Given an integer n≥2 and a digit set 𝒟⊂{0,1,…,n−1}2, there is a self-similar set K⊂ℝ2 satisfying the set equation K=(K+𝒟)/n. This set K is called a fractal square. By studying the line segments contained in K, we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with

Since the directional permeability of fractured rock masses is significantly dependent on the geometric properties of fractures, in this work, a numerical study was performed to analyze the relationships between them, in which fracture length follows a fractal distribution. A method to estimate the representative elementary volume (REV) size and directional permeability (Kf) by extracting regular polygon

This paper proposes a method to calculate the degree of fluctuation of the daily electrical load-curve using fractal dimension, which is a quantitative estimator of spatial complexity. The conventional methods for forecasting have not studied such a variable, being a new parameter that can be included to characterize the electrical load. The method of fractal dimension also allows us to propose a new

The rock cores of low permeability reservoirs have special pore structures, which are the essential factors to determine the seepage capacity and oil displacement efficiency and directly affect oil and gas reserves and oil well productivity. This paper studies 16 digital rock core samples. Based on the fractal theory for porous media, we carried out the fractal characterization of the pore structure

In this paper, the malaria transmission (MT) model under control strategies is considered using the Liouville–Caputo fractional order (FO) derivatives with exponential decay law and power-law. For the solutions we are using an iterative technique involving Laplace transform. We examined the uniqueness and existence (UE) of the solutions by applying the fixed-point theory. Also, fractal–fractional operators

If a continuous multivariate function satisfies a Lipschitz condition on its domain, Box dimension of its graph equals to the number of its arguments. Furthermore, Box dimension of the graph of its Riemann–Liouville fractional integral also equals to the number of its arguments.

This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order

As the Riemann–Liouville derivative is a derivative of a convolution of a function and the power law, the fractal–fractional derivative of a function is the fractal derivative of a convolution of that function with the power law or exponential decay. In order to further open new doors on ongoing investigations with field of partial differential equations with non-conventional differential operators

In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based

In this paper, we investigate a nonlinear coupled system of fractional pantograph differential equations (FPDEs). The respective results address some adequate results for existence and uniqueness of solution to the problem under consideration. In light of fixed point theorems like Banach and Krasnoselskii’s, we establish the required results. Considering the tools of nonlinear analysis, we develop

Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the

In this work, a novel fractal model for the laminar flow in roughened cylindrical microchannels is proposed. The average height of rough elements is derived using the fractal theory. The effects of relative roughness on the friction factor and the Poiseuille number are discussed. It is found that the Darcy friction factor and the Poiseuille number increase with the increase in the relative roughness

In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks.

Surface finish is one of the most important issues that is discussed in machining of materials. In fact, reaching the required surface finish is the key scale in analysis of the quality of machined workpiece. Chip formation is an important factor that highly affects the surface finish of machined workpiece. In this research, we analyze the relationship between surface finish and chip formation by fractal

The coupling of heat transfer and water flow in rock fractures is a key issue to geothermal energy extraction. However, this coupling in a rough fracture has not been well studied so far. This paper will study this coupling in a rock fracture with different roughness. First, multi-scale and self-affine rough fracture are constructed through the Weierstrass–Mandelbrot function and embedded into a rock

Tool wear is one of the unwanted phenomena in machining operations where tool has direct contact with the workpiece. Tool wear is an important issue in milling operation that is caused due to different parameters such as machine vibration. Tool wear shows complex structure, and machine vibration is a chaotic signal that also is complex. In this research, we analyze the correlation between tool wear

Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants SGN, N>3, proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal.174(1) (2000) 128–154]. When SGN is equipped with the standard Dirichlet form and measure μ, we show there is a full μ-measure set on which continuity of the

Magnetic resonance image (MRI) is an important tool to diagnose human diseases effectively. It is very important for research and clinical application to classify the normal and abnormal human brain MRI images automatically. In this paper, an accurate and efficient technique is proposed to extract features of MRIs and classify these images into normal and abnormal categories. We use two-dimensional

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform.151(1–4) (2017)

In this paper, by using the theory of local fractional calculus and some techniques of real analysis, the structural characteristics of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel are studied. At the same time, the necessary and sufficient conditions for these inequalities to take the best constant factor are discussed. As an application, some best constant

Hydraulic properties of rock fractures are important issues for many geoengineering practices. Previous studies have revealed that natural rock fractures have variable apertures that could significantly influence the permeability of single rock fractures, yet the effect of aperture variation on the hydraulic properties of fractured rock masses has received little attention. The present study implemented

The understanding of the diffusion process and mechanisms of harmful species (e.g. chlorides) in porous cementitious materials is important to control and improve the material durability under harsh environments. In this paper, fractal analysis on the pore structure of porous cementitious materials was conducted and involved in a diffusion model. Macro material geometric parameters were considered

In this paper, we study the boundedness character and persistence, local and global behavior, and rate of convergence of positive solutions of following system of rational difference equations xn+1=α1+β1e−yna1+b1xn,yn+1=α2+β2e−zna2+b2yn,zn+1=α3+β3e−xna3+b3zn, wherein the parameters αi,βi,ai,bi for i∈{1,2,3} and the initial conditions x0,y0,z0 are positive real numbers. Some numerical examples are given

Let Kλ be the attractor of the following iterated function system(IFS): {f1(x)=λx, f2(x)=λx+1−λ},0<λ<1/2. Given α≥0, we say the line y=αx is visible through Kλ×Kλ if {(x,αx):x∈ℝ∖{0}}∩(Kλ×Kλ)=∅. Let V={α≥0:y=αx is visible through Kλ×Kλ}. In this paper, we give a complete description of V, containing its Hausdorff dimension and topological properties.

The q-fractional differential equation usually describes the physics process imposed on the time scale set Tq. In this paper, we first propose a difference formula for discretizing the fractional q-derivative cDqαx(t) on the time scale set Tq with order 0<α<1 and scale index 0

Using multifractal detrended cross-correlation analysis (MF-DCCA) and nonlinear Granger causality test, this paper examines the return predictability of margin-trading activities. Results show that the predictive power of margin-trading activities on subsequent stock returns varies with respect to the different aspects of margin trading. In line with previous studies, we find no significant correlation

This paper studied the level-3 Sierpinski gasket. We solved Dirichlet problem of Poisson equations and proved variational principle on the level-3 Sierpinski gasket by expressing Green’s function explicitly.

Let the self-similar measure μM,D be generated by an expanding real matrix M=ρ−1I∈M2(ℝ) and a digit set D={(0,0)t,(1,0)t,(0,1)t,(−1,−1)t} in space ℝ2. In this paper, we only consider ρ>0 and the case ρ<0 is similar. We show that there exists an infinite orthogonal set of exponential functions in L2(μM,D) if and only if ρ=(q/(2p))1r for some p,q,r∈ℕ with gcd(2p,q)=1. Furthermore, for the cases that

Coal–rock dynamic disasters seriously threaten safe production in coal mines, and an effective early warning is especially important to reduce the losses caused by these disasters. The occurrence of coal–rock dynamic disasters is determined by mining-induced stress loading and unloading. Therefore, it is of great significance to analyze the precursory information of coal deformation and failure during

This paper suggests a fractal two-phase fluid model for the polymer melt filling process to deal effectively with the unsmooth front interface. An infinitesimal fluid element model in a fractal space is proposed to establish the governing equations according to the conservation laws in fluid mechanics, the fractal divergence and fractal Laplace operator are defined. The unsmooth interface is solved

COVID-19 is a pandemic disease, which massively affected human lives in more than 200 countries. Caused by the coronavirus SARS-CoV-2, this acute respiratory illness affects the human lungs and can easily spread from person to person. Since the disease heavily affects human lungs, analyzing the X-ray images of the lungs may prove to be a powerful tool for disease investigation. In this research, we

The aim of this paper is to propose the Atangana–Baleanu fractional methodology for fathoming the Van der Pol damping model by using the reproducing kernel algorithm. To this end, we discuss the mathematical structure of this new approach and some other numerical properties of solutions. Furthermore, all needed requirements for characterizing solutions by applying the reproducing kernel algorithm are

Machine joint surface has an important impact on the performance of mechanical systems. Based on fractal theory, joint surface is assumed to be the contact between absolute smooth surface and rough surface. Through the analysis of the contact process, the contact mechanics model, the contact stiffness model and the three-dimensional surface micro-contact model are studied. To obtain fractal characteristic

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem.

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

In order to address the overall properties of the Brazilian agricultural commodity market and the intricate effects of political and economic instabilities, in this work, we provide a comprehensive study of the multifractal properties of the Brazilian commodities using multifractal detrended fluctuation analysis (MFDFA). We focus on the daily price of 12 Brazilian agricultural commodities over the

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski

Microstructures dominate the physical properties of fractal porous media, which means the clarification of complexity types and their assembly are of fundamental importance for static or dynamic purposes. In this work, we identified fractal porous media to dual-complexity systems composed of stationary and scale-invariant complexities as per fractal topography theory, and proposed an open mathematical

The effective thermal conductivity of unsaturated porous media is of interest in a number of applications of heat transfer. In this paper, a novel fractal solution for effective thermal conductivity is derived based on the fractal distribution of surface roughness and pore size in unsaturated porous media with roughened surfaces. The proposed fractal model explicitly relates the effective thermal conductivity

This paper presents a modified complete normal contact stiffness model of a fractal surface considering contact friction. We use this model to study the influence of fractal dimensions and fractal roughness on normal contact stiffness. The fractal micro-contact model of an asperity and the complete length scale contact model of fractal surface (both contrasting classical mechanics) are revised. The

The present paper characterizes the resistive and inductive loads of an electric distribution system by Julia fractal sets, in order to discover other observations enabling the elevation of new theoretical approaches. The result shows that indeed the electrical load reflects a clear graphic pattern in the fractal space of the Julia sets. This result, then, is a new contribution that extends the universal

For alleviating energy shortage and environmental problems, it is of great importance to improve the energy charging and discharging efficiency of thermal energy storage systems. In this context, an innovative phase change heat exchanger (PCHE) with fractal tree-shaped fins is presented in this paper. A numerical investigation of the solidification behaviors in the PCHE with fractal tree-shaped fins

This paper focus on the compact Ahlfors–David regular subsets of fractals. For this, we introduce the Ahlfors–David regular dimension and study its basic properties. Based on these properties and some known results in dimension theory of fractals, we compute Ahlfors–David Regular dimension for Moran sets and McMullen sets.

The effect of the coal seam water infusion to reduce dust is directly related to the development degree of the coal structure under the action of the effective stress. Therefore, this paper first summarizes several basic effective-stress models and proposes the effective-stress model considering the capillary force. Thereafter, the nuclear magnetic resonance (NMR) experiment system high-voltage seepage

In this paper, the evolving networks are created from a series of Sierpinski-type polygon by applying the encoding method in fractal and symbolic dynamical system. Based on the self-similar structures of our networks, we study the cumulative degree distribution, the clustering coefficient and the standardized average path length. The power-law exponent of the cumulative degree distribution is deduced

In this paper, a 1D chain network with a reverse weighted edge is introduced. We focus on studying the relationships including the convergence rate and the length, the convergence rate and weight of adding reverse edge relationships. Laplacian characteristic determinant is calculated and subsequently, the sum of the reciprocals of all nonzero Laplacian eigenvalues is obtained. Hence, the analytic expression

This paper focuses on the design of badminton robots, and designs high-precision binocular stereo vision synchronous acquisition system hardware and multithreaded acquisition programs to ensure the left and right camera exposure synchronization and timely reading of data. Aiming at specific weak moving targets, a shape-based Brown motion model based on dynamic threshold adjustment based on singular

With the rapid development of world trade exchange, transnational and cross regional e-commerce enterprises have become the heat conductor of trade exchanges among people, organizations and related enterprises of all countries, as well as the important content of high-quality economic development of all countries. Multi-national and transregional e-commerce enterprises have the characteristics of simple

In this paper, we use the topological degree theory (TDT) to investigate the existence and uniqueness of solution for a class of evolution fractional order differential equations (FODEs) with proportional delay using Caputo derivative under local conditions. In the same line, we will also study different kinds of Ulam stability such as Ulam–Hyers (UH) stability, generalized Ulam–Hyers (GUH) stability

Study of heat and mass transfers is carried out for MHD Oldroyd-B fluid (OBF) over an infinite vertical plate having time-dependent velocity and with ramped wall temperature and constant concentration. It is proven in many already published articles that the heat and mass transfers do not really or always follow the classical mechanics process that is known as memoryless process. Therefore, the model

There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling

In this paper, the analytic and fractional solutions of governing differential equations for helical flow of cylindrical nature have been presented. The series expansions and Laplace and Hankel transforms are applied to the governing equation of generalized Burger fluid flow for generating gamma functions. The analytical solutions of velocity fields and shear stresses are obtained through Caputo fractional

Current hydraulic fracture conductivity evolution models fail to incorporate the fractures microscopic petrophysical properties, and notable discrepancies consequently exist between predictions and observations. We present a new conductivity model considering the irregular fracture undulation and channel roughness in propped fractures. The propped fracture networks are treated as bundles of tortuous

In this paper, we analyze the extreme events of non-stationary time series in the framework of horizontal visibility graph (HVG). We give a new definition of extreme events, which incorporates the temporal structure of the series and the degree of the nodes in the HVG. An advantage of the new concept is that it does not require ad hoc treatment even when the non-stationarity arises in time series.

Permeability is one of the most important parameters for accurately predicting water flow in reservoirs and quantifying underground water inrush into coal mines. This study developed a predictive permeability model by considering the microstructural parameters and tortuosity effects of low-permeability sandstone. The model incorporates the fractal geometry theory, Darcy’s law, and Poiseuille equation

The paper deals with the existence of three different weak solutions of p -Laplacian fractional for an overdetermined nonlinear fractional partial Fredholm–Volterra integro-differential system by using variational methods combined with a critical point theorem due to Bonanno and Marano.