Fractal Fract., Volume 7, Issue 7 (July 2023) – 77 articles

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ($UDET\text{-}$convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for $UDET\text{-}\mathrm{convex} FNVM$s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. Full article

In this paper, a mixed-order image denoising algorithm containing fractional-order and high-order regularization terms is proposed, which effectively suppresses the staircase effect generated by the TV model and its variants while better preserving the edges and details of the image. Adding different regularization penalties in different regions is fundamental to improving the denoising performance of the model. Therefore, a weight selection function is designed using the structure tensor to achieve a more effective selection of regularization terms in different regions. In each iteration, the regularization parameters are adaptively adjusted according to the Morozov discrepancy principle to promote the performance of the algorithm. Based on the primal–dual theory, the original algorithm is improved by using the predictor–corrector scheme to obtain a more accurate approximate solution while ensuring the convergence of the algorithm. The effectiveness of the proposed algorithm is demonstrated through simulation experiments. Full article

The present manuscript is devoted to investigating some existence and uniqueness results on fixed points by employing generalized contractions in the context of metric space endued with a weak class of transitive relation. Our results improve, modify, enrich and unify several existing fixed point theorems, The results proved in this study are utilized to find a unique solution of certain fractional boundary value problems. Full article

A fully discrete space-time finite element method for the fractional Ginzburg–Landau equation is developed, in which the discontinuous Galerkin finite element scheme is adopted in the temporal direction and the Galerkin finite element scheme is used in the spatial orientation. By taking advantage of the valuable properties of Radau numerical integration and Lagrange interpolation polynomials at the Radau points of each time subinterval ${\mathcal{I}}_n$, the well-posedness of the discrete solution is proven. Moreover, the optimal order error estimate in $L^{\infty }\left(L^2\right)$ is also considered in detail. Some numerical examples are provided to evaluate the validity and effectiveness of the theoretical analysis. Full article

The Jeffreys-type heat conduction equation with flux precedence describes the temperature of diffusive hot electrons during the electron–phonon interaction process in metals. In this paper, the deformation resulting from ultrafast surface heating on a “nanoscale” plate is considered. The focus is on the anomalous heat transfer mechanisms that result from anomalous diffusion of hot electrons and are characterized by retarded thermal conduction, accelerated thermal conduction, or transition from super-thermal conductivity in the short-time response to sub-thermal conductivity in the long-time response and described by the fractional Jeffreys equation with three fractional parameters. The recent double-strip problem, Awad et al., Eur. Phy. J. Plus 2022, allowing the overlap between two propagating thermal waves, is generalized from the semi-infinite heat conductor case to thermoelastic case in the finite domain. The elastic response in the material is not simultaneous (i.e., not Hookean), rather it is assumed to be of the Kelvin–Voigt type, i.e., $\sigma =E\left(\epsilon +{\tau }_{\epsilon }\dot{\epsilon }\right)$, where $\sigma$ refers to the stress, $\epsilon$ is the strain, $E$ is the Young modulus, and ${\tau }_{\epsilon }$ refers to the strain relaxation time. The delayed strain response of the Kelvin–Voigt model eliminates the discontinuity of stresses, a hallmark of the Hookean solid. The immobilization of thermal conduction described by the ordinary Jeffreys equation of heat conduction is salient in metals when the heat flux precedence is considered. The absence of the finite speed thermal waves in the Kelvin–Voigt model results in a smooth stress surface during the heating process. The temperature contours and the displacement vector chart show that the anomalous heat transfer characterized by retardation or crossover from super- to sub-thermal conduction may disrupt the ultrafast laser heating of metals. Full article

In this article, the security control problem of discrete-time fractional-order networked systems under denial-of-service (DoS) attacks is considered. A practically applicable finite-dimensional control strategy will be developed for fractional-order systems that possess nonlocal characteristics. By employing the Lyapunov method, it is theoretically proved that under the proposed controller, the obtained closed-loop fractional system is globally input-to-state stable (ISS), even in the presence of DoS attacks. Finally, the effectiveness of the designed control method is demonstrated by the numerical example. Full article

One of the most popular controllers for the automatic voltage regulator (AVR) in maintaining the voltage level of a synchronous generator is the fractional-order proportional–integral-derivative (FOPID) controller. Unfortunately, tuning the FOPID controller is challenging since there are five gains compared to the three gains of a conventional proportional–integral–derivative (PID) controller. Therefore, this research work presents a variant of the marine predators algorithm (MPA) for tuning the FOPID controller of the AVR system. Here, two modifications are applied to the existing MPA: the hybridization between MPA and the safe experimentation dynamics algorithm (SEDA) in the updating mechanism to solve the local optima issue, and the introduction of a tunable step size adaptive coefficient (CF) to improve the searching capability. The effectiveness of the proposed method in tuning the FOPID controller of the AVR system was assessed in terms of the convergence curve of the objective function, the statistical analysis of the objective function, Wilcoxon’s rank test, the step response analysis, stability analyses, and robustness analyses where the AVR system was subjected to noise, disturbance, and parameter uncertainties. We have shown that our proposed controller has improved the AVR system’s transient response and also produced about two times better results for objective function compared with other recent metaheuristic optimization-tuned FOPID controllers. Full article

This work investigates a weighted Banach space second order pantograph fractional differential equation. The considered equation is of second order, expressed in terms of the Caputo–Hadamard fractional operator, and constructed in a general manner to accommodate many specific situations. The asymptotic stability of the main equation’s trivial solution has been given. The primary theorem was demonstrated in a unique manner by employing the Krasnoselskii’s fixed point theorem. We provide a concrete example that supports the theoretical findings. Full article

In this investigation, weighted $psi$-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes into account the situation where $\psi \left(\mathsf{x}\right)=\mathsf{x}$. With the aid of Banach’s and Krasnoselskii’s classic fixed point results, we have established a the qualitative study. Different values of $\psi \left(\mathsf{x}\right)$ and $\mathsf{w}\left(\mathsf{x}\right)$ are discussed as special cases that are relevant to our current results. Additionally, in light of our findings, we provide a more general fractional system with the weighted $\psi$-Caputo-type that takes into account both the new problems and certain previously existing, related problems. Finally, we give two illustrations to support and validate the outcomes. Full article

In this paper, we introduce a space-fractional version of the molecular beam epitaxy (MBE) model without slope selection to describe super-diffusion in the model. Compared to the classical MBE equation, the spatial discretization is an important issue in the space-fractional MBE equation because of the nonlocal nature of the fractional operator. To approximate the fractional operator, we employ the Fourier spectral method, which gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power. And, to combine with the Fourier spectral method directly, we present a linear, energy stable, and second-order method. Then, it is possible to simulate the dynamics of the space-fractional MBE equation efficiently and accurately. By using the numerical method, we investigate the effect of the fractional power in the space-fractional MBE equation. Full article

In the present work, the main objective is to find the solution of the generalized heat and generalized Laplace equations using the fractional Fourier transform, which is a general form of the solution of the heat equation and Laplace equation using the classical Fourier transform. We also formulate its solution using a sampling formula related to the fractional Fourier transform. The fractional Fourier transform is introduced, and related theorems and essential properties are collected. Several results related to the sampling formula are derived. A few examples are presented to illustrate the effectiveness and powerfulness of the proposed method compared to the classical Fourier transform method. Full article

The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend $\mathcal{F}$-expansion approach and the Jacobi elliptic function method. Since nonlinear pulse transmission in monomode optical fibers is explained by the Fokas system, the derived solutions may be utilized to analyze a broad range of important physical processes. In order to comprehend the impacts of MTD on the solutions, the dynamic behavior of the various generated solutions are shown using 2D and 3D figures. Full article

In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained. In addition, we demonstrate the existence of ground state solutions for the corresponding limit problem. Full article

Different weakening buffer operators in a time-series model analysis usually result in different model sensitivities, which sometimes affect the effectiveness of relevant operator-based methods. In this paper, the stability of two classic fractional-order weakening buffer operator-based series models is studied; then, a new data preprocessing method based on a novel fractional-order bidirectional weakening buffer operator is provided, whose effect in improving the model’s stability is tested and utilized in prediction problems. Practical examples are employed to demonstrate the efficiency of the proposed method in improving the model’s stability in noise scenarios. The comparison indicates that the proposed method overcomes the disadvantage of many weakening buffer operators in the subjectively biased weighting of the new or old information in forecasting. These expand the application of the proposed method in time series analysis. Full article

Metaheuristic optimization algorithms (MHA) play a significant role in obtaining the best (optimal) values of the system’s parameters to improve its performance. This role is significantly apparent when dealing with systems where the classical analytical methods fail. Fractional-order (FO) systems have not yet shown an easy procedure to deal with the determination of their optimal parameters through traditional methods. In this paper, a recent, systematic. And comprehensive review is presented to highlight the role of MHA in obtaining the best set of gains and orders for FO controllers. The systematic review starts by exploring the most relevant publications related to the MHA and the FO controllers. The study is focused on the most popular controllers such as the FO-PI, FO-PID, FO Type-1 fuzzy-PID, and FO Type-2 fuzzy-PID. The time domain is restricted in the articles published through the last decade (2014:2023) in the most reputed databases such as Scopus, Web of Science, Science Direct, and Google Scholar. The identified number of papers, from the entire databases, has reached 850 articles. A Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) methodology was applied to the initial set of articles to be screened and filtered to end up with a final list that contains 82 articles. Then, a thorough and comprehensive study was applied to the final list. The results showed that Particle Swarm Optimization (PSO) is the most attractive optimizer to the researchers to be used in the optimal parameters identification of the FO controllers as it attains about 25% of the published papers. In addition, the papers that used PSO as an optimizer have gained a high citation number despite the fact that the Chaotic Atom Search Optimization (ChASO) is the highest one, but it is used only once. Furthermore, the Integral of the Time-Weighted Absolute Error (ITAE) is the best nominated cost function. Based on our comprehensive literature review, this appears to be the first review paper that systematically and comprehensively addresses the optimization of the parameters of the fractional-order PI, PID, Type-1, and Type-2 fuzzy controllers with the use of MHAs. Therefore, the work in this paper can be used as a guide for researchers who are interested in working in this field. Full article

In this study, we considered a model for novel COVID-19 consisting on five classes, namely $\mathcal{S}$, susceptible; $\mathcal{E}$, exposed; $\mathcal{I}$, infected; $\mathcal{V}$, vaccinated; and $\mathcal{R}$, recovered. We derived the expression for the basic reproductive rate ${\mathcal{R}}_0$ and studied disease-free and endemic equilibrium as well as local and global stability. In addition, we extended the nonstandard finite difference scheme to simulate our model using some real data. Moreover, keeping in mind the importance of fractional order derivatives, we also attempted to extend our numerical results for the fractional order model. In this regard, we considered the proposed model under the concept of a fractional order derivative using the Caputo concept. We extended the nonstandard finite difference scheme for fractional order and simulated our results. Moreover, we also compared the numerical scheme with the traditional RK4 both in CPU time as well as graphically. Our results have close resemblance to those of the RK4 method. Also, in the case of the infected class, we compared our simulated results with the real data. Full article

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis. Full article

Aiming at the problem of the unclear rock burst generation mechanism and anti-impact measures of a large roadway through a coal seam in a deep panel area, taking the rock burst of a large roadway in the first panel area of Gaojiapu Coal Mine as the engineering background, this paper adopts the comprehensive research methods of theoretical analysis, experiments, numerical simulation, fragmentation fractal analysis, and field monitoring, to discuss the mechanical characteristics of the loading process of the assemblage and the energy transfer law and its difference in the deformation and failure process. The possibility and strength of the impact failure of coal under the grip of rock masses with different stiffness are related to the γ value. The smaller the γ value is, the higher the impact possibility is, and the more severe the impact degree is. The assemblage under the grip of soft rock is more prone to system instability. Energy relief and impact reduction are adopted to reduce the post-peak stiffness and elastic strain energy of the coal body in a short distance, avoid the energy transfer and concentration of the roadway surrounding the rock system under the disturbance of a long-distance dynamic load, and reduce the likelihood of impact pressure occurring and the extent to which an impact manifests. Full article

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems. Full article

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models. Full article

Cryptocurrency prices have the characteristic of high volatility, which has a specific resistance to cryptocurrency price prediction. Therefore, the appropriate cryptocurrency price predictive method can help reduce the investment risk of investors. In this study, we proposed a novel prediction method using a fractional grey model (FGM (1,1)) to predict the price of blockchain cryptocurrency. Specifically, this study established the FGM (1,1) through the closing price of three representative blockchain cryptocurrencies (Bitcoin (BTC), Ethereum (ETH), and Litecoin (LTC)). It adopted the PSO algorithm to optimize and obtain the optimal order of the model, thereby conducting prediction research on the price of blockchain cryptocurrency. To verify the predictive precision of the FGM (1,1), we mainly took MAPE, MAE, and RMSE as the judging criteria and compared the model’s predictive precision with the GM (1,1) through experiments. The research results indicate that within the data range studied, the predictive accuracy of the FGM (1,1) in the closing price of BTC, ETH, and LTC has reached a “highly accurate” level. Moreover, in contrast to the GM (1,1), the FGM (1,1) outperforms predictive capability in the experiments. This study provides a feasible new method for the price prediction of blockchain cryptocurrency. It has specific references and enlightenment for government departments, investors, and researchers in theory and practice. Full article

The integrity index K_v is the quantitative index in the CHN-BQ method, which can be determined by the acoustic wave test, volume joint number J_v, or empirical judgment. However, these methods are not convenient and require the practitioner to have extensive experience. In this study, a new quantitative evaluation of K_v is proposed to determine K_v accurately and conveniently. A method for determining the fractal dimension D based on the structural plane network simulation is proposed. A quantitative relationship between fractal dimension D and integrity index K_v is established based on the geological information from 80 sampling windows in Mingtang Tunnel. To further consider the effect of structural plane conditions on K_v, a BP neural network is constructed with the fractal dimension D and structural plane condition index R₃ as input and K_v as output. The BP neural network is trained by 260 groups of tunnel data and validated by 39 groups of test data. The results show that the correlation coefficient R² between the predicted K_vp and measured K_vm is 0.93, and the average relative error is 7.51%. In addition, the predicted K_vp from the 39 groups of data is compared with the K_vd determined directly by fractal dimension D. It can be found that the K_vd has a larger error compared with the K_vp, especially in the case of a K_v less than 0.5. Finally, the BP neural network for predicting K_v is applied to the Jiulaopo Tunnel. The maximum relative error between the measured K_vm and the predicted K_vp is 5.13%, and the average relative error is 2.71%. The BP neural network is well trained and can accurately predict K_v based on the fractal dimension D and the structural plane condition index R₃. Full article

The knowledge of fractional calculus can be useful in developing models that allow us to better understand and manage some phenomena. In the present paper, we study the topological structure of the mild solution set for a semi-linear differential inclusion containing the $\tau$-Caputo fractional derivative in the presence of non-instantaneous impulses and an infinite delay. We demonstrate that this set is non-empty and an $R_{\delta }$-set. We use a recent result regarding the existence of solutions for $\tau$-Caputo fractional semi-linear differential inclusions. Unlike many results, we do not suppose that the generating semigroup is compact. An illustrative example is given as an application of our results. Full article

The modeling of biological processes has increasingly been based on fractional calculus. In this paper, a novel fractional-order model is used to investigate the epidemiological impact of vaccination measures on the co-dynamics of viral hepatitis B and COVID-19. To investigate the existence and stability of the new model, we use some fixed point theory results. The COVID-19 and viral hepatitis B thresholds are estimated using the model fitting. The vaccine parameters are plotted against transmission coefficients. The effect of non-integer derivatives on the solution paths for each epidemiological state and the trajectory diagram for infected classes are also examined numerically. An infection-free steady state and an infection-present equilibrium are achieved when ${\mathcal{R}}_0<1$ and ${\mathcal{R}}_0>1$, respectively. Similarly, phase portraits confirm the behaviour of the infected components, showing that, regardless of the order of the fractional derivative, the trajectories of the disease classes always converge toward infection-free steady states over time, no matter what initial conditions are assumed for the diseases. The model has been verified using real observations. Full article

Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model that captures the dynamics of relapse in a smoking cessation context and presents the dynamic behavior of the proposed model utilizing Caputo fractional derivatives. The model incorporates four compartments representing potential, persistent (heavy), temporally recovered, and permanently recovered smokers. The basic reproduction number $R_0$ is computed, and the local and global dynamic behaviors of the free equilibrium smoking point (${\mathcal{Y}}_0$) and the smoking-present equilibrium point (${\mathcal{Y}}^{*}$) are analyzed. It is demonstrated that the free equilibrium smoking point (${\mathcal{Y}}_0$) exhibits global asymptotic stability when $R_0\le 1$, while the smoking-present equilibrium point (${\mathcal{Y}}^{*}$) is globally asymptotically stable when $R_0>1$. Additionally, analytical results are validated through a numerical simulation using the predictor–corrector PECE method for fractional differential equations in Matlab software. Full article

Voltage-lift is a widely used technique in DC–DC converters to step-up output voltage levels. Several traditional and advanced control techniques applicable to power electronic converters (PEC) have been reported and utilized for voltage-lift applications. Similarly, in recent years the implementation of fractional-order controllers (FOC) in PEC applications has gained interest, aiming to improve system performance, and has been validated in basic converter topologies. Following this trend, this work presents an FOC for a voltage-lift converter, requiring only output voltage feedback. A third-order non-minimal phase system is selected for experimentation to verify FOC implementations for more complex PEC configurations. A simple, straightforward design and approximation methodology for the FOC is proposed. Step-by-step development of the FOC, numerical and practical results on a 50 W voltage-lift converter are reported. The results show that PEC transient and steady-state responses can be enhanced using FOC controllers when compared with classical linear controllers. Extended applications of FOC for improved performance in power conversion is also discussed. Full article

In this study, the multivalued fixed point theorem, Clarke subdifferential properties, fractional calculus, and stochastic analysis are used to arrive at the system’s mild solution (1). Furthermore, the mean square moment for the aforementioned system (1) confirms the conditions for trajectory (T-)controllability. The last part of the paper uses two numerical applications to explain the novel theoretical results that were reached. Full article

Multifractal detrended fluctuation analysis (DFA) can extract multi-scaling behavior and measure long-range correlations in climatic time series. In this study, with the help of multifractal DFA, we investigated the scaling behavior of daily minimum/maximum temperatures during the years 1989–2019 from 34 meteorological stations in Bangladesh. We revealed spatial patterns, topographic impacts and global warming impacts of long-range correlations embedded in small and large fluctuations in temperature time series. Meanwhile, we developed a multifractal DFA-based algorithm to dynamically determine thresholds to discriminate extreme and non-extreme events in climate systems and applied it to analyze the frequency and trends of temperature extremes in Bangladesh. Compared with widely-used percentile thresholds, the extreme climate events captured in our algorithm are more reliable since they are determined dynamically by the climate system itself. Full article

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization. Full article

Extensive research was conducted on the transmission dynamics of tuberculosis epidemics during its reemergence from the 1980s to the early 1990s, but this global problem of investigating tuberculosis spread dynamics remains of paramount importance. Our study utilized a fractional-order delay differential model to study tuberculosis transmission, where the time delay in the model was attributed to the disease’s latent period. What is more, this model accounts for endogenous reactivation, exogenous reinfection, and treatment of tuberculosis. The model qualitative properties and the basic reproduction number were analyzed. The primary goal of the study was to recover the important dynamic parameters of tuberculosis. Our understanding of these complex processes leverages the efficacy of efforts for controlling the disease, forecasting future dynamics, and applying further appropriate strategies to prevent its spread.The calibration itself was carried out via minimization of a quadratic cost functional. Computational simulations demonstrated that the algorithm is capable of working with noisy real data. Full article