Fractals

Let $\beta >1$ and $x\in (0,1]$ be two real numbers. For all $x\in (0,1]$, the run-length function with respect to $x$, denoted by $r_x(y,n)$, is defined as the maximal length of the prefix of the $\beta$-expansion of $x$ amongst the first $n$ digits of the $\beta$-expansion of $y$. The level set

A new type of iterated function systems is constructed based on different weights, and it is proved that this type of iterated function systems generates a class of bivariate continuous functions whose graphs are the fractal interpolation surfaces passing through the given interpolation points. Considering the influences of the fractal interpolation functions on different weights and basic functions, we give the corresponding error estimation formula. Finally, the calculation formula for the integral moments of this class of bivariate fractal interpolation functions is given.

In this study, the characterization method and evolution law of phase-point saturation based on the capacity dimension in fractal theory are studied to investigate the running-in process and state identification of a system. Tests of the running-in process under different working conditions were conducted on a ring-disc tribometer, and the friction signals were collected. The phase trajectories in different stages were then constructed and the fractal dimensions were calculated. The phase-point saturation parameter is proposed to reflect the degree of saturation of a phase space filled with phase points, and its reliability, noise-resistance, correlation with the fractal dimension, and effectiveness of the characterization of the running-in process were thoroughly investigated. The results demonstrate that the proposed phase-point saturation parameter is reliable and resistant to noise and can be used to identify changes in the system state. Furthermore, there is a negative correlation between the phase-point saturation parameter and the fractal dimension, and the phase-point saturation parameter can effectively characterize the state characteristics and describe the running-in process. During the running-in process, the phase-point saturation changes from high to low and tends to be stable when it reaches the running-in state. When entering the stable wear stage, the value of the parameter is maintained near the minimum, it fluctuates little as compared with the correlation dimension, and it exhibits good stability. Therefore, the proposed phase-point saturation parameter has important theoretical significance and application value for the quantitative characterization of the running-in process and state identification.

The thin film equation is very important in electroanalytical chemistry arising in diffusion–reaction process. Considering the effort of the surface morphology on the diffusion process, the thin film equation is modified by using the fractal derivative, and its fractal variational principle is successfully established according to the semi-inverse method and two-scale transform method. It is very helpful to study the structure of the approximate solution.

In this paper, we show that the spaces of some types of fractal interpolation functions are reproducing kernel Hilbert spaces with two different types of inner products. Then we apply these results to curve fitting problems. We establish the fractal interpolation functions that are in reproducing kernel Hilbert spaces and that minimize the regularized empirical error.

Fractal characteristic has been considered as the essential parameter to determine transport properties of porous media. Fluid flow in the real porous media is more influenced by the narrow throat system than the relatively large pore system. In this work, the concept of throat fractal dimension is proposed and applied to predict permeability in porous media. First, pore system and throat system of six sandstone rock samples and three carbonate rock samples are divided from the pore network models extracted from rock CT images. Then, fractal dimensions based on the total pore space, pore system and throat system are calculated, respectively, based on the three-dimensional (3D) box-counting method. Three groups of permeabilities are estimated, respectively, using the fractal theory based on the pore space fractal parameter, pore system fractal parameter and the proposed throat system fractal parameter. The results indicate that permeabilities estimated by the proposed throat system fractal parameter are closer to the accurate permeabilities of rock samples compared with that estimated by the pore space fractal parameter and pore system fractal parameter. Our study results reveal that throat fractal dimension should be used to accurately predict fluid transport properties in porous media.

In this paper, we make research on the approximation of functions with fractal dimension in continuous functions space. We first investigate fractal dimension of the linear combination of continuous functions with different fractal dimensions. Then, fractal winding of continuous functions has been given. Furthermore, based on Weierstrass theorem and Weierstrass function, we establish a theorem that continuous functions with the non-integer fractal dimension can be approximated by the linear combination of trigonometric polynomials with the same fractal dimension. Condition about continuous functions with fractal dimension one and two has also been discussed elementary.

The phenomenon of synchronization in coupled chaotic maps has been studied extensively by the researchers in nonlinear dynamics for several years. But, there is hardly any study of the stationary densities of coupled maps as a function of the coupling parameter. Here, we numerically analyze stationary densities of two nonlinearly coupled tent maps. In the process, we find that the emergent stationary density can become multifractal even if the stationary density of the individual maps is smooth. In this work, we study piecewise linear and everywhere expanding maps as the stationary density for these maps is described by a simple functional relation. By extensive numerical simulations we find that for this class of maps the multifractal spectrum is universal and does not seem to change with the coupling or map parameters. The existence of this multifractal nature is not surprising, a posteriori, as it is known the densities satisfying similar functional relation are known to be multifractal but the form of the multifractal spectrum here is intriguing as it does not seem to conform to the existing theory of multifractal functions.

In this work, we compare three distinct schemes for detrended fluctuation analysis (DFA) of three-dimensional (3D) $N\times N\times N$ data sets: (i) an $N\times N$ set of one-dimensional (1D) DFA runs for each vertical line of length $N$, (ii) a set of $N$ two-dimensional (2D) DFA runs for each of the $N\times N$ planes along the vertical direction, and (iii) a single 3D DFA run. The objective of this work is two-fold: (i) to compare the performance of 1D, 2D, and 3D DFA methods in terms of compatibility as well as the richness of obtained results, and (ii) to demonstrate how these three complementary approaches shed new light on the complexity of the structure of the two real soil samples, through the analysis of grayscale high-resolution X-ray computed tomography scan images.

In this paper, two different types of plasma models with variable coefficients are described by using the fractal derivative. This paper aims at establishing fractal variational principles for two different types of discontinuous plasma physics by the fractal semi-inverse method, so that it can be applied to numerical analysis of the effect of the discontinuous time and discontinuous space on the solution properties.

Horizontal drilling is the major method of shale gas extraction, and hydraulic fracturing improves the gas production effectively. However, as the main gas migration channel, shale microstructures, such as the natural fractures and pores, are extremely complicated, and it is difficult to characterize them quantitatively by conventional approaches. Besides, the microstructural evolution of the hydraulic fracture (HF) area is more intricate. In order to quantitatively characterize the evolution of shale microstructure after hydraulic fracturing, and to investigate the contribution to the gas extraction process, a fully coupled fractal thermal–hydrological–mechanical permeability model is proposed. And the interaction of shale microstructure, thermal–hydrological–mechanical effects, and gas permeability in the HF area is also quantitatively investigated. Shale thermal conduction, gas pressure, and adsorption–desorption effect are defined as functions of the effective stress responsible for shale matrix deformation, and operate directly on shale porosity, which induces changes in shale microstructure, including the shale fractal dimension (for characterizing the density of natural fractures in shale) and the maximum fracture length. Multiphysical effects have a major impact on gas seepage, and directly affecting the permeability. Furthermore, compared to the widely used cubic permeability model, this model is significantly superior in analyzing the permeability evolution in the hydraulically fractured area, and the reliability is verified by the production data from the Marcellus Shale.

This paper focuses on the Hopf bifurcation of a fractional tri-neuron network with both leakage delay and communication delay under different fractional orders. By applying fractional Laplace transform, the stability theorem of linear autonomous system and Hopf bifurcation theorem, we obtain a class of asymptotic stability criterion of zero solution as well as delay-induced Hopf bifurcation conditions for the considered system. Simultaneously, the stability and Hopf bifurcation for tri-neuron network with single fractional order are also discussed as a special case of our proposed neural network model. Finally, a simulation example is given to illustrate the efficiency of the presented theoretical results in this paper.

The simplest frequency formulation for nonlinear oscillators is introduced and proved, and a modification is suggested. A fractal vibration in a porous medium is introduced, and its low-frequency property is elucidated by the frequency formulation. It reveals that the inertia force in a fractal space is equivalent to a couple of the inertia force and the damping force for the traditional differential model.

This paper introduces evolving hierarchical networks based on hierarchical product. We discuss two important consensus behavior indexes in both un-weighted and weighted cases in those networked systems: (1) convergence speed in system without delay and noise; (2) delay robustness in system with communication time-delay. We analyze the approximative behaviors of these two consensus indexes by determining the second smallest and largest eigenvalue of Laplacian matrix, respectively, as well as how the scale of the weight factor affects the consensus performance. Moreover, the method applied in this paper is also applicable to many classic networks. Finally, two numerical simulations are given to verify the advantages of communication topology like hierarchical networks and the validity of the theoretical results.

In this paper, through finite difference and finite element coupling algorithms with structured grid, the fluid–structure coupling problems of the pressure of the fluid as well as stress and strain of the heterogeneous porous medium under the initial pressure of MPa magnitude are calculated, respectively. The porous medium is a cavity cylinder with fractal characteristics. The parameters of porosity, elastic modulus and Poisson’s ratio of porous media with different fractal dimensions ($D=2.3$, 2.4 and 2.5) are obtained by theoretical calculation. Through theoretical derivation, the amplitude of WM function is expressed by the fractal parameter porosity, which can predict the uneven characteristics caused by the nonuniformity of pore size distribution after explosion. The results show that with the increase of fractal dimensions, the porosity increases, and both the elastic modulus and the Poisson’s ratio decrease. The Poisson’s ratio has a linear relationship with porosity, which is well consistent with the modified formula in the literature. Under the same initial pressure, the stress and strain values of the material without fractal characteristics are much larger than those of the material with fractal feature which has an obvious fluctuation by extracting the average values. It indicates that the fractal porous media with rough surface can effectively reduce the pressure wave peak at the wall surface, and reduce the size of the stress and strain concentrations to prevent the vibration of the cylinder. As expected, if the initial pressure increases with an equal fractal dimension, the stress and strain values of the fluid–structure coupling will increase too. When the initial pressure is equal, the peak value of pressure in wall decreases gradually with the increase of fractal dimension, and the stress and strain values also decrease.

Recently, the consensus problem has attracted extensive attention of researchers from different scientific communities. In this paper, we study the leader–follower network coherence (i.e. the mean steady-state variance of the deviation from the static value of the leader nodes) in the weighted recursive tree networks with assigning the leaders orderly in the initial state. For the weighted recursive tree networks model, we study the characteristic polynomial of the Laplacian submatrix and obtain the relationship for the eigenvalues in two successive generations. The analytical formula of the leader–follower network coherence related by the sum of the reciprocal of all eigenvalues of this submatrix is obtained. The result shows that more number of leaders and the greater weight factor lead to better consensus.

The aim of this study is to present the numerical solutions of the Liénard nonlinear model by designing the structure of the computational Gudermannian neural networks (GNNs) along with the global/local search efficiencies of genetic algorithms (GAs) and interior-point algorithm (IPA), i.e. GNNs–GAs–IPA. A merit function in terms of differential system and its boundary conditions is designed and optimization is performed by using the proposed computational procedures of GAs–IPA to solve the Liénard nonlinear differential system. Three different highly nonlinear examples based on the Liénard differential system have been tested to check the competence, exactness and proficiency of the proposed computational paradigm of GNNs–GAs–IPA. The statistical performances in terms of different operators have been provided to check the reliability, consistency and stability of the computational GNNs–GAs–IPA. The plots of the absolute error, performance measures, results comparison, convergence analysis based on different operators, histograms and boxplots are also illustrated. Moreover, statistical gauges using minimum, mean, maximum, semi-interquartile range, standard deviation and median are also provided to authenticate the optimal performance of the GNNs–GAs–IPA.

Poisson equation is a partial differential equation with broad utility in theoretical physics. Dirichlet problem of Poisson Equations can be solved by Green’s function. It is a very attractive problem to look for analogous results of the above problem in the fractal context. This paper studies the network set Higher-Dimensional Sierpinski Gaskets, solves Dirichlet problem of Poisson Equations on them by expressing Green’s function explicitly.

Multifractals are the general form of scale invariances, where a fractal behavior is repeated by a similar object or pattern. Although the multifractal spectrum has been accepted as a measure of behavioral complexity, it cannot solely determine a fractal behavior, thus leaving the control mechanisms of multifractality unclarified. Here, we reexamine the multifractality, and discover two key features of scaling behaviors in the fractal behavior, i.e. multiplicity and repetition. Afterwards, we establish multifractal topography to unify the scale-invariance definition of arbitrary fractal behavior, clarify the physical meaning of singularity and its significance, construct a scale-invariance tree to delineate the control mechanisms in fractality, and then identify multifractals to be dual-complexity systems with the original and behavioral complexities controlling the scaling type and the scale-invariance property independently. This study gives insight into quantitative fractal theory and provides fundamental support for the mechanism exploration of nonlinear dynamics.

This paper mainly investigates fractal dimensions of Katugampola fractional integral of continuous functions. It is proven that there exist some interesting connections between the order of Katugampola fractional integral and the upper box dimension of continuous functions satisfied Hölder condition.

In the paper, we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss–Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of the Hausdorff vector calculus. The formula is proposed as a mathematical tool to describe the real-world problems for the fractal power-law flow equations with the anomalous diffusion equation. A conjecture for the fractal power-law flow equations analogous to the Smale’s 15th problem and one of Millennium Prize Problems for the Navier–Stokes equations is also addressed.

Let $A$ be a limsup random fractal with indices ${\gamma }_1,{\gamma }_2$ and $\delta$ on ${[0,1]}^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star ):{dim}_{\mathrm{\text{H}}}(G)>{\gamma }_2+\delta$, where ${dim}_{\mathrm{\text{H}}}$ denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan et al.¹ by relaxing the condition that the probability $P_n$ of choosing each dyadic hyper-cube is homogeneous and ${lim}_{n\to \infty }\frac{{log}_2{P}_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\star )$ cannot be replaced by the packing dimension.

In this work, we mainly focus on the fractal variant Boussinesq–Burgers equation which can well describe the motion of shallow water traveling along an unsmooth boundary. First, we construct its fractal variational principle and prove its strong minimum condition by the fractal Weierstrass theorem. Then two types of soliton solutions are acquired according to the constructed fractal variational principle. We find that the order of the fractal derivative hardly affects the whole shape of the solitary waves, but it remarkably affects its propagation process.

In this paper, we propose an anomalous chaotic system of the scaling-law (SL) ordinary differential equations involving the Mandelbrot SL. This chaotic behavior shows the “Wukong” effect. The comparison among the Lorenz and SL attractors is discussed in detail. We also suggest the conjecture for the fixed point theory for the fractal SL attractor. The SL chaos may be open a new door in the study of the chaos theory.

The complex rock fracture structures in reservoirs play an important role during methane extraction. However, there is still a challenge to elucidate the impacts of adsorption–desorption, rock expansion, and thermal conduction on the microstructures under thermal–hydrological–mechanical interactions. In this paper, fractal theory for porous media was applied to characterize the structures of rock fractures, and the fracture fractal dimension $(D_f)$ was adopted to analyze the density of fractures and the microstructural evolution. We developed a coupled thermal–hydrological–mechanical model enabling simultaneous analysis of rock fracture microstructures and multi-physical field effects. Furthermore, we analyzed the evolution of the fracture fractal dimension with the reservoir parameter effects, including: (1) methane extraction process; (2) reservoir stress; (3) pore pressure; and (4) reservoir temperature. We also calculated the effects of physical and mechanical factors on the above parameters, including (1) adsorption constant; (2) the in-situ stress; and (3) thermal expansion coefficient. The present results indicate that various characteristic parameters have multiple effects on the microstructures of rock fractures. It was found that the fractal dimension is inversely proportional to the reservoir stress, the gas pressure, and the reservoir temperature.

In this paper, we use a kind of Sierpiński-like carpet to construct a special evolving network, whose nodes are all the solid squares in the construction of the carpet up to stage t. In our network, two nodes are neighbors if and only if the intersection of their corresponding squares is a line segment. Using the encoding method in fractal geometry, we show that our evolving network is scale-free and has the small-world effect, but is not fractal scaling.

Archie’s law is commonly applied to predict electrical conductivity of rock by correlating the homogeneous, water-wet reservoir rock conductivity with its porosity and saturation. However, Archie’s law is an empirical model consisting of unknown cementation exponent and saturation exponent. In this study, we propose analytical electrical conductivity models for both single-phase and multi-phase porous media based on the fractal theory. The established electrical conductivity model for fractal porous media saturated with a single fluid phase is the function of several factors including pore fluid electrical conductivity, porosity, pore fractal dimension and tortuosity fractal dimension. The established electrical conductivity model for fractal porous media containing multiple fluid phases is the function of wetting fluid saturation, contact angle and pore shape apart from the factors for saturated fractal porous media condition. The calculated formation factor based on the proposed model correlates well with the sandstone experimental data.

In this paper, the analytical solution that describes the evolution in time of the fractal damped Duffing equation subjected to external forces of elliptic type is derived using He’s two-scale fractal transform and the elliptic balance method (EBM). This solution predicts the evolution in time of the Duffing equation and unveils qualitative and quantitative system behavior when the values of the fractal parameter varies, and how these affect the frequency, the wavelength, and the oscillation amplitude from the start of the motion. Comparison of the amplitude–time response curves over the selected time-interval with those obtained from numerical simulations confirms the accuracy of the derived analytical solution.

In this work, the fractal nature of Selective Laser Melting (SLM) additive manufacturing process (AM) is elucidated. Fractal dimension and lacunarity of metallic powders are calculated from Scanning Electron Microscopy (SEM) images adapted from literature. The complexity and homogeneity of the textures of the powder beds are also studied through the comparison of fractal dimension and lacunarity. It is found that better densification results are obtained when the powder bed’s fractal dimension is closer to the golden mean number of 1.618. Furthermore, this finding is extended to expressions for predicting the component’s bulk density produced via SLM by setting the $\alpha$ exponent equal to the golden mean value and finding the proportionality constant, $C$, using a nonlinear least squares method. The proposed approach works well since theoretical prediction and experimental data compare well with root-mean-square-error (RMSE) values that do not exceed $7\times 10^{-08}$. This work sheds new light on enhancing additive manufacturing technologies considering the fractal nature of SLM since its process mathematical models are constructed around Euclidian space-time with continuous smooth assumptions that should be adapted to include the fractal nature of the manufacturing process aiming to improve their precision. The underlying interweaving of SLM, as a fractal process, and the golden mean number is revealed.

The classic single-fiber efficiency model based on ordered packing of fiber composites has been widely used to predict filtration efficiency of fibrous filters for 40 years. However, the simplified single-fiber model often overestimates the filtration efficiency as most fibrous composite filters are composed of randomly distributed fibers. The numerical methods have been successfully employed to re-construct the realistic fiber composites, but simulation of the multi-mechanism filtration process in the complex fibrous architecture is computationally expensive and case-based and derivation of a compact versatile model for broad applications remain challenging. In this work, a fractal-based homogenization model is developed to predict the filtration efficiency of the fibrous composites, by considering the spatially randomly distributed fibers and the quasi-random distribution of pore size. A comparison with available experimental results collected from references shows that the proposed model is of high accuracy in predicting the filtration of submicron aerosol particles. The calculated results show that the Kuwabara hydrodynamic factor is sensitive to the pore fractal dimension. The filtration efficiency for different particle diameters can be divided into three stages, which increase as the proportion of fibers warped by the fluid decreases. In addition, the total efficiency decreases with the increasing pore fractal dimension in a nonlinear trend. For the given fractal dimension, the filtration efficiency increases with the increase of pore size ratio, indicating that the more uniform the pore size distribution, the higher the efficiency.

Based on the local fractional derivative, we develop a new local fractional (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. With the help of the traveling wave transform of the nondifferentiable type, we convert the local fractional equation into a nonlinear local fractional ordinary differential equation (ODE). Then the Mittag-Leffler function-based method is presented here for the first time to construct the abundant nondifferentiable exact traveling wave solutions, and three families (six sets) of the exact solutions are obtained. In addition, the performances of different solutions on the Cantor set are presented via numerical results in the form of 3D plots. It reveals that the proposed method is simple and straightforward, which can be used to study the local fractional partial differential equations (PDEs).

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants’ fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere.

This paper examines the populational impact of the COVID-19 vaccinations for Brazil. Therefore, our analysis takes into account the time series of the daily number of deaths related to COVID-19 from March 17, 2020 until October 19, 2021 with 582 observations. Specifically, we apply the permutation entropy ($H_s$), statistical complexity ($C_s$) and Fisher information measure ($F_s$) to investigate the predictability of the daily deaths for COVID-19 considering two pandemic scenarios (until and after the extreme day). Based on these complexity measures, we construct the Complexity-Entropy causality plane (CECP) and Shannon–Fisher causality plane (SFCP), which allows us to assess the disorder and estimate randomness inherent to the time series of the daily deaths for COVID-19 concerning these two pandemic scenarios. Our empirical results indicate that after the extreme day, the increase in the vaccinated population contingent led to a lower entropy, higher predictability, and lower death cases. Given this, we conclude that the COVID-19 vaccines in Brazil were a highly effective public health action. In the most extreme situation, Brazil had 4249 records of daily deaths on April 8, 2021, approximately 3.5 months after the first dose of the vaccine. After this extreme situation on April 9, 2021, the daily records of deaths decrease to a minimum of 130 deaths on October 19, 2021. Thus, there is a percentage variation of −96.44% in records of daily deaths. To the best of our knowledge, this work is the first to provide empirical evidence for the populational impact related to COVID-19 vaccines.

The internal model control (IMC) has a good control effect for large time-delay systems, but the parameter tuning of the controller is a challenge for researchers. As a typical fractal (strange) attractor, a strange nonchaotic attractor has the dynamic characteristics of randomness and ergodicity. Based on these characteristics, an optimization method is proposed to tune the parameters of an internal model controller for a large time-delay process, which is named a strange nonchaotic optimization algorithm (SNOA). SNOA has the advantages of easy implementation, short execution time and having robust mechanisms to avoid local optimization. It can find the optimal parameters of the internal model controller in large time-delay systems. The results show that SNOA can effectively improve the dynamic stability of the large time-delay systems. It has also proved to be superior to chaotic optimization (CO) algorithm and particle swarm optimization (PSO) algorithm.

Thermoelectric effects have attracted wide attention in recent years from physicists and engineers. In this work, we explore the self-similar patterns in the thermoelectric effects of monolayer graphene-based structures, by using the quantum relativistic Dirac equation. The transfer matrix method has been used to calculate the transmission coefficient. The Landauer–Büttiker formalism and the Cutler–Mott formula were used to calculate the conductance, the Seebeck coefficient, and the power factor. We find self-similar behavior and the scale factors between generations in transport and thermoelectric properties. Furthermore, we implement these scale invariances as general scaling rules. We present a new analytical demonstration of self-similarity in the Seebeck coefficient. These findings can open outstanding perspectives for experimentalists to develop thermoelectric devices.

The dissimilar multi-scale structures of shale to conventional reservoirs make it a challenge to understand the fluid flow and heat transfer through unsaturated shale formations. In this paper, the pore structure and moisture content of shale samples are measured by low-field nuclear magnetic resonance technique and thermogravimetric differential scanning calorimetry test, respectively. A pore-scale model is accordingly developed for the immiscible two-phase fluid flow and heat conduction through unsaturated shale based on the statistically self-similar fractal scaling law of pore size distribution. The analytical expressions of effective and relative permeability, as well as effective thermal conductivity (ETC), are proposed, which indicate good agreement with experimental results. It has been shown that the capillary pressure and gas slippage play important role in multiphase flow through unsaturated shale. Both pore and tortuosity fractal dimensions show significant influence on the relative permeability for nonwetting phase (RPNW), while they indicate the marginal effect on the relative permeability for the wetting phase (RPW). The ETC decreases with the increase of pore and tortuosity fractal dimensions, and it is positively and negatively correlated with RPW and RPNW, respectively. The correlation between ETC and relative permeability is found to follow a logistic function. The present fractal model can characterize the multiscale structures of shale reservoirs and may help understand transport mechanisms of immiscible multiphase flow and heat transfer through unsaturated shale.

Cryptocurrencies play a pivotal role in the financial market. Given this, we perform the asymmetric multifractal cross-correlation analysis to examine the weak form of the Efficient Market Hypotheses (EMH) considering two temporal scales. In the daily scale, we find that the pair Bitcoin–Litecoin displays the largest multifractal spectrum. While, in the hourly scale, the pair Bitcoin–Ethereum presents the largest multifractal spectrum. Our empirical evidence has rejected the weak form of the EMH and clearly suggests that the dynamics of the analyzed cryptocurrency pairs are in line with the Fractal Market Hypothesis (FMH). Cross-correlation asymmetries are more persistent for small fluctuations than for large fluctuations. The results are essential for investors, portfolio and risk managers, and policymakers.

A generalized local fractional Riccati differential equation (LFRDE) method, which is a combination of the local fractional Riccati differential equation method and the transformed rational function approach, is presented to obtain the non-differentiable exact traveling wave solutions for n-dimensional fractional partial differential equations. In order to verify the effectiveness of the method, the seventh-order local fractional Sawada–Kotera–Ito equation, the local fractional sine-Gordon equation and the local fractional Kadomtsev–Petviashvili equation in fractal domain are first considered. The obtained results show that the presented method involving fractal special functions, are powerful and effective for obtaining exact solutions of nonlinear fractional partial differential equations in fractal domains.

The study characterizes power-law fluid through charged fibrous porous media with spatial fractional-derivative and fractal geometry. Seepage flow of power-law fluid across fractal fibrous porous media in the presence of electric double layers (EDLs) is investigated based on the capillary bundle model. The acquired velocity distribution equation in a narrow capillary is then transformed into the form of series with appropriate Taylor approximation. After that, an analytical formula for dimensionless permeability is derived based on the generalized Darcy’s law. The effects of diverse parameters, including the fractal dimension of pore area, porosity, fractional order and Zeta potential on dimensionless permeability, are discussed. It can be seen from the results that lower fractional order has an amplification effect on dimensionless permeability with the change in Zeta potential. The results provide some theoretical guidance for revealing the seepage mechanism of a power-law fluid in charged porous media.

We study the heat equation on a bounded open set $U\subset {ℝ}^d$ supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. The structures of these measures are in general complicated and intractable. However, for a class of such measures that we call essentially of finite type, important information about the structure of the measures can be obtained. We make use of this information to set up a framework to study the associated heat equations in one dimension. We show that the heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study the propagation speed problem.

Let $X$ be a Banach space with a Schauder basis ${(x_m)}_{m=0}^{\infty }$, and $I$ be the identity operator on $X$. It is known, at least in essence, that if ${(T_m)}_{m=0}^{\infty }$ is a sequence of bounded linear operators on $X$ such that ${\sum }_{m=0}^{\infty }\parallel I-T_m\parallel <\infty ,$ then ${(T_m(x_m))}_{m=0}^{\infty }$ is also a basis. The first part of this work acts as an expository note to formally record the aforementioned stability result. In the second part, we apply this stability result to construct a Schauder basis consisting of bivariate fractal functions for the space of continuous functions defined on a rectangle. To this end, fractal perturbations of the elements in the classical bivariate Faber–Schauder system are formulated using a sequence of bounded linear fractal operators close to the identity operator in accordance with the stability result mentioned above. This illustration, although emphasized only for the bivariate case, can easily be extended to higher dimensions. Further, the perturbation technique used here acts as a companion for a few researches on fractal bases in the univariate setting.