Fractal Fract., Volume 7, Issue 8 (August 2023) – 71 articles

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion. Full article

Sequential minimal optimization (SMO) method is an algorithm for solving optimization problems arising from the training process of support vector machines (SVM). The SMO algorithm is mainly used to solve the optimization problem of the objective function of SVM, and it can have high accuracy. However, its optimization accuracy can be improved. Fractional order calculus is an extension of integer order calculus, which can more accurately describe the actual system and get more accurate results. In this paper, the fractional order sequential minimal optimization (FOSMO) method is proposed based on the SMO method and fractional order calculus for classification. Firstly, an objective function is expressed by a fractional order function using the FOSMO method. The representation and meaning of fractional order terms in the objective function are studied. Then the fractional derivative of Lagrange multipliers is obtained according to fractional order calculus. Lastly, the objective function is optimized based on fractional order Lagrange multipliers, and then some experiments are carried out on the linear and nonlinear classification cases. Some experiments are carried out on two-classification and multi-classification situations, and experimental results show that the FOSMO method can obtain better accuracy than the normal SMO method. Full article

A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented. Full article

In the current work, a fast $\theta$ scheme combined with the Legendre spectral method was developed for solving a fractional Klein–Gordon equation (FKGE). The numerical scheme was provided by the Legendre spectral method in the spatial direction, and for the temporal direction, a $\theta$ scheme of order $O\left({\tau }^2\right)$ with a fast algorithm was taken into account. The fast algorithm could decrease the computational cost from $O\left(M^2\right)$ to $O(MlogM)$, where M denotes the number of time levels. In addition, correction terms could be employed to improve the convergence rate when the solutions have weak regularity. We proved theoretically that the scheme is unconditionally stable and obtained an error estimate. The numerical experiments demonstrated that our numerical scheme is accurate and efficient. Full article

In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method. Full article

A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a phase diagram, bifurcation diagram, Lyapunov Exponent spectrum (LEs), and spectral entropy (SE) complexity algorithm. Changing the template parameters and the order values has an impact on the dynamical behaviors. The results indicate that rich dynamical properties exist in the system, such as hyperchaotic attractors, chaotic attractors, asymptotic periodic loops, complex coexisting attractors, and interesting state transition phenomena. In addition, the digital circuit implementation of this fractional-order CNN system is completed on a digital signal processing (DSP) platform, which proves the accuracy of ADM and the physical feasibility of the CNN system. The study in this paper offers a fundamental theory for the fractional-order CNN system as it applies to secure communication and image encryption. Full article

The study of fractional partial differential equations is often plagued with complicated models and solution processes. In this paper, we tackle how to simplify a specific parabolic model to facilitate its analysis and solution process. That is, we investigate a general time-fractional pricing equation, and propose new transformations to reduce the underlying model to a different but equivalent problem that is less challenging. Our procedure leads to a conversion of the model to a fractional 1 + 1 heat transfer equation, and more importantly, all the transformations are invertible. A significant result which emerges is that we prove such transformations yield solutions under the Riemann–Liouville and Caputo derivatives. Furthermore, Lie point symmetries are necessary to construct solutions to the model that incorporate the behaviour of the underlying financial assets. In addition, various graphical explorations exemplify our results. Full article

This paper mainly proposes some improved stochastic gradient descent (SGD) algorithms with a fractional order gradient for the online optimization problem. For three scenarios, including standard learning rate, adaptive gradient learning rate, and momentum learning rate, three new SGD algorithms are designed combining a fractional order gradient and it is shown that the corresponding regret functions are convergent at a sub-linear rate. Then we discuss the impact of the fractional order on the convergence and monotonicity and prove that the better performance can be obtained by adjusting the order of the fractional gradient. Finally, several practical examples are given to verify the superiority and validity of the proposed algorithm. Full article

In fully developed turbulence, there is a flux of energy from large to small scales in the inertial range until the dissipation at small scales. It is associated with irreversibility, i.e., a breaking of the time reversal symmetry. Such turbulent flows are characterized by scaling properties, and we consider here how irreversibility depends on the scale. Indicators of time-reversal symmetry for time series are tested involving triple correlations in a non-symmetric way. These indicators are built so that they are zero for a time-reversal symmetric time series, and a departure from zero is an indicator of irreversibility. We study these indicators applied to two fully developed turbulence time series, from flume tank and wind tunnel databases. It is found that irreversibility occurs in the inertial range and has scaling properties with slopes close to one. A maximum value is found around the injection scale. This confirms that the irreversibility is associated with the turbulent cascade in the inertial range and shows that the irreversibility is maximal at the injection scale, the largest scale of the turbulent cascade. Full article

This article investigates the effects of thermal memory and the moving line thermal shock on heat transfer in biological tissues by employing a generalized form of the Pennes equation. The mathematical model is built upon a novel time-fractional generalized Fourier’s law, wherein the thermal flux is influenced not only by the temperature gradient but also by its historical behavior. Fractionalization of the heat flow via a fractional integral operator leads to modeling of the finite speed of the heat wave. Moreover, the thermal source generates a linear thermal shock at every instant in a specified position of the tissue. The analytical solution in the Laplace domain for the temperature of the generalized model, respectively the analytical solution in the real domain for the ordinary model, are determined using the Laplace transform. The influence of the thermal memory parameter on the heat transfer is analyzed through numerical simulations and graphic representations. Full article

This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the higher-order Lane-Emden-Fowler equations. We have also discussed the convergence rate and stability analysis of our technique. We have explained the applications and numerically simulated the examples graphically and in tabular format to elaborate on the accuracy and efficiency of this approach. Full article

The main purpose of this paper is to show the existence of a sequence of infinitely many small energy solutions to the nonlinear elliptic equations of Kirchhoff–Schrödinger type involving the fractional p-Laplacian by employing the dual fountain theorem as a key tool. Because of the presence of a non-local Kirchhoff coefficient, under conditions on the nonlinear term given in the present paper, we cannot obtain the same results concerning the existence of solutions in similar ways as in the previous related works. For this reason, we consider a class of Kirchhoff coefficients that are different from before to provide our multiplicity result. In addition, the behavior of nonlinear terms near zero is slightly different from previous studies. Full article

Since the introduction of the grey forecasting model, various improvements have been developed in the field of grey accumulated generating operators (AGOs). Fractional accumulated generating operator (FAGO) and other novel AGOs have enriched the grey theory and expanded its application scope. Nevertheless, limited attention has been given to interrelationships and contributions of new and old information. To fill this research gap, this study employed the DEMATEL method to calculate the influence degree of samples under different grey AGOs. Additionally, the pattern of influence degree variation with respect to the accumulation order was determined. The results demonstrate that, compared to traditional first-order AGO, FAGO and its corresponding grey forecasting models can effectively utilize the advantages of new information by altering the accumulation order. Full article

In this paper, the new representations of optical wave solutions to fiber Bragg gratings with cubic–quartic dispersive reflectivity having the Kerr law of nonlinear refractive index structure are retrieved with high accuracy. The residual power series technique is used to derive power series solutions to this model. The fractional derivative is taken in Caputo’s sense. The residual power series technique (RPST) provides the approximate solutions in truncated series form for specified initial conditions. By using three test applications, the efficiency and validity of the employed technique are demonstrated. By considering the suitable values of parameters, the power series solutions are illustrated by sketching 2D, 3D, and contour profiles. The analysis of the obtained results reveals that the RPST is a significant addition to exploring the dynamics of sustainable and smooth optical wave propagation across long distances through optical fibers. Full article

The convergence and robustness rejecting parameters variations and external disturbance of the system are crucial for repetitive processes. In this paper, a two-dimensional robust fractional-order iterative learning control (FOILC) is proposed for the repetitive motion process to enhance the convergence and robustness. A fractional-order proportional derivative function (FOPDF) is designed as the control variable to replace the tracking error of the integer-order iterative learning control (IOILC) algorithm. The required dynamic output fractional-order iterative learning controller is constructed by solving a set of linear matrix inequalities (LMI), and the control parameters are adjusted according to the given specifications. Simulation and experimental results in robot torque control are given to prove the effectiveness and feasibility of the proposed design method. Full article

This paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann–Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts. Full article

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results. Full article

In this paper, two types of fractional-order damping are proposed for a single flexible link: internal and external friction, related to the material of the link and the environment, respectively. Considering these dampings, the Laplace transform is used to obtain the exact model of a slewing flexible link by means of the Euler–Bernoulli beam theory. The model obtained is used in a sensing antenna with the aim of accurately describing its dynamic behavior, thanks to the incorporation of the mentioned damping models. Therefore, experimental data are used to identify the damping phenomena of this system in the frequency domain. Welch’s method is employed to estimate the experimental frequency responses. To determine the best damping model for the sensing antenna, a cost function with two weighting forms is minimized for different model structures (i.e., with internal and/or external dampings of integer- and/or fractional-order), and their robustness and fitting performance are analyzed. Full article

The purpose of this study was to assess the effects of transverse friction massage (TFM) on the electromechanical delay components and complexity of the surface electromechanical activity in the rectus femoris (RF) and vastus medialis (VM) muscles and to identify possible mechanisms behind TFM-induced alterations in the dynamics of RF and VM activity. Seven female and five male healthy subjects participated in this study. The subjects generated five maximal voluntary isometric contractions (MVICs) consecutively before and after TFM. Meanwhile, electromyography (EMG), mechanomyography (MMG), and force were recorded. The onset times of the recorded signals were detected offline by setting the threshold to three times the SD of the baseline. The delays between EMG and MMG (Δt(EMG–MMG)), MMG and force (Δt(MMG–Force)), and EMG and force (Δt(EMG–Force)) were computed from the detected onsets. The fractal dimension (FD) of the EMG time series was computed using the correlation dimension method. TFM increased Δt(MMG–Force) and Δt(EMG–Force) significantly in the RF but decreased Δt(EMG–MMG) and increased Δt(MMG–Force) in the VM. TFM decreased the FD in the RF and increased it in the VM. The results imply that TFM decreased the stiffness of both the RF and VM and decreased the duration of the electrochemical processes in the VM. It is proposed that the decrease in EMG complexity in the RF may be associated with the decreased stiffness of the RF, and the increase in EMG complexity in the VM may be associated with the decreased electrochemical processes in this muscle. It is also suggested that the opposite changes in EMG complexity in the RF and VM can be used as a discriminating parameter to search for the effects of an intervention in the quadriceps muscles. The present study also demonstrates how to discriminate the nonlinear dynamics of a complex muscle system from a noisy time series. Full article

Inflammatory bowel disease (IBD), which encompasses two different phenotypes—Crohn’s disease (CD) and ulcerative colitis (UC)—consists of chronic, relapsing disorders of the gastrointestinal tract. In 20–30% of cases, the disease begins in the pediatric age. There have been just a few studies that used fractals for IBD investigation, but none of them analyzed intestinal cell chromatin. The main aim of this study was to assess whether it is possible to differentiate between the two phenotypes in pediatric patients, or either of the phenotypes versus control, using the fractal dimension and lacunarity of intestinal cell chromatin. We analyzed nuclei from at least seven different intestinal segments from each group. In the majority of colon segments, both the fractal dimension (FD) and the lacunarity significantly differed between the UC group and CD group, and the UC group and control group. In addition, the ileocecal valve and rectum were the only segments in which CD could be differentiated from the controls based on the FD. The potential of the fractal analysis of intestinal cell nuclei to serve as an observer-independent histological tool for ulcerative colitis diagnosis was identified for the first time in this study. Our results pave the way for the development of computer-aided diagnosis systems that will assist the physicians in their clinical practice. Full article

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. Full article

This paper aims to study the bounds of k-integral operators with the Mittag-Leffler kernel in a unified form. To achieve these bounds, the definition of exponentially $(\alpha ,h-m)-p$-convexity is utilized frequently. In addition, a fractional Hadamard type inequality which shows the upper and lower bounds of k-integral operators simultaneously is presented. The results are directly linked with the results of many published articles. Full article

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied. Full article

We focus our study on a co-variational inequality problem involving two generalized Yosida approximation operators in real uniformly smooth Banach space. We show some characteristics of a generalized Yosida approximation operator, which are used in our main proof. We apply the concept of nonexpansive sunny retraction to obtain a solution to our problem. Convergence analysis is also discussed. Full article

Currently, residual useful life (RUL) prediction models for insulated-gate bipolar transistors (IGBT) do not focus on the multi-modal characteristics caused by the pulse-width modulation (PWM). To fill this gap, the Markovian stochastic process is proposed to model the mode transition process, due to the memoryless properties of the grid operation. For the estimation of the mode transition probabilities, transfer learning is utilized between different control signals. With the continuous mode switching, fractional Weibull motion (fWm) of multiple modes is established to model the stochasticity of the multi-modal IGBT degradation. The drift and diffusion coefficients are adaptively updated in the proposed RUL prediction model. In the case study, two sets of the real thermal-accelerated IGBT aging data are used. Different degradation modes are extracted from the meta degradation data, and then fused to be a complex health indicator (CHI) via a multi-sensor fusion algorithm. The RUL prediction model based on the fWm of multiple modes can reach a maximum relative prediction error of 2.96% and a mean relative prediction error of 1.78%. The proposed RUL prediction model with better accuracy can reduce the losses of the power grid caused by the unexpected IGBT failures. Full article

In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities in the literature. As applications, the reverse versions of weighted Radon-, Jensen- and power mean-type inequalities for modified unified generalized FIOs with extended unified MLFs are also investigated. Full article

Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from IFS fractals by adding minimal modifications to their defining rule. In this work, we propose two modifications to a fractal recently introduced by the authors: the so-called $\sqrt{2}$-ball fractal dust, which consists of a set of balls diminishing in size along an iterative process and delimited by an enclosing square. The proposed modifications are (a) adding a resizer parameter to introduce an interaction between the generator and generated ball elements and (b) a new fractal embedded into the $\sqrt{2}$-ball fractal dust, having the characteristic of filling zones not covered by the previous one. We study some numerical properties of both modified resulting sets to gain insights into their general properties. The resulting sets are geometrical forms with potential applications. Notably, the first modification generates an algorithm capable of producing geometric structures similar to those in mandalas and succulent plants; the second modification produces shapes similar to those found in nature, such as bubbles, sponges, and soil. Then, although a direct application of our findings is beyond the scope of this research, we discuss some clues of possible uses and extensions among which we can remark two connections: the first one between the parametrization we propose and the mandala patterns, and the second one between the embedded fractal and the grain size distribution of rocks, which is useful in percolation modeling. Full article

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered $\Psi$-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered $\Psi$-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered $\Psi$-Caputo fractional derivative of order $0<\alpha <1$, where the right side may depend on time, the solution, or its tempered $\Psi$-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the $\Psi$-exponential stability of trivial solutions are proven for these kinds of differential equations. Full article

An eco-epidemiological model involving competition regarding the predator and quarantine on infected prey is studied. The prey is divided into three compartments, namely susceptible, infected, and quarantine prey, while the predator only attacks the infected prey due to its weak condition caused by disease. To include the memory effect, the Caputo fractional derivative is employed. The model is validated by showing the existence, uniqueness, non-negativity, and boundedness of the solution. Three equilibrium points are obtained, namely predator-disease-free, predator-free-endemic, and predator-endemic points, which, respectively, represent the extinction of both predator and disease, the extinction of predator only, and the existence of all compartments. The local and global stability properties are investigated using the Matignon condition and the Lyapunov direct method. The numerical simulations using a predictor–corrector scheme are provided not only to confirm the analytical findings but also to explore more the dynamical behaviors, such as the impact of intraspecific competition, memory effect, and the occurrence of bifurcations. Full article

The main idea in this research is introducing another approximate method to calculate solutions of the fractional Torsion equation, which is one of the applied equations in civil engineering. Since the fractional order is closed to an integer, we convert the fractional Torsion equation to a perturbed ordinary differential equation involving a small parameter epsilon. Then we can find the exact solutions and approximate symmetries for the alternative approximation equation. Also, with help of the definition of conserved vector and the concept of nonlinear self-adjointness, approximate conservation laws(ACL) are obtained without approximate Lagrangians by using their approximate symmetries. In order to apply the presented theory, we apply the Lie symmetry analysis (LSA) and concept of nonlinear self-adjoint Torsion equation, which are very important in mathematics and engineering sciences, especially civil engineering. Full article