Fractal Fract., Volume 6, Issue 10 (October 2022) – 86 articles

In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving $\psi$-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered system. For the existence and uniqueness of the solution, we use the Banach contraction principle. With the help of Schaefer’s fixed-point theorem and some assumptions, we also obtain at least one solution of the mentioned system. Finally, the main results are verified with an appropriate example. Full article

We studied a Zener-type model of a viscoelastic body within the context of general fractional calculus and derived restrictions on coefficients that follow from the dissipation inequality, which is the entropy inequality under isothermal conditions. We showed, for a stress relaxation and a wave propagation, that the restriction that follows from the entropy inequality is sufficient to guarantee the existence and uniqueness of the solution. We presented numerical data related to the solution of a wave equation for several values of parameters. Full article

In this paper, the Peyrard–Bishop–Dauxois model of DNA dynamics is discussed along with the fractional effects of the M-truncated derivative and $\beta$-derivative. The Kudryashov’s R method was applied to the model in order to obtain a solitary wave solution. The obtained solution is explained graphically and the fractional effects of the $\beta$ and M-truncated derivatives are also shown for a better understanding of the model. Full article

This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with measures of noncompactness. The obtained result is illustrated by an example at the end. Full article

Fractional calculus has been opening new doors in terms of better modeling and control of several phenomena and processes. Biomedical engineering has seen a lot of combined attention from clinicians, control engineers and researchers in their attempt to offer individualized treatment. A large number of medical procedures require anesthesia, which in turn requires a closely monitored and controlled level of hypnosis, analgesia and neuromuscular blockade, as well maintenance of hemodynamic variables in a safe range. Computer-controlled anesthesia has been given a tremendous amount of attention lately. Hemodynamic stabilization via computer-based control is also a hot topic. However, very few studies on automatic control of combined anesthesia–hemodynamic systems exist despite the fact that hemodynamics is strongly influenced by hypnotic drugs, while the depth of hypnosis is affected by drugs used in hemodynamic control. The very first multivariable fractional-order controller is developed in this paper for the combined anesthesia–hemodynamic system. Simulation studies on 24 patients show the effectiveness of the proposed approach. Full article

In most geophysical flows, vortices (or eddies) of all sizes are observed. In 1941, Kolmogorov devised a theory to describe the hierarchical organization of such vortices via a homogeneous self-similar process. This theory correctly explains the universal power-law energy spectrum observed in all turbulent flows. Finer observations however prove that this picture is too simplistic, owing to intermittency of energy dissipation and high velocity derivatives. In this review, we discuss how such intermittency can be explained and fitted into a new picture of turbulence. We first discuss how the concept of multi-fractality (invented by Parisi and Frisch in 1982) enables to generalize the concept of self-similarity in a non-homogeneous environment and recover a universality in turbulence. We further review the local extension of this theory, and show how it enables to probe the most irregular locations of the velocity field, in the sense foreseen by Lars Onsager in 1949. Finally, we discuss how the multi-fractal theory connects to possible singularities, in the real or in the complex plane, as first investigated by Frisch and Morf in 1981. Full article

This paper is devoted to studying a class of fractional differential equations (FDEs) with the Prabhakar fractional derivative of Caputo type in an analytical manner. At first, an estimate of the Prabhakar fractional derivative of a function at its extreme points is obtained. This estimate is used to formulate and prove comparison principles for related fractional differential inequalities. We then apply these comparison principles to derive pre-norm estimates of solutions and to obtain a uniqueness result for linear FDEs. The solution of linear FDEs with constant coefficients is obtained in closed form via the Laplace transform. For linear FDEs with variable coefficients, we apply the obtained comparison principles to establish an existence result using the method of lower and upper solutions. Two well-defined monotone sequences that converge uniformly to the actual solution of the problem are generated. Full article

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii’s fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results. Full article

In this paper, we study the existence of positive solutions for a system of fractional differential equations with $\rho$-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type. Full article

Carbon nanotubes (CNTs) are considered among the ideal modifiers for cement-based materials. This is because CNTs can be used as a microfiber to compensate for the insufficient toughness of the cement matrix. However, the full dispersion of CNTs in cement paste is difficult to achieve, and the strength of cement material can be severely degraded by the high air-entraining property of CNT dispersion. To analyze the relationship between the gas entrainment by CNT dispersion and mortar strength, this study employed data obtained from strength and micropore structure tests of CNT dispersion-modified mortar. The fractal dimensions of the pore volume and pore surface, as well as the box-counting dimension of the pore structure, were determined according to the box-counting dimension method and Menger sponge model. The relationship between the fractal dimensions of the pore structure and mortar strength was investigated by gray correlation. The results showed that the complexity of the pore structure could be accurately reflected by fractal dimensions. The porosity values of mortar with 0.05% and 0.5% CNT content were 15.5% and 43.26%, respectively. Moreover, the gray correlation between the fractal dimension of the pore structure and strength of the CNT dispersion-modified mortar exceeded 0.95. This indicated that the pore volume distribution, roughness, and irregularity of the pore inner surface were the primary factors influencing the strength of CNT dispersion-modified mortar. Full article

When performing fractional factorial experiments in a completely random order is impractical, fractional factorial split-plot designs are suitable options as an alternative. It is well recognized that the more there are lower order effects of interest at lower order confounding, the better the designs. From this viewpoint, this paper considers the construction of optimal regular two-level fractional factorial split-plot designs. The optimality criteria for two different design scenarios are proposed. Under the newly proposed optimality criteria, the theoretical construction methods of optimal regular two-level fractional factorial split-plot designs are then proposed. In addition, we also explore the theoretical construction methods of some optimal regular two-level fractional factorial split-plot designs under the widely adopted general minimum lower order confounding criterion. Full article

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle. Full article

Probabilistic machine learning and data-driven methods gradually show their high efficiency in solving the forward and inverse problems of partial differential equations (PDEs). This paper will focus on investigating the forward problem of solving time-dependent nonlinear delay PDEs with multi-delays based on multi-prior numerical Gaussian processes (MP-NGPs), which are constructed by us to solve complex PDEs that may involve fractional operators, multi-delays and different types of boundary conditions. We also quantify the uncertainty of the prediction solution by the posterior distribution of the predicted solution. The core of MP-NGPs is to discretize time firstly, then a Gaussian process regression based on multi-priors is considered at each time step to obtain the solution of the next time step, and this procedure is repeated until the last time step. Different types of boundary conditions are studied in this paper, which include Dirichlet, Neumann and mixed boundary conditions. Several numerical tests are provided to show that the methods considered in this paper work well in solving nonlinear time-dependent PDEs with delay, where delay partial differential equations, delay partial integro-differential equations and delay fractional partial differential equations are considered. Furthermore, in order to improve the accuracy of the algorithm, we construct Runge–Kutta methods under the frame of multi-prior numerical Gaussian processes. The results of the numerical experiments prove that the prediction accuracy of the algorithm is obviously improved when the Runge–Kutta methods are employed. Full article

The accurate characterization of the surface microstructure of ultra-high temperature ceramics after thermal shocks is of great practical significance for evaluating their thermal resistance properties. In this paper, a fractal reconstruction method for the surface image of Ultra-high temperature ceramics after repeated thermal shocks is proposed. The nonlinearity and spatial distribution characteristics of the oxidized surfaces of ceramics were extracted. A fractal convolutional neural network model based on deep learning was established to realize automatic recognition of the classification of thermal shock cycles of ultra-high temperature ceramics, obtaining a recognition accuracy of 93.74%. It provides a novel quantitative method for evaluating the surface character of ultra-high temperature ceramics, which contributes to understanding the influence of oxidation after thermal shocks. Full article

The objective of this study is to examine numerical evaluations of the mosquito dispersal mathematical system (MDMS) in a heterogeneous atmosphere through artificial intelligence (AI) techniques via Bayesian regularization neural networks (BSR-NNs). The MDMS is constructed with six classes, i.e., eggs, larvae, pupae, host, resting mosquito, and ovipositional site densities-based ODEs system. The computing BSR-NNs scheme is applied for three different performances using the data of training, testing and verification, which is divided as 75%, 15%, 10% with twelve hidden neurons. The result comparisons are provided to check the authenticity of the designed AI method portrayed by the BSR-NNs. The AI based BSR-NNs procedure is executed to reduce the mean square error (MSE) for the MDMS. The achieved performances are also presented to validate the efficiency of BSR-NNs scheme using the process of MSE, correlation, error histograms and regression. Full article

A model-free fractional-order sliding mode control (MFFOSMC) method based on a non-linear disturbance observer is proposed for the electric drive system in this paper. Firstly, the ultra-local model is established by using the mathematical model of electric drive system under parameter perturbation. Then, aiming at reducing the chattering of the sliding mode controller and improving the transient response, a model-free fractional-order sliding mode controller is designed based on fractional-order theory. Next, considering that the traditional sliding mode control can only suppress matched disturbance and that it is sensitive to mismatched disturbance, a non-linear disturbance observer is used to estimate disturbance, and the estimated variables are used in the design of a sliding mode surface to improve the tracking accuracy of the system. Finally, the experiment is completed on an asynchronous motor drive platform. Compared with the model-free integer-order sliding mode control (MFIOSMC), the results show that the proposed method has good dynamic response and strong robustness. Meanwhile, the proposed method reduces the dependence on mathematical models. Full article

An iterated function system that defines a fractal interpolation function, where ordinate scaling is replaced by a nonlinear contraction, is investigated here. In such a manner, fractal interpolation functions associated with Matkowski contractions for finite as well as infinite (countable) sets of data are obtained. Furthermore, we construct an extension of the concept of $\alpha$-fractal interpolation functions, herein called R-fractal interpolation functions, related to a finite as well as to a countable iterated function system and provide approximation properties of the R-fractal functions. Moreover, we obtain smooth R-fractal interpolation functions and provide results that ensure the existence of differentiable R-fractal interpolation functions both for the finite and the infinite (countable) cases. Full article

As large-sized spacecraft have been developed, they have been equipped with flexible appendages, such as solar cell plates and mechanical flexible arms. The attitude control of spacecraft with flexible appendages has become more complex, with higher requirements. In this paper, a fractional-order PD attitude control method for a type of spacecraft with flexible appendages is presented. Firstly, a lumped parameter model of a spacecraft with flexible appendages is constructed, which provides the transfer function of the attitude angle and external moment. Then, a design method for the fractional-order PD controller for the attitude control of a spacecraft with flexible appendages is provided. Based on the designed steps, a numerical example is provided to compare the control performances between the fractional-order and integer-order PD controllers. Finally, the obtained numerical results are presented to verify the effectiveness of the proposed control method. Full article

In this research, we investigate a novel class of granular type optimality guidelines for the fuzzy multi-objective optimizations based on guidelines of vector granular convexity and granular differentiability. Firstly, the concepts of vector granular convexity is introduced to the vector fuzzy-valued function. Secondly, several properties of vector granular convex fuzzy-valued functions are provided. Thirdly, the granular type Karush-Kuhn-Tucker(KKT) optimality guidelines are derived for the fuzzy multi-objective optimizations. Full article

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order $2-\alpha$ is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers. Full article

Hydraulic concrete in cold regions is necessary for good frost resistance. The utilization of finely ground PS (FGPS) in the construction of hydropower projects could solve the pollution issue and the fly ash shortage problem. In this work, the influence of FGPS and fly ash on frost resistance, pore structure and fractal features of hydraulic concretes was investigated and compared. The main results are: (1) The inclusion of 15–45% FGPS reduced the compressive strength of plain cement concretes by about 21–52%, 7–23% and 0.4–8.2% at 3, 28 and 180 days, respectively. (2) The inclusion of FGPS less than 30% contributed to the enhancement of 180-day frost resistance. At the same dosage level, the FGPS concrete presented larger compressive strengths and better frost resistance than fly ash concrete at 28 and 180 days. (3) At 3 days, both the addition of FGPS and fly ash coarsened the pore structures. FGPS has a much stronger pore refinement effect than fly ash at 28 and 180 days. The correlation between frost resistance of hydraulic concrete and pore structure is weak. (4) At 28 days, the incorporation of FGPS and fly ash weakened the air void structure of hydraulic concrete. At 180 days, the presence of FGPS and fly ash was beneficial for refining the air void structure. The optimal dosage for FGPS and fly ash in terms of 180-day air void refinement was 30% and 15%, respectively. The frost resistance of hydraulic concretes is closely correlated with the air void structure. (5) The pore surface fractal dimension (D_s) could characterize and evaluate the pore structure of hydraulic concretes, but it was poorly correlated with the frost resistance. Full article

To obtain high-quality Pareto optimal solutions and to enhance the searchability of the biogeography-based optimization (BBO) algorithm, we present an improved BBO algorithm based on hybrid migration and a dual-mode mutation strategy (HDBBO). We first adopted a more scientific nonlinear hyperbolic tangent mobility model instead of the conventional linear migration model which can obtain a solution closer to the global minimum of the function. We developed an improved hybrid migration operation containing a micro disturbance factor, which has the benefit of strengthening the global search ability of the algorithm. Then, we used the piecewise application of Gaussian mutation and BBO mutation to ensure that the solution set after mutation was also maintained at a high level, which helps strengthen the algorithm’s search accuracy. Finally, we performed a convergence analysis on the improved BBO algorithm and experimental research based on 11 benchmark functions. The simulation results showed that the improved BBO algorithm had superior advantages in terms of optimization accuracy and convergence speed, which showed the feasibility of the improved strategy. Full article

We propose a discrete-time, finite-state stationary process that can possess long-range dependence. Among the interesting features of this process is that each state can have different long-term dependency, i.e., the indicator sequence can have a different Hurst index for different states. Furthermore, inter-arrival time for each state follows heavy tail distribution, with different states showing different tail behavior. A possible application of this process is to model over-dispersed multinomial distribution. In particular, we define a fractional multinomial distribution from our model. Full article

SEM micrographs of the fracture surface for UO₂ ceramic materials have been analysed. In this paper, we introduce some algorithms and develop a computer application based on the time-series method. Utilizing the embedding technique of phase space, the attractor is reconstructed. The fractal dimension, lacunarity, and autocorrelation dimension average value have been calculated. Full article

In this paper, a collocation method based on the Jacobi polynomial is proposed for a class of optimal-control problems of a fractional distributed system. By using the Lagrange multiplier technique and fractional variational principle, the stated problem is reduced to a system of fractional partial differential equations about control and state functions. The uniqueness of this fractional coupled system is discussed. For spatial second-order derivatives, the proposed method takes advantage of Jacobi polynomials with different parameters to approximate solutions. For a temporal fractional derivative in the Caputo sense, choosing appropriate basis functions allows the collocation method to be implemented easily and efficiently. Exponential convergence is verified numerically under continuous initial conditions. As a particular example, the relation between the state function and the order of the fractional derivative is analyzed with a discontinuous initial condition. Moreover, the numerical results show that the integration of the state function will decay as the order of the fractional derivative decreases. Full article

The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation has a positive solution by utilising the Schauder fixed point theorem (SFPT) and the upper and lower solutions method. Another method for proving the existence of a singular positive solution is the Banach fixed point theorem (BFPT). Finally, we provide an example that illustrates and explains our conclusions. Full article

Distributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, A-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space-fractional, distributed-order derivatives. The considered problem was transformed into a multi-term, space-fractional problem using Simpson’s three-eighths rule. The method is based on an approximation of matrix exponential functions using fourth-order diagonal Padé approximation. The Gaussian quadrature approach is used to approximate the integral matrix exponential function, along with the inhomogeneous term. Partial fraction splitting is used to address the issues regarding stability and computational efficiency. Convergence of the method was proved analytically and demonstrated through numerical experiments. CPU time was recorded in these experiments to show the computational efficiency of the method. Full article

This study presents an extended dissipative analysis of fractional order fuzzy networked control system with uncertain parameters. First, we designed the network-based fuzzy controller for the considered model. Second, a novel Lyapunov-Krasovskii functional (LKF) approach, inequality techniques, and some sufficient conditions are established, which make the proposed system quadratically stable under the extended dissipative criteria. Subsequently, the resultant conditions are expressed with respect to linear matrix inequalities (LMIs). Meanwhile, the corresponding controller gains are designed under the larger sampling interval. Finally, two numerical examples are presented to illustrate the viability of the obtained criteria. Full article

Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point. Full article

A pseudorandom sequence is a repeatable sequence with random statistical properties that is widely used in communication encryption, authentication and channel coding. The pseudorandom sequence generator based on the linear feedback shift register has the problem of a fixed sequence, which is easily tracked. Existing methods use the secret linear feedback shift register (LFSR) and built-in multiple LFSRs and is difficult to prevent cracking based on the hardware analysis. Since the plaintext depends on a specific language to be generated, using pseudo-random sequence encryption, it faces the problem that the encryptor cannot hide the characteristics of the plaintext data. Fractal functions have the following properties: chaotic, unpredictable and random. We propose a novel pseudorandom sequence generator based on the nonlinear chaotic systems, which is constructed by the fractal function. Furthermore, we design a data processing matrix to hide the data characteristics of the sequence and enhance the randomness. In the experiment, the pseudo-random sequences generator passed 16 rigorous test items from the National Institute of Standards and Technology (NIST), which means that the nonlinear pseudorandom sequence generator for the fractal function is effective and efficient. Full article