The close link between the roughness of a surface and its adsorptive properties in Cerofolini’s model yields, with an adequate choice of adsorption energy, the well-known Dubinin-Radushkevich or Freundlich adsorption isotherms. Assuming fractal behavior concerning both energetic and geometric surface heterogeneities described by the power-law expressions and fractal dimensions, the paper will develop some fractal adsorption isotherms. Using our theoretical approach, fractal isotherms will provide insights not only into the fractal behavior of the surface geometry but also into the fractal energetic heterogeneities, implying that a sorbent does not need to be porous to apply a fractal isotherm: adsorption on “flat” surfaces can also be described by fractal isotherms and fractal dimensions related to energetic disorders. For example, the theory will be applied to computing the energetic fractal dimensions of some nanoparticle catalysts, Rh/Al₂O₃, Rh/TiO₂, and Rh/WO₃. Full article

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed, and the effects of three typical calculation methods in different test scenarios are compared and studied in this paper. It is proved that Oustaloup’s rational approximation method usually overestimates the peak-to-peak current and brings in the pulse–voltage phenomenon in piecewise smooth test scenarios, while the results of the two iterative recurrence-form numerical methods are in good agreement with the experimental results. The study results are dedicated to provide a reference for efficiently deploying calculation methods in fractional-order piecewise smooth circuit systems. Some quantitative analysis results are concluded in this paper. Full article

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to an ordinary continuous medium, a randomly wandering particle moves away from the reference point more slowly since not all directions of motion become available to it. The slowdown of the diffusion process in fractal media is so significant that physical quantities begin to change more slowly than in ordinary media.This effect can only be taken into account with the help of integral and differential equations containing a fractional derivative with respect to time. Here, the problem of heat and mass transfer in media with a fractal structure was posed and analytically solved when a heat flux was specified on one of the boundaries. The second initial boundary value problem for the heat equation with a fractional Caputo derivative with respect to time and the Riesz derivative with respect to the spatial variable was studied. A theorem on the semigroup property of the fractional Riesz derivative was proved. To find a solution, the problem was reduced to a boundary value problem with boundary conditions of the first kind. The solution to the problem was found by applying the Fourier transform in the spatial variable and the Laplace transform in time. A computational experiment was carried out to analyze the obtained solutions. Graphs of the temperature distribution dependent on the coordinate and time were constructed. Full article

In the article, the existence of solutions for the Van der Pol differential equation is proved, and the approximate structure of such solutions in the analyticity domain is obtained. In the proof, the majorant method was applied not to the right side of the differential equation, as per usual, but to the solution to the nonlinear differential equation under consideration. Results of the numerical study are presented. Full article

This paper is concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional $r(·)$-Laplacian equations of Kirchhoff–Schrödinger type with concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, under suitable assumptions on a nonlinear term, is to discuss the multiplicity result of non-trivial solutions by using the dual fountain theorem as the main tool. Full article

As the fourth fundamental circuit element, the memristor may execute computations while storing data. Fracmemristor takes advantage of the fractional calculate’s long-term memory, non-locality, weak singularity, and the memristor’s storage–computational integration. Since the physical structure of the fracmemristor is similar to the topology of the ant transfer probability flow in ACO, we propose the fractional-order memristive ant colony algorithm (FMAC), which uses the fracmemristor physical system to record the probabilistic transfer information of the nodes that the ant will crawl through in the future and pass it to the current node of the ant, so that the ant acquires the ability to predict the future transfer. After instigating the optimization capabilities with TSP, we discovered that FMAC is superior to PACO-3opt, the best integer-order ant colony algorithm currently available. FMAC operates substantially more quickly than the fractional-order memristor ant colony algorithm due to the transfer probability prediction module based on the physical fracmemristor system (FACA). Full article

This paper deals with the problem of group consensus for a fractional-order multi-agent system (FOMAS) without considering the intergroup balance condition. By adopting a dynamic event-triggered mechanism, the updating frequency of control input is significantly reduced while the consensus performance is maintained. By utilizing the Lyapunov direct method and the properties of a fractional-order derivative, several novel criteria are presented for analyzing the Mittag–Leffler stability of error systems and excluding the Zeno behavior in the triggering mechanism. An example and its simulations are demonstrated to prove the validity of the theoretical results. Full article

Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms. Full article

Road traffic networks are chaotic and highly complex systems. In this paper, we introduce a dynamic gravity model that characterizes the behaviors of the O-D (origin-destination) traffic, such as equilibrium, period-doubling, chaos, and fractal in discrete time. In cases where the original cost function is used, the trip distribution model might degenerate into an all-or-nothing problem without the capacity constraints. To address this shortcoming, we propose substituting the original cost function with an improved conical volume-delay function. This new function retains some of the properties of the original cost function, and its parameters have the same meaning as those in the original function. Our analysis confirms that the double-constrained dynamic gravity model successfully characterizes complex traffic behavior because of the improved conical volume-delay function. Our analysis further shows that the three-parameter bifurcation diagram based on the period characteristics provides deep insight into the actual state of the road traffic networks. Investigating the properties of the model solutions, we further show that the new model is more effective in addressing the all-or-nothing problem. Full article

In this paper, we study some relations between different weights in the classes ${\mathcal{B}}_p,$${\mathcal{B}}_p^{*},$${\mathcal{M}}_p$ and ${\mathcal{M}}_p^{*}$ that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space ${\Lambda }_p$. These results will be proven by using the properties of classes ${\mathcal{B}}_p,$${\mathcal{B}}_p^{*},$${\mathcal{M}}_p$ and ${\mathcal{M}}_p^{*}$, including the self-improving properties and also the properties of the generalized Hardy operator ${\mathcal{H}}_p$, the adjoint operator ${\mathcal{S}}_q$ and some fundamental relations between them connecting their composition to their sum. Full article

In this study, we proposed a sliding mode control method based on fixed-time sliding mode surface for the synchronization of uncertain fractional-order hyperchaotic systems. In addition, we proposed a novel self-evolving non-singleton-interval type-2 probabilistic fuzzy neural network (SENSIT2PFNN) to estimate the uncertain dynamics of the system. Moreover, an adaptive compensator was designed to eliminate the influences of random uncertainty and fuzzy uncertainty, thereby yielding an asymptotically stable controlled system. Furthermore, an adaptive law was introduced to optimize the consequence parameters of SENSIT2PFNN. The membership layer and rule base of SENSIT2PFNN were optimized using the self-evolving algorithm and whale optimization algorithm, respectively. The simulation results verified the effectiveness of the proposed methods for the synchronization of uncertain fractional-order hyperchaotic systems. Full article

Fractional differential operators have recently been linked with numerous other areas of science, technology and engineering studies. For a real variable, the class of fractional differential and integral operators is evaluated. In this study, we look into the Prabhakar fractional differential operator, which is the most applicable fractional differential operator in a complex domain. In terms of observing a group of normalized analytical functions, we express the operator. In the open unit disc, we deal with its geometric performance. Applying the Prabhakar fractional differential operator ${}_d^c{\theta }_{\alpha ,\beta }^{\gamma ,\omega }$ to a subclass of analytic univalent function results in the creation of a new subclass of mathematical functions: $\mathcal{W}$($\gamma ,\omega ,\alpha ,\beta ,\theta$,m,c,z,p,q). We obtain the characteristic, neighborhood and convolution properties for this class. Some of these properties are extensions of defined results. Full article

This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to authenticate the assumptions in our main result. At the end, we use our new method to solve a fractional delay differential equation in the sense of Caputo. Our main results improve and generalize the results of many prominent authors in the existing literature. Full article

In this work, we perform univariate approximation with rates, basic and fractional, of continuous functions that take values into an arbitrary Banach space with domain on a closed interval or all reals, by quasi-interpolation neural network operators. These approximations are achieved by deriving Jackson-type inequalities via the first modulus of continuity of the on hand function or its abstract integer derivative or Caputo fractional derivatives. Our operators are expressed via a density function based on a q-deformed and $\lambda$-parameterized hyperbolic tangent activation sigmoid function. The convergences are pointwise and uniform. The associated feed-forward neural networks are with one hidden layer. Full article

The primary goal of this study is to create a nonlinear fractional boundary element method (BEM) model for magneto-thermo-visco-elastic ultrasound wave problems in temperature-dependent functionally graded anisotropic (FGA) rotating granular plates in a constant primary magnetic field. Classical analytical methods are frequently insufficient to solve the governing equation system of such problems due to nonlinearity, fractional order heat conduction, and strong anisotropy of mechanical properties. To address this challenge, a BEM-based coupling scheme that is both reliable and efficient was proposed, with the Cartesian transformation method (CTM) used to compute domain integrals and the generalized modified shift-splitting (GMSS) method was used to solve the BEM-derived linear systems. The calculation results are graphed to show the effects of temperature dependence, anisotropy, graded parameter, and fractional parameter on nonlinear thermal stress in the investigated plates. The numerical results validate the consistency and effectiveness of the developed modeling methodology. Full article