Fractal Fract., Volume 6, Issue 5 (May 2022) – 55 articles

In this paper, the problem of the uniform stability for a class of fractional-order fuzzy impulsive complex-valued neural networks with mixed delays in infinite dimensions is discussed for the first time. By utilizing fixed-point theory, theory of differential inclusion and set-valued mappings, the uniqueness of the solution of the above complex-valued neural networks is derived. Subsequently, the criteria for uniform stability of the above complex-valued neural networks are established. In comparison with related results, we do not need to construct a complex Lyapunov function, reducing the computational complexity. Finally, an example is given to show the validity of the main results. Full article

The amount by which the artificial neural network weights are updated during the training process is called the learning rate. More precisely, the learning rate is an adjustable parameter used in training neural networks in which small values, often in the interval [0, 1], are handled. The learning rate determines how quickly the model updates its weights to adapt to the problem. Smaller learning rates require more training periods due to small changes to the weights per refresh cycle, while larger learning rates lead to faster changes and require fewer training periods. In this paper, the effect of changing the learning rate value in the artificial neural network designed to solve the inverse problem of fractals was studied. Some results were obtained showing the impact of this change, whether when using large values of the learning rate or small values based on the type of fractal shape required to identify the recursive functions that generate it. Full article

Some researchers believe fractional differential operators should not have a non-singular kernel, while others strongly believe that due to the complexity of nature, fractional differential operators can have either singular or non-singular kernels. This contradiction in thoughts has led to the publication of a few papers that are against differential operators with non-singular kernels, causing some negative impacts. Thus, publishers and some Editors-in-Chief are concerned about the future of fractional calculus, which has generally brought confusion among the vibrant and innovative young researchers who desire to apply fractional calculus within their respective fields. Thus, the present work aims to develop a model based on a stochastic process that could be utilized to portray the effect of arbitrary-order derivatives. A nonlinear perturbation is used to study the proposed stochastic model with the help of white noises. The required condition(s) for the existence of an ergodic stationary distribution is obtained via Lyapunov functional theory. The finding of the study indicated that the proposed noises have a remarkable impact on the dynamics of the system. To reduce the spread of a disease, we imposed some control measures on the stochastic model, and the optimal system was achieved. The models both with and without control were coded in MATLAB, and at the conclusion of the research, numerical solutions are provided. Full article

In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is driven by a fractional Lévy process. The option price is given by a deterministic representation by means of a real valued function satisfying some fractional PDE. The numerical scheme of the fractional PDE is obtained by means of a weighted and shifted Grunwald approximation. Full article

Over time and across space, the hydraulic conductivity, fractal dimension, and porosity of embankment soil have strong randomness, which makes analyzing seepage fields difficult, affecting embankment risk analysis and early disaster warning. This strong randomness limits the application of fractal theory in embankment engineering and sometimes keeps it in the laboratory stage. Based on the capillary model of porous soil, an analytical formula of the fractal relationship between hydraulic conductivity and fractal dimension is derived herein. It is proposed that the influencing factors of hydraulic conductivity of embankment soil mainly include the capillary aperture, fractal dimension, and fluid viscosity coefficient. Based on random field theory and combined with the embankment parameters of Shijiu Lake, hydraulic conductivity is discretized, and then the soil fractal dimension is approximately solved to reveal the internal relationship between hydraulic gradient, fractal dimension, and hydraulic conductivity. The results show that an increased fractal dimension will reduce the connectivity of soil pores in a single direction, increase the hydraulic gradient, and reduce the hydraulic conductivity. A decreased fractal dimension will lead to consistency of seepage channels in the soil, increased hydraulic conductivity, and decreased hydraulic gradient. Full article

This paper concerns a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features. The hidden long-range dependence and the multifractal property are extracted to determine the fractional order. A fractional autoregressive integrated moving average model (FARIMA) is then proposed considering innovations with stable infinite variance. The existence and convergence of the model solutions are discussed in depth. Ensemble learning with an autoregressive moving average model (ARMA) is used to further improve upon accuracy and generalization. The proposed method is used to predict the energy consumption in a real cooling system, and superior prediction results are obtained. Full article

In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with $x^{\alpha }$, with positive $\alpha$. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if $\alpha =1$. Full article

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid. Full article

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, $\psi$-Caputo, $\psi$-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. Full article

The purpose of this paper is to introduce several generalized F-contractions in b-metric-like spaces and establish some fixed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, we find the existence and uniqueness of a solution to a class of integral equations in the context of b-metric-like spaces. Full article

In this paper, we introduce a special family ${\mathfrak{M}}_{{\sigma }_m}(\tau ,\nu ,\eta ,\phi )$ of the function family ${\sigma }_m$ of m-fold symmetric bi-univalent functions defined in the open unit disc $\mathfrak{D}$ and obtain estimates of the first two Taylor–Maclaurin coefficients for functions in the special family. Further, the Fekete–Szegö functional for functions in this special family is also estimated. The results presented in this paper not only generalize and improve some recent works, but also give new results as special cases. Full article

Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies. Full article

As we know one of the most important equations which have many applications in various areas of physics, mathematics, and financial markets, is the Sturm–Liouville equation. In this paper, by using the $\alpha$-$\psi$-contraction technique in fixed point theory and employing some functional inequalities, we study the existence of solutions of the partial fractional hybrid case of generalized Sturm–Liouville-Langevin equations under partial boundary value conditions. Towards the end, we present two examples with numerical and graphical simulation to illustrate our main results. Full article

First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained. Full article

As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our findings is provided. Full article

The purpose of this article is to discuss the existence, uniqueness, and Ulam–Hyers stability of solutions to a coupled system of fractional differential equations with Erdélyi–Kober and Riemann–Liouville integral boundary conditions. The Banach fixed point theorem is used to prove the uniqueness of solutions, while the Leray–Schauder alternative is used to prove the existence of solutions. Furthermore, we conclude that the solution to the discussed problem is Hyers–Ulam stable. The results are illustrated with examples. Full article

This article examines a natural convection viscous unsteady fluid flowing on an oscillating infinite inclined plate. The Newtonian heating effect, slip effect on the boundary wall, and constant mass diffusion conditions are also considered. In order to account for extended memory effects, the semi-analytical solution of transformed governed partial differential equations is attained with the help of a recent and more efficient fractional definition known as Prabhakar, like a thermal fractional derivative with Mittag-Leffler function. Fourier and Fick’s laws are also considered in the thermal profile and concentration field solution. The essentials’ preliminaries, fractional model, and execution approach are expansively addressed. The physical impacts of different parameters on all governed equations are plotted and compared graphically. Additionally, the heat transfer rate, mass diffusion rate, and skin friction are examined with different numerical techniques. Consequently, it is noted that the variation in fractional parameters results in decaying behavior for both thermal and momentum profiles while increasing with the passage of time. Furthermore, in comparing both numerical schemes and existing literature, the overlapping of both curves validates the attained solution of all governed equations. Full article

In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretized by a fourth-order Riesz fractional difference operator in spatial discretization and the collocation methods in the temporal direction. Not only can the present method achieve fourth-order accuracy in the spatial direction and arbitrarily high-order accuracy in the temporal direction, but it also has long-time computing stability. Then, the unconditional discrete energy dissipation law of the present numerical schemes is proved. Finally, some numerical experiments are provided to certify the efficiency and the structure-preserving properties of the proposed schemes. Full article

A novel computational approach is developed to investigate the mixed convection, boundary layer flow over a nonlinear elastic (stretching or shrinking) surface. The viscous fluid is electrically conducting, incompressible, and propagating through a porous medium. The consequences of viscous dissipation, Joule heating, and heat sink/source of the volumetric rate of heat generation are also included in the energy balance equation. In order to formulate the mathematical modeling, a similarity analysis is performed. The numerical solution of nonlinear differential equations is accomplished through the use of a robust computational approach, which is identified as the Spectral Local Linearization Method (SLLM). The computational findings reported in this study show that, in addition to being simple to establish and numerically implement, the proposed method is very reliable in that it converges rapidly to achieve a specified goal and is more effective in resolving very complex models of nonlinear boundary value problems. In order to ensure the convergence of the proposed SLLM method, the Gauss–Seidel approach is used. The SLLM’s reliability and numerical stability can be optimized even more using Gauss–Seidel approach. The computational results for different emerging parameters are computed to show the behavior of velocity profile, skin friction coefficient, temperature profile, and Nusselt number. To evaluate the accuracy and the convergence of the obtained results, a comparison between the proposed approach and the bvp4c (built-in command in Matlab) method is presented. The Matlab software, which is used to generate machine time for executing the SLLM code, is also displayed in a table. Full article

Image dehazing is a traditional task, yet it still presents arduous problems, especially in the removal of haze from the texture and edge information of an image. The state-of-the-art dehazing methods may result in the loss of some visual informative details and a decrease in visual quality. To improve dehazing quality, a novel dehazing model is proposed, based on a fractional derivative and data-driven regularization terms. In this model, the contrast constrained adaptive histogram equalization method is used as the data fidelity item; the fractional derivative is applied to avoid over-enhancement and noise amplification; and the proposed data-driven regularization terms are adopted to extract the local and non-local features of an image. Then, to solve the proposed model, half-quadratic splitting is used. Moreover, a dual-stream network based on Convolutional Neural Network (CNN) and Transformer is introduced to structure the data-driven regularization. Further, to estimate the atmospheric light, an atmospheric light model based on the fractional derivative and the atmospheric veil is proposed. Extensive experiments display the effectiveness of the proposed method, which surpasses the state-of-the-art methods for most synthetic and real-world images, quantitatively and qualitatively. Full article

The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, we first study some well-known classes of starlike functions and then determine the third Hankel determinant bound for the logarithmic coefficients of certain subclasses of starlike functions that also involve the sine functions. We also obtain a number of coefficient estimates. Some of our results are shown to be sharp. Full article

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a $\delta$-sink and an asymmetric space-fractional derivative operator with stable index $\alpha$ and asymmetry (skewness) parameter $\beta$. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with $\beta \le 0$ (with a rightward bias) for short initial distances, while for $\beta >0$ (with a leftward bias) Lévy flights with $\alpha \to 1$ are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for $\alpha =1$ with $\beta =0$) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices $\alpha$ close to unity. Full article

The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to $O({\tau }^{2-\alpha }+h^2)$ under strong regularity and $O({\tau }^{\alpha }+h^2)$ under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE. Full article

This study is built on the relationship between inequality theory and fractional analysis. Thanks to the new fractional operators and based on the proportional Caputo-hybrid operators, integral inequalities containing new approaches are obtained for differentiable convex functions. In the findings section, firstly, an integral identity is obtained and various integral inequalities are obtained based on this identity. The peculiarity of the results is that a hybrid operator has been used in inequality theory, which includes the derivative and integral operators together. Full article

The aim of this work is to describe the dynamics of a fractional-order coupled FitzHugh–Nagumo neuronal model. The equilibrium states are analyzed in terms of their stability properties, both dependently and independently of the fractional orders of the Caputo derivatives, based on recently established theoretical results. Numerical simulations are shown to clarify and exemplify the theoretical results. Full article

In contrast to previous research on periodic averaging principles for various types of impulsive stochastic differential equations (ISDEs), we establish an averaging principle without periodic assumptions of coefficients and impulses for impulsive stochastic fractional differential equations (ISFDEs) excited by fractional Brownian motion (fBm). Under appropriate conditions, we demonstrate that the mild solution of the original equation is approximately equivalent to that of the reduced averaged equation without impulses. The obtained convergence result guarantees that one can study the complex system through the simplified system. Better yet, our techniques dealing with multi-time scales and impulsive terms can be applied to improve some existing results. As for application, three examples are worked out to explain the procedure and validity of the proposed averaging principles. Full article

In the present article, we study a class of Kirchhoff-type equations driven by the $\left(p\right(x),q(x\left)\right)$-Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not applicable. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions. One of the main difficulties and innovations of the present article is that we consider competing $\left(p\right(x),q(x\left)\right)$-Laplacian, convective terms, and logarithmic nonlinearity with variable exponents, another one is the weaker assumptions on nonlocal term $M_{\upsilon \left(x\right)}$ and nonlinear term g. Full article

The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox–Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived. Full article

This article focuses on designing an adaptive sliding mode controller via state and output feedback for nonlinear singular fractional-order systems (SFOSs) with mismatched uncertainties. Firstly, on the basis of extending the dimension of the SFOS, a new integral sliding mode surface is constructed. Through this special sliding surface, the sliding mode of the descriptor system does not contain a singular matrix E. Then, the sufficient conditions that ensure the stability of sliding mode motion are given by using linear matrix inequality. Finally, the control law based on an adaptive mechanism that is used to update the nonlinear terms is designed to ensure the SFOS satisfies the reaching condition. The applicability of the proposed method is illustrated by a practical example of a fractional-order circuit system and two numerical examples. Full article

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics. Full article