Fractals

In this work, we investigate a nonlinear fractional order Lotka–Voltera prey–predator model. It is also worth mentioning that we develop some results about the existence of the solution through fixed point theory. After investigating the existence results, we also develop a suitable technique with the help of Laplace transform coupled with Adomain decomposition in order to obtain the approximate solutions for the underlying model. Here, it should be kept in mind that we study the proposed model subject to variable coefficients. The results obtained via the aforesaid techniques are illustrated graphically via Matlab-16 corresponding to different fractional order by taking two cases.

Reviewers list:

• Prof. Dumitru Baleanu (Cancaya University Ankara Turkey), Email: dumitru@cankaya.edu.tr,
• Dr. Anwar Zeb (COMSAT, University Pakistan), Email: anwar@cuiatd.edu.pk,
• Prof. J. F. Gómez-Aguilar (CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C. P. 62490, Cuernavaca, Morelos, México), Email: jgomez@cenidet.edu.mx.

In this paper, we focus on the periodic points during the iteration process to understand the convergence characteristics of the Mandelbrot set. A new internal structure of the Mandelbrot set is obtained. In addition, the influences of the number of initial iteration points and computational accuracy on this structure are investigated. The results show that the new internal structure is stable no matter how the computational accuracy and the number of initial iteration points change. However, the boundaries of subsets of the Mandelbrot set are sensitive to the computational accuracy. This finding reveals the true convergence structure of the Mandelbrot set during numerical simulations, which differs from the theoretical convergence point distribution. Thus, it helps us further understand the convergence mechanism of the Mandelbrot set.

The capillary pressure and relative permeability are the key parameters to affect the transport properties of low-permeability reservoirs. In this paper, the Fractal-Monte Carlo technique is applied to predict the oil and water relative permeabilities of low-permeability reservoirs with rough surfaces by considering the effect of tortuosity of capillaries and capillary pressure as well as roughness of capillary walls. The oil and water relative permeabilities are expressed as a function of the pore size, the area fractal dimension of pores, porosity, the water saturation, the capillary pressure, the relative roughness, the tortuosity fractal dimension and the microstructural parameters of low-permeability reservoirs with rough surfaces. It is observed that the average capillary pressure for water phase decreases with the increase of water saturation. On the other hand, it is found that the average capillary pressure for water phase increases with tortuosity fractal dimension and relative roughness. It can be found that the oil relative permeability of low-permeability reservoirs with rough surfaces increases with the increase of driving pressure. On the contrary, it can be observed that the oil and water relative permeabilities of low-permeability reservoirs with rough surfaces decrease with the increase of tortuosity fractal dimension and relative roughness. The predicted oil and water relative permeabilities obtained by the proposed Fractal-Monte Carlo simulation are shown to have a good agreement with the available experimental results. The proposed probability models for capillary pressure and relative permeability well revealed the physical mechanisms of transport properties of low-permeability reservoirs with rough surfaces.

Fractal Calculus is designed to reveal the study of waves. Most of the waves are very grim to model correctly. Fractal representation of waves helps to better understand complex wave phenomena. In this paper, a Maclaurin series method (MSM) is proposed to obtain the exact and approximate solutions of fractal nonlinear differential-difference models produced by coupled nonlinear optical waveguides. The method is very simple, easy to understand, and minimizes calculation compared to existing approximate methods.

We consider a family of fractal squares, denoted as ${{ℱ}}_{3,7}$. Each of them satisfies the set equation $K=\frac{1}{3}(K+{𝒟})$ for some ${𝒟}\subset {\{0,1,2\}}^2$ with $\#{𝒟}=7$. It is known that two of these fractal squares are Lipschitz equivalent if and only if they are isometrically equivalent. The aim of our study is to improve this by replacing Lipschitz equivalence with topological equivalence. To this end, we shall investigate the group $G_{\mathrm{\text{aut}}}(K)$ of all homeomorphisms of a fractal square $K\in {{ℱ}}_{3,7}$ that has a cut point and show that $\#G_{\mathrm{\text{aut}}}(K)=2$ or $8$.

In this paper, we establish some local fractional Ostrowski-type integral inequalities for generalized $h$-convex functions on real linear fractal set $R^{\gamma }(0<\gamma \le 1)$. We present two examples to illustrate our main results. As applications, we establish some local fractional integral inequalities involving generalized moments of continuous random variables to estimate the bounds of generalized moments.

The current study achieves a melting improvement of the latent thermal energy storage (LTES) system using fractal-branched fins (i.e., Y-type and T-type fins). A transient melting model with free convection in LTES systems is developed and numerically simulated with the enthalpy-porosity method. The effects of branched fins on the melting performance are discussed and compared to the traditional plate fin with emphasis on the roles of fin parameters. Besides, the optimized configuration is obtained by the response surface methodology (RSM) in terms of the total charging time. The results indicate that the branched fin expedites the melting process, especially for the T-type branched fin. Due to a more desirable fin layout, the LTES system using T-type branched fins presents a more uniform temperature field and faster melting rate, manifesting as the reduced total melting time by 5.6% compared to the Y-type branches. The melting process includes the pre-heat conduction dominated, free convection dominated, and post-heat conduction dominated stages. A T-type branched fin with moderate branch level ($n)$, larger length index ($\alpha )$, and lower width fractal dimension ($\mathrm{\Delta })$ is favorable for the coordinated enhancement of heat convection and conduction. The RSM optimization implies the length index and branch level play more remarkable roles than the width fractal dimension in melting enhancement. To maximize the charging efficiency of LTES systems, the optimal parameters of T-type fin are $\alpha =0.9$, $n=4$, and $\mathrm{\Delta }=1$, which shortens the total charging time by 52.9% compared to the traditional plate fin.

In this paper, the fractional order model for the binge of pests in tea plants is studied numerically. This model consists of three compartments such as tea plant, pest and predator. The local and non-local stability investigation of the system is also deliberated. The Grunwald–Letnikov (GL coefficients) method and generalized Euler method (GEM) are used to elucidate and simulate the proposed system. We have obtained stability conditions for equilibrium points, provided a numerical example, and proved our results. The results illustrate the concentrations of tea plants, pests, and predators all reach their equilibrium values as time passes. An imperative feature of this model is that it controls the motion at which the solution to equilibrium is grasped.

In this paper, a new algorithm to select the relevant nodes — those that maintain the cohesion of the network — of the complex network is presented. The experiments on most of the real complex networks show that the proposed approach outperforms centrality measures as node degree, PageRank algorithm and betweenness centrality. The rationale of the algorithm for extracting relevant nodes is to discover the self-similarity of the network. As seen in the algorithm, throughout the extraction sequence of relevant nodes, differences are advised with node degree, PageRank algorithm and betweenness centrality. Finally, empirical evidence is considered to show that complex network robustness is a nonlinear function of the small-worldness measure.

In this paper, we investigate the efficiency of gold market through time evolution of multifractal spectrum parameters (position of maximum, width and asymmetry), by applying multifractal detrended fluctuation analysis in sliding windows of 4, 10 and 20 years. We find that geopolitical and economic events can decrease market efficiency by inducing changes in gold price dynamics toward increased persistency and stronger multifractality with dominance of large fluctuations. For all analyzed geopolitical events and for all windows sizes, we found stronger multifractality of underlying stochastic process, indicated by the increase in the spectrum width $W$.

In this paper, we structure the idea of the fractal memristor to simulate the nonlinear self-similarity dopant drift effect for the first time. We investigate the fractal memristor device with the use of the local fractional Laplace transform, and the dynamic fractal behavior of the fractal HP TiO₂ memristor model is simulated by changing the integer order of traditional memristor model to $\frac{\mathrm{\text{ln}}2}{\mathrm{\text{ln}}3}$. Compared with the traditional model, the fractal memristor can be well-presented in describing the non-differentiable dopant drift effect.

Since the ability to control the investor’s income or loss within a certain range, barrier option has been among the most popular path-dependent options where its payoff depends on whether or not the underlying asset’s price reaches a given “barrier”. First, assuming the underlying asset as an uncertain variable for the case that the Caputo fractional-order derivative is adopted instead of the ordinary derivative, the real financial market is better modeled by the uncertain fractional-order differential equation with Caputo type. Then, a first-hitting time model which can measure the exercise ability is innovatively presented. Second, based on the first-hitting time theorem of the uncertain fractional-order differential equation, the reliability index (including validity and survival index) for the proposed model is obtained, and four types of European barrier option (including up-and-in call, down-and-in put, up-and-out put, and down-and-out call options) pricing formulas are obtained accordingly. Lastly, applying the predictor–corrector method, numerical algorithms are provided for calculating European barrier and the reliability index, numerical experiments and corresponding sensitivity analysis are also illustrated concerning various conditions.

In this paper, we provide extensions to hyperbolic numbers plane of the classical Chaos game algorithm and the Shannon entropy. Both notions connected with that of probability with values in hyperbolic number, introduced by Alpay et al. [Kolmogorov’s axioms for probabilities with values in hyperbolic numbers, Adv. Appl. Clifford Algebras 27(2) (2017) 913–929]. Within this context, particular attention has been paid to the interpretation of the hyperbolic valued probabilities and the hyperbolic extension of entropy as well.

This paper studies the problem of bifurcation for a fractional-order predator–prey system with two different delays by considering fractional interval $(0,2]$. Multiple delays-depended stability domains of the developed model are procured and the bifurcation points are exactly established by taking advantage of two different delays as bifurcation parameters, respectively. It detects that the stability performance can be extremely varied with the changes of control parameter upon one delay is established. System possesses excellent stability performance when choosing the smaller control parameter, and Hopf bifurcation emerges from system once control parameter passes through the critical values, which degrades the performance of the system. Numerical simulations are finally performed to check our theoretical analysis.

In this paper, we mainly investigate fractal dimension of fractional calculus of certain continuous functions. It has been proved that upper Box dimension of Riemann–Liouville fractional integral of fractal functions whose upper Box dimension is greater than one of certain positive order is at least linearly decreasing. Fractal dimension of Riemann–Liouville fractional integral of any one-dimensional continuous functions with unbounded variation has been proved to be still one-dimensional too. An example about fractal linear interpolation functions shows that estimation is optimal.

We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space.

In this paper, a fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media is derived. The determined imbibition height and imbibition mass of capillary rise are in satisfying agreement with the existing models reported in the literature. It is found that the imbibition height and imbibition mass of capillary decreases with increasing relative roughness. Besides, it is observed that the equilibrium time in a single tortuous capillary with roughened surfaces decreases with an increase in relative roughness. In addition, it is seen that the imbibition height and imbibition mass of capillary rise increases with imbibition time. With the proposed fractal model, the physical mechanisms of capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media are better elucidated. One advantage of our fractal analytical model is that it contains no empirical constant, which is usually required in previous models.

In this paper, we study solutions to the max–min mean value problem

In this paper, an equivalent power-form transformation method with a weighted function is applied for solving the one-dimensional fractal Bratu’s boundary value equation. Numerical integration solutions obtained from the equivalent Bratu’s equation as well as those from its approximate Taylor’s series solution reveal that the proposed methodology yields highly accurate solutions. Therefore, it is believed that by applying the power-form transformation, various fractal differential equations in which two scales are needed because of the physical laws involved in modeling the observed phenomena, can be solved by treating the nonlinear terms as equivalent power-form terms.

In this paper, new model on novel coronavirus disease (COVID-19) with four compartments including susceptible, exposed, infected, and recovered class with fractal-fractional derivative is proposed. Here, Banach and Leray–Schauder alternative type theorems are used to establish some appropriate conditions for the existence and uniqueness of the solution. Also, stability is needed in respect of the numerical solution. Therefore, Ulam–Hyers stability using nonlinear functional analysis is used for the proposed model. Moreover, the numerical simulation using the technique of fundamental theorem of fractional calculus and the two-step Lagrange polynomial known as fractional Adams-Bashforth (AB) method is proposed. The obtained results are tested on real data of COVID-19 outbreak in Malaysia from 25 January till 10 May 2020. The numerical simulation of the proposed model has performed in terms of graphs for different fractional-order $q$ and fractal dimensions $p$ via number of considered days of disease spread in Malaysia. Since COVID-19 transmits rapidly, perhaps, the clear understanding of transmission dynamics of COVID-19 is important for countries to implement suitable strategies and restrictions such as Movement Control Order (MCO) by the Malaysian government, against the disease spread. The simulated results of the presented model demonstrate that movement control order has a great impact on the transmission dynamics of disease outbreak in Malaysia. It can be concluded that by adopting precautionary measures as restrictions on individual movement the transmission of the disease in society is reduced. In addition, for such type of dynamical study, fractal-fractional calculus tools may be used as powerful tools to understand and predict the global dynamics of the mentioned disease in other countries as well.

This paper gives a short remark on variational principle for the fractal Telegraph equation [K. L. Wang, S. W. Yao, Y. P. Liu et al., Fractals 28(4) (2020) 2050058], the emphasis is put on temporal and spatial fractal derivatives and the derivation process of the fractal variational principle.

Bacterial light harvesters (LH) are interpreted as Vicsek fractals, based upon their morphology, even though such fractals contain only one, or perhaps two, generations. After using fractal dimensions to describe the geometry, we progress to use connectivity matrices to make an improved description of the light harvesters and make connections with our previous studies. Such inspirations of metamaterial from the biological world are useful for building quantum computers, solar cells, heat radiators, and optical sensors.

For pattern recognition, natural scenes and medical images are often modeled as two-dimensional fractional Brownian motion (2D FBM), which can be easily described by the Hurst exponent ($H)$, a real number between 0 and 1. The Hurst exponent is directly related to the fractal dimension ($D$) by $D=3$–$H$, and hence it very suitably serves as a characteristic index. Therefore, how to estimate the Hurst exponent effectively and efficiently is very important in pattern recognition.

In this paper, a partially iterative algorithm, simply called an iterative maximum likelihood estimator (MLE) for 2D DFBM is first proposed and then a more sufficiently iterative algorithm, simply called an efficient MLE for 2D DFBM, is further proposed via a perfect structure of the log-likelihood function related to the Hurst exponent. Except for theoretical knowledge, two practical algorithms are also correspondingly provided for easy applications. Experimental results show that the MLE for 2D DFBM is effective and workable; its accuracy gets higher as the image size increases, and hence its recognition resolution is much finer to identify small difference among patterns. Furthermore, the efficient MLE is much quicker than the iterative MLE, especially at larger image sizes.

Let $f(t)$ be $\alpha$-Hölder continuous on $ℝ$ and well-defined about the Weyl fractional integral. Then,

Blockchain technology apparently is a trivial innovation, but this technology has attracted huge investors in a very short period compared to other technologies, and it is still having a lot of potential applications. Smart contracts are making possible execution in an automated and safe way by using blockchain technology. Therefore, smart contracts are applied in this research for the expert system. This paper is about an expert system working with smart contracts and neural networks as the inference machine to decide on the sensors optimal distribution and taking actions when sensor readings are out of range: control lights, activating fire alarms, temperature alarms, etc. for all spaces (parks, schools, hospitals, etc.) in a smart city based on the needs, and likes of the expert system user. This expert system works using a blockchain structure on the EOSIO ecosystem with all data gathered by the sensors being saved in cloud online making internet of things environment and essential data saved in a blockchain node.

Evaluation of the correlation of the activities of various organs is an important area of research in physiology. In this paper, we evaluated the correlation among the brain and facial muscles’ reactions to various auditory stimuli. We played three different music (relaxing, pop, and rock music) to 13 subjects and accordingly analyzed the changes in complexities of EEG and EMG signals by calculating their fractal exponent and sample entropy. Based on the results, EEG and EMG signals experienced more significant changes by presenting relaxing, pop, and rock music, respectively. A strong correlation was observed among the alterations of the complexities of EMG and EEG signals, which indicates the coupling of the activities of facial muscles and brain. This method could be further applied to investigate the coupling of the activities of the brain and other organs of the human body.