Fractal Order Research Articles

Fractional and fractal order effects in soft elastomers: Strain rate and temperature dependent nonlinear mechanics

Soft dielectric elastomers are commonly used in applications where large deformation is prevalent such as energy harvesters, sensors, and soft actuators. The thermo-mechanical response of these elastomers is important in order to accurately describe the viscoelastic behavior over a broad range of operating conditions. Fractal characteristics of the dielectric elastomer VHB 4905 and their connection to fractional order viscoelasticity are analyzed to better understand viscoelasticity over a range of elevated temperatures starting at room temperature. We extend prior work in viscoelasticity to include excluded volume effects as a function of temperature. A fractal hyperelastic model is combined with a fractional order viscoelastic model and validated experimentally. Bayesian uncertainty methods are used to quantify the material parameters and their influence on temperature dependent viscoelasticity measurements. The model fits are compared to previously collected temperature dependent viscoelastic measurements on VHB 4905 for temperatures ranging from 23 °C to 60 °C. Based on fractional order viscoelasticity, we infer that the excluded volume parameter is negative and initially decreases before reaching a constant value near 50 °C.

A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis

The pandemic caused by coronaviruses (SARS-COV-2) is a zoonotic disease targeting the respiratory tract of active humans. Few mild symptoms of fever and tiredness get cured without any medicinal aid, whereas some severe symptoms of dry cough with breathing illness led to perceived risk of secondary transmission. This paper studies the effectiveness of vaccination in Covid-19​ pandemic disease by modelling three compartments susceptible, vaccinated and infected (SVI) of Atangana Baleanu of Caputo (ABC) type derivatives in non-integer order. The disease dynamics is analysed and its stability is performed. Numerical approximation is derived using Adam’s Moulton method and simulated to forecast the results for controllability of pandemic spread.

Nonlocal magneto-thermoelastic infinite half-space due to a periodically varying heat flow under Caputo–Fabrizio fractional derivative heat equation

Abstract This article deals with a new modified heat conduction model with fractional order that includes the Caputo–Fabrizio differential operator (CF) and the thermal relaxation time. This new approach to the CF fractional derivative has attracted many researchers because it includes a nonsingular kernel. The nonlocal theory proposed by Eringen has also been applied to demonstrate the effect of scale-dependent thermoelastic materials. The problem of thermal isotropic semi-infinite space is addressed as an application of the presented model. The medium is exposed to regularly changing heat sources and is initially placed in a continuous external magnetic field. The system of governing equations was expressed in the field of the Laplace transform, and the problem in this field was solved by the state-space operation. The inverse of the transformed expressions of physical quantities is found numerically using Zakian’s algorithm. The effects of the nonlocal parameter, the fractal order parameter, and the magnetic field were graphically presented and analyzed in detail. Some of the previous investigations were extracted in some special cases.

Active near-field plasmonic switches based on Sierpiński-fractal nanoantennas on VO2 films

We propose active near-field plasmonic switches based on a Sierpiński-fractal contour-bowtie plasmonic nanoantenna on top of a VO2 (vanadium-dioxide) thin film. The near-field intensity of the proposed fractal plasmonic nanoantenna can be switched by the application of heat, voltage, or optical energy to the underlying VO2 layer, which leads to a phase change of VO2 from the semiconductor state to the metallic state. This phase transition of the underlying VO2 film leads to an overall change in the optical properties of the nanoantenna system, hence driving the switch from an OFF state (with low near-field intensity) to an ON state (with high near-field intensity). The near-field switching ability of the proposed switch is quantified by the intensity switching ratio (ISR), i.e. the ratio of the intensity between the two arms of plasmonic nanoantenna in its ON state to its OFF state (I ON/I OFF). Finite difference time domain simulations were employed to calculate the ISR of the proposed near-field plasmonic switches. As the fractal order of the Sierpiński-fractal contour-bowtie nanoantenna is increased, the intensity in the ‘ON’ state of the switch is enhanced along with a reduction of intensity in the ‘OFF’ state of the switch. Thus, higher fractal orders of the Sierpiński-fractal contour-bowtie plasmonic nanoantenna lead to very high values of the ISR for the proposed near-field switch. We demonstrate an ISR of ∼900 with a fractal order of 2 for the proposed switch which is the highest value of near-field ISR reported thus far for a near-field plasmonic switch. Further, we also demonstrate that the ISR and the spectral response of the proposed near-field switch can be controlled by changing its structural parameters such as the length of the nanoantenna arm, the contour thickness, and the thickness of VO2 layer.

Analysis of a TB and HIV co-infection model under Mittag-Leffler fractal-fractional derivative

In this paper, the nonlocal operator with the Mittag-Leffler kernel is used to analyze a TB-HIV co-infection model with recurrent TB and exogenous reinfection. The non-negative invariant region and basic reproduction number of the proposed model are demonstrated. By using the Krasnoselskii fixed result, we investigate that the TB-HIV co-infection model possesses at least one solution. We look at the existence of a unique solution using Banach’s fixed point theorem. Functional analysis is used to demonstrate Ulam-Hyres stability. The numerical solution of the given model is obtained using the Adams-Bashforth technique. We illustrate the achieved results by studying the co-infection of TB and HIV for different fractional and fractal orders.

A case study of fractal-fractional tuberculosis model in China: Existence and stability theories along with numerical simulations

In this article, a fractal-fractional order tuberculosis mathematical model is presented for the existence results, numerical simulations and stability analysis. The model has six classes S1,S2,S3,E,I,R. The first three classes S1, S2, S3 represent the population of susceptible children, middle-aged, and senior adults, respectively. While I is the class of active infected individuals who can transmit the tuberculosis, E stands for non-active infected class. The population of recovered individuals is represented by R. For the existence criterion of the given model, successive iterative sequences are defined whose limit points are the solutions of our proposed tuberculosis model. After investigation of uniqueness property, the Hyers–Ulam (HU)-stability is established in the sequel. With the help of two-step Lagrange polynomials, we provide numerical solutions and we give a comparative numerical analysis for different values of the fractional order and fractal order based on the obtained algorithms. The numerical simulations show the applicability of the schemes and the future prediction.

New Type Modelling of the Circumscribed Self-Excited Spherical Attractor

The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder’s fixed point and Banach fixed existence theory are used to guarantee that there are solutions to the model. Approximate solutions to the problem are presented by an effective method. To prove the efficiency of the given technique, different values of fractal and fractional orders as well as initial conditions are selected. Figures of the approximate solutions are provided for each case in different dimensions.

Fractal, diffraction-encoded space-division multiplexing for FSO with misalignment-robust, roaming transceivers

For free space optical (FSO) communication, a small misalignment of the transceivers may result in link failure or severe performance degradation. It can be difficult to track the narrow optical beams over long distances. Here, we propose “diffractal space-division multiplexing” (DSDM), an FSO transmission system capable of supporting misaligned roaming transceivers. This system enables spatial multiplexing for enhanced data capacity with partial off-axis beam reception. We numerically simulate and analyze the performance of the DSDM system with a particular focus on the divergence angle, roaming area, kernel bit-error-rate (K-BER), and fractal order. Our simulation results achieve K-BERs of 10^{-3} with 81times 81-pixel fractal beams at link distances of 2.5 km when the receiver sizes are 30% of the effective beam diameter.

A novel fractal-fractional order model for the understanding of an oscillatory and complex behavior of human liver with non-singular kernel

Scientists and researchers are increasingly interested in numerical simulations of infections with non-integer orders. It is self-evident that conventional epidemiological systems can be given in a predetermined order, but fractional-order derivative systems are not stable orders. The fractional derivative proves increasingly effective in representing real-world issues when it has a non-fixed order. Various novel fractional operator notions, including special functions in the kernel, have been presented in recent decades, which transcend the constraints of prior fractional order derivatives. These novel operators have been shown to be useful in simulating scientific and technical challenges. The fractal-fractional operator is a relatively modern fractional calculus operator that has been proposed. Besides that, we propose a new technique and implement it in a human liver model and want to investigate its dynamics. In the context of this novel operator, we demonstrate certain interesting findings for the human liver model. The findings of the uniqueness and existence will be revealed. We describe modeling estimates for the proposed model using an innovative numerical method that has never been used before for a human liver model of this type. Additionally, graphical illustrations are demonstrated for both fractal and fractional orders. It is expected that the fractal-fractional approach is more invigorating and effective for epidemic models than the fractional operator.

Modeling the dynamics of coronavirus with super-spreader class: A fractal-fractional approach

Super-spreaders of the novel coronavirus disease (or COVID-19) are those with greater potential for disease transmission to infect other people. Understanding and isolating the super-spreaders are important for controlling the COVID-19 incidence as well as future infectious disease outbreaks. Many scientific evidences can be found in the literature on reporting and impact of super-spreaders and super-spreading events on the COVID-19 dynamics. This paper deals with the formulation and simulation of a new epidemic model addressing the dynamics of COVID-19 with the presence of super-spreader individuals. In the first step, we formulate the model using classical integer order nonlinear differential system composed of six equations. The individuals responsible for the disease transmission are further categorized into three sub-classes, i.e., the symptomatic, super-spreader and asymptomatic. The model is parameterized using the actual infected cases reported in the kingdom of Saudi Arabia in order to enhance the biological suitability of the study. Moreover, to analyze the impact of memory index, we extend the model to fractional case using the well-known Caputo–Fabrizio derivative. By making use of the Picard-Lindelöf theorem and fixed point approach, we establish the existence and uniqueness criteria for the fractional-order model. Furthermore, we applied the novel fractal-fractional operator in Caputo–Fabrizio sense to obtain a more generalized model. Finally, to simulate the models in both fractional and fractal-fractional cases, efficient iterative schemes are utilized in order to present the impact of the fractional and fractal orders coupled with the key parameters (including transmission rate due to super-spreaders) on the pandemic peaks.

Construction of Fractal Order and Phase Transition with Rydberg Atoms.

We propose the construction of a many-body phase of matter with fractal structure using arrays of Rydberg atoms. The degenerate low energy excited states of this phase form a self-similar fractal structure. This phase is analogous to the so-called "type-II fracton topological states." The main challenge in realizing fractonlike models in standard condensed matter platforms is the creation of multispin interactions, since realistic systems are typically dominated by two-body interactions. In this work, we demonstrate that the van der Waals interaction and experimental tunability of Rydberg-based platforms enable the simulation of exotic phases of matter with fractal structures, and the study of a quantum phase transition involving a fractal ordered phase.

A study on the fractal-fractional tobacco smoking model

<abstract><p>In this article, we consider a fractal-fractional tobacco mathematical model with generalized kernels of Mittag-Leffler functions for qualitative and numerical studies. From qualitative point of view, our study includes; existence criteria, uniqueness of solution and Hyers-Ulam stability. For the numerical aspect, we utilize Lagrange's interpolation polynomial and obtain a numerical scheme which is further illustrated simulations. Lastly, a comparative analysis is presented for different fractal and fractional orders. The numerical results are divided into four figures based on different fractal and fractional orders. We have found that the fractional and fractal orders have a significant impact on the dynamical behaviour of the model.</p></abstract>

Numerical analysis of some partial differential equations with fractal-fractional derivative

<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.</p> </abstract>

Finite Difference Simulation of Fractal-Fractional Model of Electro-Osmotic Flow of Casson Fluid in a Micro Channel

In this work, an evolving definition of the fractal-fractional operator with exponential kernel was employed to examine Casson fluid flow with the electro-osmotic phenomenon. Electrically conducted Casson fluid flow with the effect of the electro-osmotic phenomenon has been assumed in a vertical microchannel. With the help of relative constitutive equations, the local mathematical model is formulated in terms of partially coupled partial differential equations along with appropriate physical initial and boundary conditions. The dimensional governing equations have been non-dimensional by using relative similarity variables to encounter the units and reduce the variables. The local mathematical model has been transformed to a fractal-fractional model by using a fractal-fractional derivative operator with exponential kernel and then analyze numerically with the discretization of finite difference (Crank-Nicolson) scheme. For an insight view of the proposed phenomena, various plots are drawn in respect of inserted parameter. From the graphical analysis, it has been observed that the electro-kinetic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> parameter retards the fluid’s motion. It is also worth noting that graphs for the fractal-fractional, fractional, and classical order parameters have been drawn. Due to the fractal order parameter, it was revealed that the fractal-fractional order model has a larger memory effect than the fractional-order and classical models.

SOLITARY WAVES OF THE FRACTAL REGULARIZED LONG-WAVE EQUATION TRAVELING ALONG AN UNSMOOTH BOUNDARY

The unsmooth boundary has a great influence on the solitary wave form of a nonlinear wave equation. It this work, we for the first time ever propose the fractal regularized long-wave equation which can describe the shallow water waves under the unsmooth boundary (such as the fractal seabed). The fractal variational principle is established and is proved to have a strong minimum condition by the He–Weierstrass theorem. Then, the solitary wave solution is obtained by the fractal variational method which can reduce the order of differential equation and obtain the optimal solution by the stationary condition. Finally, the impact of the unsmooth boundary on the solitary wave is presented. It shows that the fractal order can affect the wave morphology, but cannot affect its peak value. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences.

RESEARCH ON THE NONLINEAR VIBRATION OF CARBON NANOTUBE EMBEDDED IN FRACTAL MEDIUM

The vibration of carbon nanotubes embedded in elastic medium is a hot topic and many research findings are available. But when the medium is a fractal medium such as chips, ceramics, etc., its vibration characteristics are difficult to explain with the existing theories. So in this paper, we mainly study the nonlinear vibration of carbon nanotube embedded in a fractal medium. The fractal variational principle of the problem is successfully established, and is proved to have a strong minimum condition according to the He–Weierstrass Theorem. Then the Hamiltonian theory combined with the residual equation is implemented to find the solution of the fractal nonlinear vibration. Finally, the impacts of different fractal orders on the vibration characteristics are also presented.

Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study

This research study consists of a newly proposed Atangana–Baleanu derivative for transmission dynamics of the coronavirus (COVID-19) epidemic. Taking the advantage of non-local Atangana–Baleanu fractional-derivative approach, the dynamics of the well-known COVID-19 have been examined and analyzed with the induction of various infection phases and multiple routes of transmissions. For this purpose, an attempt is made to present a novel approach that initially formulates the proposed model using classical integer-order differential equations, followed by application of the fractal fractional derivative for obtaining the fractional COVID-19 model having arbitrary order Ψ and the fractal dimension Ξ. With this motive, some basic properties of the model that include equilibria and reproduction number are presented as well. Then, the stability of the equilibrium points is examined. Furthermore, a novel numerical method is introduced based on Adams–Bashforth fractal-fractional approach for the derivation of an iterative scheme of the fractal-fractional ABC model. This in turns, has helped us to obtained detailed graphical representation for several values of fractional and fractal orders Ψ and Ξ, respectively. In the end, graphical results and numerical simulation are presented for comprehending the impacts of the different model parameters and fractional order on the disease dynamics and the control. The outcomes of this research would provide strong theoretical insights for understanding mechanism of the infectious diseases and help the worldwide practitioners in adopting controlling strategies.

Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels

In this paper, three new models of fractal–fractional Michaelis–Menten enzymatic reaction (FFMMER) are studied. We present these models based on three different kernels, namely, power law, exponential decay, and Mittag-Leffler kernels. We construct three schema of successive approximations according to the theory of fractional calculus and with the help of Lagrange polynomials. The approximate solutions are compared with the resulting numerical solutions using the finite difference method (FDM). Because the approximate solutions in the classical case of the three models are very close to each other and almost matches, it is sufficient to compare one model, and the results were good. We investigate the effects of the fractal order and fractional order for all models. All calculations were performed using Mathematica software.

Theoretical and numerical analysis of fractal fractional model of tumor-immune interaction with two different kernels

Fractal fractional operators in Caputo and Caputo-Fabrizio sense are being used in this manuscript to explore the interaction between the immune system and cancer cells. The tumour-immune model has been investigated numerically and theoretically by the singular and non-singular fractal fractional operators. Via fixed point theorems, the existence and uniqueness of the model under the Caputo fractal fractional operator have been demonstrated. Using the fixed point theory, the existence of a unique solution has been derived under the Caputo-Fabrizio case. Through nonlinear analysis, the Ulam-Hyres stability of the model has been derived. For the singular and nonsingular fractal fractional operators, numerical results have been developed by Lagrangian-piece wise interpolation. We simulate the numerical results for the various sets of fractional and fractal orders to describe the relationship between immune and cancer cells under the novel operators with two different kernels. We compared the dynamics of the tumor-immune model using a power law and an exponential-decay kernel to explore that the nonsingular fractal fractional operator provides better dynamics for the considered model.

A study of a modified nonlinear dynamical system with fractal-fractional derivative

PurposeThis paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control.Design/methodology/approachThe systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems.FindingsSimulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved.Originality/valueThis paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.