Let (tn)n≥0 be the well-known ±1 Thue–Morse sequence +1,−1,−1,+1,−1,+1,+1,−1,…. Since the 1982–1983 work of Coquet and Dekking, it is known that ∑k

The calculus of the mixed Weyl–Marchaud fractional derivative has been investigated in this paper. We prove that the mixed Weyl–Marchaud fractional derivative of bivariate fractal interpolation functions (FIFs) is still bivariate FIFs. It is proved that the upper box dimension of the mixed Weyl–Marchaud fractional derivative having fractional order γ=(p,q) of a continuous function which satisfies μ-Hölder

It is very difficult to characterize the transport properties of porous media due to the disorder of pores distribution in porous media. In this paper, we study heat conduction through porous media with randomly fractures which are considered as tree-like networks. An expression for effective thermal conductivity (ETC) of saturated dual-porosity media is derived based on the fractal characteristics

Recently, wavelets are playing a very important role in the numerical analysis. In this paper, an investigation is made for numerical solution of a class of nonlinear fractional differential equations (FDEs) with error analysis using Haar wavelet collocation method. The proposed method is illustrated through presenting different kinds of FDEs, which gives the approximate solution and is in good agreement

The stretched Cantor product is not self-similar but rectifiable. Using the method of finite pattern, we study the mean geodesic distance on the stretched Cantor product.

In this paper, we introduce touching networks including skeleton networks of Sierpinski gasket, Sierpinski tetrahedron, Sierpinski hexagon and Lindstr öm Snowflake. For touching networks, we establish the self-similar theory of limit metric space.

This paper relates to the field of Artificial Intelligence, specifically to image recognition, and provides an innovative method to take advantage of Blockchain Convolutional Neural Networks (BCNNs) in Emotion Recognitions (ERs) using audio–visual emotion patterns to determine a healthcare emergency to be attended. BCNN architectures were used to identify emergency patterns. The results obtained indicate

The impact of crack–cocaine dependence on the quality and microarchitecture of the mandibular bone tissue requires further investigation. This cross-sectional study evaluated the fractal dimension and panoramic radiomorphometric indices in crack–cocaine-addicted men. Panoramic radiographs were obtained from 24 addicted and 24 nonaddicted men (control individuals) between the ages of 18 years and 60

Based on high-frequency data, we study the difference in cryptocurrency market before and during the COVID-19. We analyze the multifractality of three major cryptocurrencies via the multifractal detrended fluctuation analysis (MFDFA). To investigate the source of multifractality, we construct shuffled, surrogated and truncate data. The results show that market efficiency of cryptocurrency has decreased

In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional

The purpose of this work is to investigate some inequalities for generalized s-convexity on fractal sets ℝα, which are associated with Simpson-like inequalities. To this end, an improved version of Hölder inequality and a Simpson-like identity on fractal sets are established, in view of which we give several estimation-type results involving Simpson-like inequalities for the first-order differentiable

In this paper, considering the structure of the shape factor and the rough element, combined with fractal geometry theory, the shape factor, the relative increase in the heat transfer between the rough surface and the smooth surface in a fractal unit, and the fractal analytical expression of the dimensionless equivalent thermal conductivity caused by convection thermal conductivity are derived. In

There are still mathematical predictions in the fight against epidemics. Speedy expansion, ways and procedures for the pandemic control require early understanding when solutions with better computer-based mathematical modeling and prognosis are developed. Despite high uncertainty in each of these models, one of the important tools for public health management system is epidemiology models. The fractional

The well-known coupled Boussinesq–Burger equation can be used to describe the flow of the shallow water of the harbor. But when the boundary is nonsmooth, it becomes powerless. So, the fractal calculus is needed to be applied to it and the fractal coupled Boussinesq–Burger equation (FCBBe) is presented for the first time in this paper. By using the semiinverse method, we have successfully established

Analysis of the variations of heart activity during different human activities is an important area of research in sport sciences. Therefore, in this paper, we evaluated the variations of heart activity for 23 subjects while sitting, hand biking, walking, and running. Since the obtained R-R time series (as the indicator of heart rate variability (HRV)) has a complex structure that contains information

Attention deficit hyperactivity disorder (ADHD) is a mental health disorder that is very common among children and may last into their adulthood. It is known that ADHD affects the attention of patients due to problems with short-term memory. Therefore, analysis of attention and memory of these patients should come into consideration. In this study, the complexity and memory of Electroencephalogram

In this study, the scaling characteristics of root-mean-squared roughness (Rq) was investigated for both fractal and non-fractal profiles by using roughness scaling extraction (RSE) method proposed in our previous work. The artificial profiles generated through Weierstrass–Mandelbrot (W–M) function and the actual profiles, including surface contours of silver thin films and electroencephalography signals

The effective thermal conductivity (ETC) of roughened porous media (RPM) is of interest in a number of applications of heat transfer. In this work, a fractal analytical model for the ETC of RPM with microscale effect is proposed. The proposed fractal model is expressed in terms of relative roughness, the molecular mean free path, porosity, fractal dimensions (pore area fractal dimension and tortuosity

The unsmooth boundary will greatly affect the motion morphology of ion-acoustic waves in plasma, so a modified equal width equation with fractal derivatives is proposed. The fractal variational formulation of the problem is established by using the semi-inverse method, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation

A concrete beam is a basic element for many applications in architectural engineering, and its vibration property plays an important role in safety and life-span. Previous vibration models were based on continuum beam theories, which could not take into account the porous property of the concrete beam. This paper adopts a two-scale fractal vibration model, and its low frequency property is revealed

The paleoenvironment influences trace-making behavior significantly, and progressively more evidence shows that this behavior follows scale-invariance properties. In this study, we focused on the definition of behavioral complexity in the trace fossil Zoophycos to explore its paleo-environmental significance. Firstly, the fractal topography in Zoophycos was established by introducing virtual tertiary

Fractal coding has been widely used as an image compression technique in many image processing problems in the past few decades. On the other hand side, most of the natural images have the characteristic of nonlocal self-similarity that motivates low-rank representations of them. We would employ both the fractal image coding and the nonlocal self-similarity priors to achieve image compression in image

In this paper, a multi-AUV dynamic maneuver decision-making algorithm is studied based on intuitionistic fuzzy game and fractional-order Particle Swarm Optimization (PSO). Because of the weak communication condition and complex marine environment, a maneuver decision-making algorithm is usually hard to realize in real-time multi-AUV couter-game process. First, the weak communication condition is analyzed

This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the blacksolution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are blackevaluated in

The fractional derivative holds historical dependence or non-locality and it becomes a powerful tool in many real-world applications. But it also brings error accumulation of the numerical solutions as well as the theoretical analysis since many properties from the integer order case cannot hold. This paper defines the tempered fractional derivative on an isolated time scale and suggests a new method

The present work presents a simplified mathematical model to calculate the flowrate of the combined electroosmotic flow (EOF) and pressure driven laminar flow (PDLF) in the symmetric tree-like microchannel network under the assumptions of small zeta potential and thin electrical double layer. A numerical analysis of the combined EOF and PDLF in symmetric Y-shaped microchannel is also carried out to

Let K be a self-similar set in ℝ. Generally, if the iterated function system (IFS) of K has some overlaps, then some points in K may have multiple codings. If an x∈K has a unique coding, then we call x a univoque point. We denote by 𝒰 (univoque set) the set of points in K having unique codings. In this paper, we shall consider the following natural question: if two self-similar sets are bi-Lipschitz

In the current study, we propose a bifractal stratified characterization method to evaluate the surface profile evolution of a plateau honing cylinder liner during the wear process. This method is suitable for the characterization of two-process and worn surface profiles with bifractal and stratified characteristics. We develop a bifractal truncated separation structure function method to calculate

We analytically investigate a nonlinear fractional-order sine-Gordon (sG) equation. The derivatives considered herein, are taken in Caputo’s sense. The Laplace transform together with the Adomian decomposition method (LADM) is applied to attain analytical approximation of the aforesaid equation. The sG equation having Caputo derivative is solved in the order of series solutions and the results are

In this paper, we develop the multifractal detrending weighted average algorithm of historical volatility (MF-DHV) for one-dimensional multifractal measure based on the classical multifractal detrended fluctuation analysis (MF-DFA). In the calculation process of getting a local trend for MF-DHV, historical volatility is taken to develop an moving average algorithm, which is different from the simple

In this work, the nonlinear fractal K(p,q) model is defined via the fractal derivative. The variational principle of the fractal model is constructed by using the fractal semi-inverse method, and it is of great significance to study the structure and properties of its soliton solution. Based on the established variational principle, we successfully proposed a new method to solve the fractal model.

The definitions of derivatives as delta and nabla in time scale theory are kept to follow the notion of the classical derivative. The jump operators are used to transfer the notion from the classical derivative to the derivatives in the time scale theory. The jump operators can be determined by analyst to model phenomena. In this study, the definitions of derivatives in the time scale theory are transferred

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption

We used multifractals to analyze the Lebanese topography focusing on Mount Lebanon. The elevation data were obtained from NASA STRM Global Digital Elevation of Earth Land, spaced at 80m in the East-West direction, and at 90m in the North-South direction. After transforming the grid to be perpendicular and parallel to the range, we found anisotropic scaling from 500m to 10,000m, and it reflected the

One-dimensional (1D) map-based neuron models are of significant interest according to their simplicity of simulation and ability to mimic real neurons’ complex behaviors. A fractional-order 1D neuron map is proposed in this paper. Dynamical characteristics of the model are analyzed by obtaining bifurcation diagrams and the Lyapunov exponents’ diagram. Furthermore, emerging the spiral wave as one of

Some complex systems may suffer from failure processes arising from soft failures and hard failures. The existing researches have shown that the reliability of a dynamic system is not constant under uncertain random environments. First, two types of uncertain random optimization models are proposed where reliability index is quantified by chance measure based on whether soft and hard failures are independent

This work is devoted to explore the dynamics of the proposed discrete fractional-order prey–predator model. The model is the generalization of the conventional discrete prey–predator model to its corresponding fractional-order counterpart. The fixed points of the proposed model are first found and their stability analyses are carried out. Then, the nonlinear dynamical behaviors of the model, including

As a kind of classical fractal network models, Sierpinski Gasket and its extended network models have attracted the attention of many scholars because of their many applications in real-world networks. Based on the classical Sierpinski Gasket, this paper proposes a joint Sierpinski Gasket model, which can be used not only to study the topology and dynamic properties of Sierpinski Gasket itself, but

In this paper, a novel direct scheme to solve a set of time-delay fractional optimal control problems is introduced. The method firstly uses a set of Dickson polynomials as basis functions to approximate the states and control variables of the system. Next, the context of these basis functions and the use of a collocation method allow to transform the problem into a system of nonlinear algebraic equations

In this work, the fractal cubic–quintic Duffing’s equation analytical solution is obtained using the two-scale transform and elliptic functions. Then, the analytical solution is used to study wave propagation in a fractal medium. Since the value of the fractal parameter adjusts the pulse frequency and wavelength propagation velocity, depending upon the fractal medium physical properties, it is found

Generalized synchronization is a typical dynamical phenomenon in nonlinear systems, for which the real-valued setting has been widely investigated. The complex-valued functions relationship in generalized synchronization is equally important for complex-valued dynamical systems, which however are seldom studied. Complex parameters identification on the synchronization manifold remains an open problem

This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered

This study experimentally estimated the fractal pore size distribution of granites, which is then linked to the hydro-mechanical properties of rocks treated by temperatures that range from 25∘C to 900∘C. The mercury intrusion experiment was carried out to characterize the pore size distributions and the MTS815.02 triaxial testing system was used to investigate hydro-mechanical properties of rocks.

The main objective of this paper is to present weighted k-fractional integral and derivative operators of a function with respect to another function and to uncover their properties. In addition to this, we find the weighted Laplace transform of the newly defined operators. As applications of the weighted k-fractional operators in mathematical physics, we study the fractional forms of kinetic differintegral

Cooperative behavior is an extensive mechanism in nature, and seen in several organisms, from bacterial cells to primates. In this paper, we have investigated a spatial predator–prey model with hunting cooperation in predators. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive

In this paper, we derive the probability distribution of linear fractal interpolation function (FIF) for a random dataset on ℝ2, where randomness in the data is induced, using α-stable noise, in the second coordinate. In particular, we show that the distribution of linear FIF, at any point (on the first coordinate), is also stable, but with different parameters which can be estimated. Finally, we also

We identify a countable infinity of parameter regimes with strange nonchaotic attractors (SNAs). At the edge of each arc parameter area, there is an uncountable infinity of SNAs with torus intermittency. The mechanism for the creation of SNAs in different regime is induced by an n-frequency quasiperiodic orbit through a quasiperiodic analog of saddle-node bifurcation (Type-I intermittent route). We

The apparent liquid permeability (ALP) of shale is challenging to be characterized due to complex wettability and nanopore size distribution. The nanopores in organic matter of shale are usually hydrophobic and the nanopores in inorganic matter are hydrophilic. The flow behaviors in these two different nanopores are quite different, and accurately predicting the ALP of shale is difficult. This paper

Coal failure behavior under dynamic loading is significant for dealing with the failure issues encountered in underground coal mines. In this study, the crack propagation and internal fracture process of coal under dynamic loading were investigated by a split Hopkinson pressure bar (SHPB) experimental system and numerical approach, respectively. The experimental and numerical results show that the

Dürer’s pentagon is known to the artist Albrecht Dürer. In this paper, we consider directed networks generated by Dürer-type polygons. We aim to present a study on some properties of these networks, such as degree distribution, clustering coefficient and average path length. We show that such networks have the scale-free effect, but do not have the small-world effect.

The geodesic structures on self-similar fractals are interesting. One feature of geodesic structures is average distance on fractal. We investigate the Sierpinski-like carpet and obtain the average distance in terms of self-similar measure.

In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With f(t,x) being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function V(t,x(t)) along solutions is continuous and moreover, t0CDtαV(t,x(t))≤∂V(t,x)∂t(t,x(t))t0CDtαt+∂V(t,x)∂x⋅f(t,x)(t,x(t))

In this work, the Schrödinger equation is described by the fractal derivative, and its variational principle is obtained by using the fractal semi-inverse method. The variational principle is helpful to research the construction of the solution. The approximate analytical solution of the fractal Schrödinger equation is obtained based on the proposed variational approach and the fractal two-scale transform

We investigate a class of self-similar networks modeled on the three-dimensional Vicsek fractal, and obtain the asymptotic formula of node-weighted average geodesic distances, which is an integral of geodesic distance in terms of the self-similar measure.

Fractal analysis was used in the study to determine a set of feature descriptors which could be applied in the process of diagnosing bone damage caused by osteoporosis. The subject of the research was CT images of vertebrae on the thoraco-lumbar region. The dataset contained images of healthy patients and patients diagnosed with osteoporosis. On the basis of fractal analysis and feature selection by

We investigate the n-level Sierpinski carpets including the classical one for n=3, which are non-p.c.f. self-similar fractals. By using the finitely, many geometric patterns and the self-similar measure, we obtain the mean geodesic distance on the n-level Sierpinski carpets and their skeleton networks.

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order

In this paper, the definition of generalized s-preinvex function on Yang’s fractal sets ℝγ(0<γ≤1) is proposed, and the generalized Hermite–Hadamard’s inequality for this class of functions is established. By using this convexity, some generalized Hermite–Hadamard type integral inequalities with parameters are established. For these inequalities, the absolute values of twice local fractional order derivative

Let [a1(x),a2(x),…,an(x),…] be the continued fraction expansion of an irrational x∈(0,1). For any n≥1, write Tn(x)=max1≤k≤n{ak(x)}. This paper is concerned with the Hausdorff dimension of the set E(ψ):=x∈(0,1):limn→∞Tn(x)ψ(n)=1, where ψ:ℕ→ℝ+ is a function such that ψ(n)→∞ as n→∞. We calculate the Hausdorff dimension of E(ψ) for a very large class of functions with certain growth rates, which improves

Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto