The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the ‘thickest’ and ‘thinnest’ parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, 𝜃-Assouad spectrum, and Φ-dimensions. In this paper, we study the analogue

In this paper, we study geodesics in the Sierpinski carpet and Menger sponge, as well as in a family of fractals that naturally generalize the carpet and sponge to higher dimensions. In all dimensions, between any two points we construct a geodesic taxicab path, namely a path comprised of segments parallel to the coordinate axes and possibly limiting to its endpoints by necessity. These paths are related

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an

Talking is the most common type of human interaction that people have in their daily life. Besides all conducted studies on the analysis of human behavior in different conditions, no study has been reported yet that analyzed how the brain activity of two persons is related during their conversation. In this research, for the first time, we investigate the relationship between brain activities of people

The multiphase flow behavior in shale porous media is known to be affected by multiscale pore size, dual surface wettability, and nanoscale transport mechanisms. However, it has not been fully understood so far. In this study, fractal model of gas–water relative permeabilities (RP) in dual-wettability shale porous media for both injected water spontaneous imbibition and the flow back process are proposed

Surface finish of machined workpiece is one of the factors to evaluate the performance of machining operations. There are different factors such as machining parameters that affect the surface finish of machined workpiece. Tool wear is an unwanted machining issue that highly affects the surface finish of machined workpiece. In a similar way, different parameters (e.g. cutting speed, feed rate and depth

This paper extends the (2+1)-dimensional Broer–Kaup equations in continuum mechanics to its fractional partner, which can model a lot of nonlinear waves in fractal porous media. Its derivation is demonstrated in detail by applying He’s fractional derivative. Using the semi-inverse method, two variational principles are established for the nonlinear coupled equations, which up to now are not discovered

Superfast diffusion exists in various complex anisotropic systems. Its mean square displacement is an exponential function of time proved by several theoretical and experimental investigations. Previous studies have studied the superfast diffusion based on the time-space scaling local structural derivatives without considering the memory of dynamic behavior. This paper proposes a nonlocal time structural

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder

In this paper, a comparative study of natural transform decomposition method and new iterative method is presented. The proposed methods are tested upon nonlinear fractional order foam drainage problem and fractional order modified regularized long-wave equation. The solutions obtained by the proposed methods have been compared with the classical solutions and the solution obtained by Adomian decomposition

Fractal model of gas diffusion in porous nanofibers with rough surfaces is derived, in which the porous structure is assumed to be composed of a bundle of tortuous capillaries whose pore size distribution and surface roughness follow the fractal scaling laws. The analytical expression for gas relative diffusion coefficient is a function of the relative roughness and the other microstructural parameters

During the last years, civil infrastructure has experienced an increasing development to satisfy the society’s demands such as communication, transportation, work and living spaces, among others. In this sense, the development and application of methods to guarantee the structure optimal operation, known as Structural Health Monitoring schemes, are necessary in order to avoid economic and human losses

In this paper, we mainly research on Hadamard fractional integral of Besicovitch function. A series of propositions of Hadamard fractional integral of sinx have been proved first. Then, we give some fractal dimensions of Hadamard fractional integral of Besicovitch function including Box dimension, K-dimension and Packing dimension. Finally, relationship between the order of Hadamard fractional integral

Coronavirus disease (COVID-19) is a pandemic disease that has affected almost all around the world. The most crucial step in the treatment of patients with COVID-19 is to investigate about the coronavirus itself. In this research, for the first time, we analyze the complex structure of the coronavirus genome and compare it with the other two dangerous viruses, namely, dengue and HIV. For this purpose

Let I be the unit matrix and D=00,10,01,−1−1. In this paper, we consider the self-similar measure μρI,D on ℝ2 generated by the iterated function system {ϕd(x)=ρI(x+d)}d∈D where 0<|ρ|<1. We prove that there exists Λ such that EΛ=e2πi<λ,x>:λ∈Λ is an orthonormal basis for L2(μρI,D) if and only if |ρ|=1/(2q) for some integer q>0.

This paper deals with the problem of Lyapunov inequalities for local fractional differential equations with boundary conditions. By using analytical method, a novel Lyapunov-type inequalities for the local fractional differential equations is provided. A systematic design algorithm is developed for the construction of Lyapunov inequalities.

The homogeneous perfect sets introduced by Wen and Wu [Hausdorff dimension of homogeneous perfect sets, Acta. Math. Hungar.107 (2005) 35–44] is an important class of Moran sets. In this paper, we obtain the Assouad dimension and Assouad spectrum formulas for homogeneous perfect set under suitable condition. In the proof an Assouad spectrum formula for a large class of fractal sets is established.

In study of gas migration process in coal seam, the permeability evolution rule with time has been one of hot topics in the area of coal seam for decades. At present, in view of the time-varying rule of coal seam permeability, the influence of microstructure parameters and adsorption effect are seldom considered simultaneously. In this paper, the fractal seepage model coupled with coal deformation

Coronavirus disease (COVID-19) is a pandemic disease that has had a deadly effect on all countries around the world. Since an essential step in developing a vaccine is to consider genomic variations of a virus, in this research, we analyzed the variations of the coronavirus genome between different countries. For this purpose, we benefit from complexity and information theories. We analyzed the variations

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears

The spread of infectious disease, virus epidemic, fashion, religion and rumors is strongly affected by the nearest neighbor hence underlying morphologies of the colonies are crucial. Likewise, the morphology of naturally grown patterns ranges from fractal to compact with lacunarity. We analyze the contact process on the fractal clusters simulated by generalized Diffusion-limited Aggregation (g-DLA)

Complicated gas–water transport behaviors in nanoporous shale media are known to be influenced by multiple transport mechanisms and pore structure characteristics. More accurate characterization of the fluid transport in shale reservoirs is essential to macroscale modeling for production prediction. This paper develops the analytical relative permeability models for gas–water two-phase in both organic

In this paper, we mainly discuss continuous functions with certain fractal dimensions on [0,1]. We find space of continuous functions with certain Box dimension is not closed. Furthermore, Box dimension of linear combination of two continuous functions with the same Box dimension maybe does not exist. Definitions of fractal functions and local fractal functions have been given. Linear combination of

For any x∈(0,1], let the infinite series Σn=1∞1d1(x)d2(x)⋯dn(x) be the Engel expansion of x. Suppose ψ:ℕ→ℝ+ is a strictly increasing function with limn→∞ψ(n)=∞ and let E(ψ), Esup(ψ) and Einf(ψ) be defined as the sets of numbers x∈(0,1] for which the limit, upper limit and lower limit of logdn(x)ψ(n) is equal to 1. In this paper, we qualify the size of the set E(ψ), Esup(ψ) and Einf(ψ) in the sense

This paper presents a thorough study of a time-dependent nonlinear Schrödinger (NLS) differential equation with a time-fractional derivative. The fractional time complex transform is used to convert the problem into its differential partner, and its nonlinear part is then discretized using He’s polynomials so that the homotopy perturbation method (HPM) can be applied powerfully. The two-scale concept

Under the realistic background that the capital market nowadays is a fractal market, this paper embeds the detrended cross-correlation analysis (DCCA) into the return-risk criterion to construct a Mean-DCCA portfolio model, and gives an analytical solution. Based on this, the validity of Mean-DCCA portfolio model is verified by empirical analysis. Compared to the mean-variance portfolio model, the

Currently, sparse code multiple access (SCMA) is a commonly used multiple-access technique, and it is a strong candidate for implementation as part of the fifth generation (5G) of wireless mobile communications. Although several design methods are available for SCMA codebooks, we propose a new method that optimizes point-to-point distances within the same codeword and from codebook-to-codebook for

SARS-CoV-2 is a deadly virus that has affected human life since late 2019. Between all the countries that have reported the cases of patients with SARS-CoV-2 disease (COVID-19), the United States of America has the highest number of infected people and mortality rate. Since different states in the USA reported different numbers of patients and also death cases, analyzing the difference of SARS-CoV-2

Severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) is the most dangerous type of coronavirus and has infected over 25.3 million people around the world (including causing 848,000 deaths). In this study, we investigated the similarity between the genome walks of coronaviruses in various animals and those of human SARS-CoV-2. Based on the results, although bats show a similar pattern of coronavirus

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

Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this

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

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

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.

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 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

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

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

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

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.