article

In this paper, we derive the coincidence fixed-point and common fixed-point results for ℑ-type mappings satisfying certain contractive conditions and containing fewer conditions imposed on function ℑ with regard to generalized metric spaces (in terms of Jleli Samet). Finally, a fractional boundary value problem is reduced to an equivalent Volterra integral equation, and the existence results of common solutions are obtained with the use of proved fixed-point results. Full article

Rubella is a viral disease that can lead to severe health complications, especially in pregnant women and their unborn babies. Understanding the dynamics of the Rubella disease model is crucial for developing effective strategies to control its spread. This paper introduces a major innovation by employing a novel piecewise approach that incorporates two different kernels. This innovative approach significantly enhances the accuracy of modeling Rubella disease dynamics. In the first interval, the Caputo operator is employed to address initial conditions, while the Atangana–Baleanu derivative is utilized in the second interval to account for anomalous diffusion processes. A thorough theoretical analysis of the piecewise derivative for the problem is provided, discussing mathematical properties, stability, and convergence. To solve the proposed problem effectively, the piecewise numerical Newton polynomial technique is employed and the numerical scheme for both kernels is established. Through extensive numerical simulations with various fractional orders, the paper demonstrates the approach’s effectiveness and flexibility in modeling the spread of the Rubella virus. Furthermore, to validate the findings, the simulated results are compared with real data obtained from Rubella outbreaks in Uganda and Tanzania, confirming the practical relevance and accuracy of this innovative model. Full article

A contaminant spreading affected by a random field at boundaries in the comb geometry is considered. The physical effect of the random boundary conditions results in increasing a transport exponent such that the mean squared displacement increases with time from $t^{\frac{1}{2}}$ to $t^{\frac{1}{2}+5\alpha /2}$ for real $0\le \alpha \le 1$. This stochastic acceleration due to these space-time-dependent boundary conditions leads to a transition from subdiffusion to superdiffusion. Experimentally, it can be realized by controlling the boundary conditions of $2D$ diffusion in the comb geometry. Full article

Due to the significant value that hotel reviews hold for both consumers and businesses, the development of an accurate sentiment classification method is crucial. By effectively distinguishing the authenticity of reviews, consumers can make informed decisions, and businesses can gain insights into customer feedback to improve their services and enhance overall competitiveness. In this paper, we propose a partial differential equation model based on phase-field for sentiment analysis in the field of hotel comment texts. The comment texts are converted into word vectors using the Word2Vec tool, and then we utilize the multifractal detrended fluctuation analysis (MF-DFA) model to extract the generalized Hurst exponent of the word vector time series to achieve dimensionality reduction of the word vector data. The dimensionality reduced data are represented in a two-dimensional computational domain, and the modified Allen–Cahn (AC) function is used to evolve the phase values of the data to obtain a stable nonlinear boundary, thereby achieving automatic classification of hotel comment texts. The experimental results show that the proposed method can effectively classify positive and negative samples and achieve excellent results in classification indicators. We compared our proposed classifier with traditional machine learning models and the results indicate that our method possesses a better performance. Full article

In this paper, we first use the quantum Fourier transform (QFT) and quantum phase estimation (QPE) to realize the quantum fractional Fourier transform (QFrFT). As diverse definitions of the discrete fractional Fourier transform (DFrFT) exist, the relationship between the QFrFT and a classical algorithm is then established; that is, we determine the classical algorithm corresponding to the QFrFT. Second, we observe that many definitions of the multi-fractional Fourier transform (mFrFT) are flawed: when we attempt to propose a design scheme for the quantum mFrFT, we find that there are many invalid weighting terms in the definition of the mFrFT. This flaw may have very significant impacts on relevant algorithms for signal processing and image encryption. Finally, we analyze the circuit of the QFrFT and the reasons for the observed defects. Full article

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too. Full article