Fractals

In this paper, we studied the time fractional generalized $(2+1)$-dimensional Zakharov–Kuznetsov (ZK) equation with single-power law nonlinearity, which can be used to describe weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions. Through the fractional Lie transformation group idea, the symmetries, similarity reduction, one-parameter Lie transformation group and conservation laws of these considered equations were obtained. In addition, some new exact solutions were also constructed for them. These novel results provide a new perspective for studying the evolutionary mechanism of these considered models.

This paper proposed a method for the fractal characterization of the three-dimensional (3D) fracture tortuosity ($D_{T3})$ in coal based on CT scanning experiment. The methodology was deduced in detail, and the values of $D_{T3}$ of four coal samples were calculated by the rigorous derivation equation established by Feng and Yu. The values of $D_{T3}$ by the proposed method fit the relation of $D_{T3}$ versus the fractal dimension for 3D fracture number $D_{f3}$, and the relation of $D_{T3}$ versus the 3D fracture porosity, indicating the rationality and accuracy of the proposed method on estimation of the $D_{T3}$. The results show that the proposed $D_{T3}$ can comprehensively character the fractal characteristics of fractures tortuosity in 3D space. It is worth to further study for establishing an analytical fractal equation for fluid mass transfer in 3D fractures of porous media based on the $D_{T3}$.

In this paper, we introduce the fractal version of Zagreb eccentricity index which is a singular integral of the eccentricity squared in terms of self-similar measure. Based on the geometric characteristics of the farthest point, we calculate the fractal version of Zagreb eccentricity index on the (3,1) flower fractal to be $\frac{1829}{360}$ by using the technique of finite pattern.

In this paper, for the first time in pass records, we create the fractal fourth-order nonlinear Ablowitz–Kaup–Newell–Segur (FFONAKNS) shoal water wave mold under an unsmooth boundary or in microgravity of space. With the aid of fractal traveling wave variation (FTWV) and fractal semi-inverse technology (FSIT), the fractal variational principle (FVP) is achieved, and then, using He–Weierstrass function, the strong minimum necessary condition is proved. Afterwards, the solitary wave solution is attained by FVP and minimum stationary conditions. Finally, the effect of a non-smooth border on solitary wave is deliberated and demeanors of solutions are displayed via 3D isohypse. The fractal dimension can impact the waveform, but not its apex value. The solitary wave solution (SWS) is given, and the exhibition of the technology used is not only creditable but also significant.

Let ${\sum }_{i=1}^{\infty }{c}_i{R}_i(x)$ be the Rademacher series, where ${\{R_i(x)\}}_{i=1}^{\infty }$ is the classical Rademacher function system and ${\{c_i\}}_{i=1}^{\infty }$ is an arbitrary real number sequence. In this paper, we first show that the value range of the Rademacher series at any subinterval of $[0,1]$ is $ℝ\cup \{\pm \infty \}$ when ${\{c_i\}}_1^{\infty }\in {\ell }^2\setminus {\ell }^1$. This result provides us with the basic facts that when ${\{c_i\}}_1^{\infty }\in {\ell }^2\setminus {\ell }^1$, the Rademacher series cannot converge to an approximate continuous function, and there is no approximate limit at any point of $[0,1]$. Further, when ${\{c_i\}}_1^{\infty }\in {\ell }^2\setminus {\ell }^1$, we show various dimensions of the level set of Rademacher series on any subinterval of $[0,1]$. Finally, we give the relationship between the box dimension and the coefficient of Rademacher series when ${\{c_i\}}_1^{\infty }\in {\ell }^1$, and the exact values of box dimension, packing dimension and Hausdorff dimension are obtained in some special cases.

Applying the local fractional integrals, a generalized identity involving the local second-order differentiable mappings is first developed in this paper. A series of fractal integral inequalities pertaining to Simpson type, for the mappings whose local second-order derivatives are generalized $(s,P)$-convex in absolute value at some power, are then deduced by the discovered identity. Finally, from an application perspective, a range of fractal outcomes with regard to $\beta$-type special means, Simpson numerical integrations, midpoint numerical integrations and wave equations are presented, correspondingly.

This paper investigates the fractal dimension of the Weyl–Marchaud (W–M) fractional derivative of a type of self-affine curves. We first define the W–M fractional derivative of a general self-affine functions, then calculate the Box dimension of them, finally prove a linear relationship between the order of the W–M fractional derivative and the fractal dimension.

This paper provides a new idea for interpolating complex curves and surfaces by fractal interpolation. A type of bivariate fractal interpolation functions with function vertical scaling factors for the special interpolation data set is presented. The interpolation algorithm is proposed for complex curves by contour lines of two types of fractal interpolation functions which are proposed by this paper and Navascués et al. [Construction of fractal surfaces, Fractals 28(2) (2020) 2050033], respectively. Meanwhile, some properties of the interpolation algorithm are introduced. Then, the two classes of fractal interpolation functions are extended to the cases of trivariate functions. We also introduce the interpolation algorithm for complex surfaces based on iso-surfaces of fractal interpolation functions constructed. Finally, several examples are provided.

This paper investigates the Weyl–Marchaud fractional derivative of affine and non-affine fractal interpolation functions with function scaling factors. The dependence of fractal interpolation function on the scaling factor is mainly explored by choosing the scaling factor as a function instead of a constant. In addition, for some fixed order $v$, the Weyl–Marchaud fractional derivative of a linear fractal interpolation function is estimated by predefining the fractional derivative values at the end points. Similarly, the Weyl–Marchaud fractional derivative of a $\alpha$-fractal function is investigated for some fixed order $v$ with additional constraints on the derivative of prescribed continuous function and base function.

This paper has investigated the predictability of the top 10 cryptocurrencies’ price dynamics, ranked by their daily market capitalization and trade volume, via the information theory quantifiers. Our analysis considers the Complexity-entropy causality plane to study the temporal evolution of the price of these cryptocurrencies and their respective locations along this 2D map, bearing in mind after and during the Russia–Ukraine war. Moreover, we apply the permutation entropy and the Jensen–Shannon statistical complexity measure to rank these cryptocurrencies similarly to a complexity hierarchy. Our findings reflect that the Russian–Ukraine war affects the informational efficiency of cryptocurrency dynamics. Specifically, the cryptocurrencies notably showed a decrease in informational inefficiency (USD-coin, Binance-USD, BNB, Dogecoin, and XRP). At the same time, the cryptocurrencies with more expressiveness for the financial market, considering the volume traded and the capitalized market, were strongly impacted, presenting an increase in informational inefficiency (Tether, Cardano, Ethereum, and Bitcoin). It clarifies the potential of cryptocurrencies to mitigate exogenous shocks and their capability to use with portfolio selection, risk diversification and herding behavior.

The microstructures and transport properties of fibrous porous material are significant for chemical catalysts, textile engineering, electronic devices, etc. In this paper, a fractal Monte Carlo method (FMCM) is developed to reconstruct the random microstructure of fibrous porous material based on the fractal scaling laws of fiber columns. And, the two-point correlation function of reconstructed fibrous material is calculated, which shows the effectiveness of the FMCM reconstruction. Also, the single-phase fluid flow through the reconstructed random fibrous porous material is simulated by the finite element method. The predicted permeability indicates good agreement with available empirical formulas. It has been found that the effective permeability of fibrous porous material decreases with the increase of fractal dimensions for fiber column. However, the fractal dimension of fiber column width has a greater influence on the effective permeability of fibrous porous material compared with that of fiber column length. The proposed numerical method provides an effective tool to reconstruct the irregular microstructure and understand the complex transport mechanisms of fibrous porous material.

In this paper, we make research on the fractal structure of the space of continuous functions and explore how the fractal dimension of continuous functions under certain operations changes. We prove that any nonzero real power and the logarithm of a positive continuous function can keep the fractal dimensions of bi-Lipschitz invariance closed. For continuous functions having finite zero points, the relationship between its global behavior and the local behavior of its square on zero points has been given. Further, we discuss the fractal dimension of the product of continuous functions and provide the product decomposition of a continuous function in terms of the lower and upper box dimensions. Some special properties of the space of one-dimensional continuous functions have also been shown.

In this paper, we calculate electrical resistance scaling exponents for analogues of “carpet” and “sponge” fractals in higher-dimensional embedding spaces. The calculation idealizes the voids that define such fractals as spherical, and exploits the elementary theory of dielectrics. Possible applications include models of elementary particles with “extra” dimensions, and new large-dimension methods for the theory of fractals.

Heat transport in tree-like bifurcation networks has been widely studied in various fields. In this work, we investigate heat conduction in the dual porous media embedded with asymmetric tree-like bifurcation networks. In addition, considering the effects of nonuniform tube shape, we assume that the bifurcated tube shows sinusoidal fluctuations. Based on the fractal distribution of pore size and bifurcation structure, we established a dimensionless effective thermal conductivity (ETC) model of the dual porous media. The dimensionless ETC (${\lambda }^{+})$ obtained is related to the porosity ($\varphi )$, the fluid–solid thermal conductivity ratio (${\lambda }_f/{\lambda }_s)$, the pore area fractal dimension $D_f$ and the structural parameters of the bifurcation network (bifurcation level $i$, length ratio $\gamma$, radius ratio $\eta$, fluctuation amplitude factor $\xi$, bifurcation angle $𝜃)$. To verify the validity of this model, a comparison of the present dimensionless ETC model with available experimental data was carried out and the results were in good agreement. We have discussed the effects of each parameter on the dimensionless thermal conductivity in detail and constructed parametric planes to evaluate the structural parameters more directly. The model has positive implications for revealing the heat transport mechanism in asymmetric tree-like bifurcation dual porous media.

In this paper, we derive a new fractal modified Benjamin–Bona–Mahony equation (MBBME) that can model the long wave in the fractal dispersive media of the optical illusion field based on He’s fractal derivative. First, we apply the semi-inverse method (SIM) to develop its fractal generalized variational principle with the aid of the fractal two-scale transforms. The obtained fractal generalized variational principle reveals the conservation laws via the energy form in the fractal space. Second, Wang’s Bäcklund transformation-based method, which combines the Bäcklund transformation and the symbolic computation with the ansatz function schemes, is used to study the abundant exact solutions. Some new solutions in the form of the rational function-type, double-exp function-type, Sin-Cos function-type and the Sinh-Cosh function-type are successfully constructed. The impact of the fractal orders on the behaviors of the different solutions is elaborated in detail via the 3D plots, 2D contours and 2D curves, where we can find that: (1) When the fractal order $𝜀>\eta$, the direction of wave propagation tends to be more vertical to the $x$-axis, on the other hand, it tends to be more parallel to the $x$-axis when $𝜀<\eta$; (2) The fractal order cannot impact the peak amplitude of the waveform; (3) For the periodic waveform, the fractal orders can affect its period, that is, the period becomes smaller when the fractal order $𝜀,\eta <1$. The obtained results show that the proposed methods are effective and powerful, and can construct the abundant exact solutions, which are expected to give some new enlightenment to study the variational theory and traveling wave solutions of the fractal partial differential equations.

In this paper, we study the approximation orders of a real number $x\in (0,1)$ by the partial sums of its $\beta$-expansions as $\beta$ varies in the parameter space $\{\beta \in ℝ:\beta >1\}$. More precisely, letting $S_n(x,\beta )$ be the partial sum of the first $n$ items of the $\beta$-expansion of $x$, we prove that for any real number $x\in (0,1)$, the approximation order of $x$ by $S_n(x,\beta )$ is ${\beta }^{-n}$ for Lebesgue almost all $\beta >1$. Moreover, we obtain the size of the set of $\beta >1$ for which $x$ can be approximated with a more general order ${\beta }^{-\phi (n)}$, where $\phi :\mathbb{N}\to {ℝ}^{+}$ is a positive function. We also determine the Hausdorff dimension of the set

Much information on the structural properties and some relevant dynamical aspects of a graph can be provided by its normalized Laplacian spectrum, especially for those related to random walks. In this paper, we aim to present a study on the normalized Laplacian spectra and their applications of weighted level-$4$ Sierpiński graphs. By using the spectral decimation technique and a theoretical matrix analysis that is supported by symbolic and numerical computations, we obtain a relationship between the normalized Laplacian spectra for two successive generations. Applying the obtained recursive relation, we then derive closed-form expressions of Kemeny’s constant and the number of spanning trees for the weighted level-$4$ Sierpiński graph.

In this study, we proposed a novel approach for modeling the dynamics of a three-agent financial bubble using the fractal-fractional (FF) derivative of the Caputo sense. This new concept was developed to deal with the complex geometry of any dynamical system, and it utilizes both the fractional derivative for the order and the fractal term for the order of the independent variables. The model was investigated using the conformable order derivative of the Caputo operator, with a focus on the fractal dimension and fractional order. The existence and uniqueness of the solution were tested using a FF global derivative, and the approximate root of the system was calculated using the numerically iterative technique of the Newton polynomial. To verify the accuracy of the approximate root scheme, we applied the power singular law with two fractional parameters in the developed numerical technique. The curve representation of the system was also verified by applying the data for the fractals and different conformable orders. Additionally, we tested the sensitivities of the fractional parameters and dynamical system parameters by varying the parameter values. This allowed us to gain a better understanding of how changes in these parameters affect the system’s behavior and stability. As a result, this study presents an innovative and effective approach for modeling the dynamics of financial bubbles using the FF derivative of the Caputo sense. The results of this research contribute to the ongoing efforts to develop more accurate and comprehensive models of complex systems in economics and finance.

In this paper, we discuss the thickness and quasisymmetric equivalence of a class of fractal sets based on the binary expansion of numbers. We give the sufficient and necessary conditions for the thickness of this kind of sets to be positive, and the necessary and sufficient condition for their quasisymmetric equivalence with the standard Cantor ternary set. This paper also studies the relationship between the thickness and the uniform completeness about the product sets of two sets defined by digit restrictions.

In this work, we analyze the high-frequency time series of the deposits collected from the systemic biggest Greek banks during the recent economic crisis in Greece. Our focus has been to reveal hidden fractal and periodic patterns using a hybrid approach, which combines correlation and frequency analysis of the original and difference series in a synergistic manner. We find that during the first period of the recorded series featured by the dramatic decrease of deposits, the short time behavior exhibits Brownian motion characteristics with fractal dimension close to 1.5, while a cyclical pattern of monthly repetition (21 days) is detected in the difference series to bound fractal behavior. The Brownian property is also observed in the second uprising segment of deposit series, but it lasts more (50 days) and no well-defined cyclical pattern is detected. Our work reveals that the economic crisis in Greece has gradually eroded the cyclical behavior of daily differences in bank deposits, while maintaining a short-term Brownian motion pattern, albeit with an increased duration.

This study proposes a new method, called the Fractal Yang transform method (F$\mathcal{𝒴}$TM), for obtaining the fractal solution of the modified Camassa–Holm (mCH) and Degasperis–Procesi (mDP) models with fractal derivatives. The authors use the two-scale fractal approach to convert the fractal problem into its differential components and implement the Yang transform ($\mathcal{𝒴}$T) to achieve the recurrence iteration. We then apply the homotopy perturbation method (HPM) to overcome the difficulty of nonlinear elements in the recurrence iteration, which makes it simple to acquire further iterations. The most advantage of this fractal approach is that it has no restriction on variables and provides successive iterations. The fractal results are presented in the sense of a series that converges to the exact solution only after a few iteration. Graphical behavior demonstrates that this fractal approach is a very fast and remarkable solution, particularly with fractal derivatives.

In this study, the local fractional derivative is employed to build the fractional Bogoyavlensky–Konopelchenko model, which is then used to develop the interaction between long wave propagation and Riemann wave propagating under particular conditions. The major goal of this study is to obtain some new solitary wave solutions of the local fractional Bogoyavlensky–Konopelchenko model using two effective methods, the Yang–Machado–Baleanu–Cattain wave method (YMBCWM) and fractional sech function method (FSFM). These obtained solitary wave solutions are unique from those found in the literature. Several 3D simulation figures show the dynamic behavior of these new solitary wave solutions. The two novel approaches bring new perspectives for resolving the same class of fractional wave equations.

In this work, we derived the (2+1)-dimensional Schrödinger equation from the (2+1)-dimensional Klein–Gordon equation. We also obtained the fractional order form of this equation at the same time so as to discover the connection between them. For the (2+1)-dimensional Klein–Gordon equation, symmetries and conservation laws are pres ented. For different gauge constraint, from the perspective of conservation laws, the corresponding symmetries are obtained. After that, based on the fractional complex transform, soliton solutions of the time fractional (2+1)-dimensional Schrödinger equation are displayed. Some figures are showed behaviors of soliton solutions. It is important to discover the relationships between these equations and to obtain their explicit solutions. These solutions will perhaps provide a theoretical basis for the explanation of complex nonlinear phenomena. From the results of this paper, it is clear that the Lie symmetry method is a particularly important tool for dealing with differential equations.

In this paper, the integral of classical fractal interpolation function (FIF) and A-fractal function is explored for both the cases of constant and variable scaling factors. The definite integral for the classical FIF in the closed interval of $ℝ$ is estimated. The novel notion of affine-quadratic FIF is introduced and integrated for both constant and variable scaling factors. It is demonstrated that its integral is not an affine-quadratic FIF, however it is a FIF. Similarly, by choosing the vertical scaling factors as constants and variables, A-fractal function is integrated. Further, by assuming certain condition on the block matrix, it is shown that like the original A-fractal function its integral is also an attractor for the iterated function system.

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of $ℝ.$ We show that the Weyl–M derivative of order $\gamma$ of a continuous function satisfying Hölder condition of order $\mu$ also satisfies Hölder condition of order $\mu -\gamma$ and the upper box dimension of the Weyl–M derivative increases at most linearly with the order $\gamma$. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order $\gamma$. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one.

We analyze the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighborhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighborhoods that are not equivalent under similitudes. We show that at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure $\nu$ at $\nu$-a.e. point, where $\nu$ is any self-similar measure. We characterize the spectrum of asymptotic densities of metric measures in terms of the packing and centered Hausdorff measures. As an example, we compute the spectrum of asymptotic densities of the Sierpiński gasket.

The main goal of this paper is to study the new solitary wave behaviors of the fractional Jaulent–Miodek hierarchy model (FJMHE) with M-truncated fractional derivative. First, we use the fractional sech-function method (FSFM) to obtain some new solitary wave solutions of the fractional Jaulent–Miodek hierarchy equation. The new method is simple and effective, which provides a more powerful mathematical technique for exploring solitary wave solutions of the fractional evolution equations in mathematical physics. Finally, some 3D and 2D graphs are employed to illustrate the physical properties of the obtained new solitary wave solutions.

Stress field is a key external factor affecting the thermodynamic properties of rock in many scientific and engineering fields ranging from geoscience to practical problems in geothermal resources, oil and gas exploitation, mineral exploitation, etc. In this work, the stress-dependent models for the effective thermal conductivity of fractured rock are proposed by combining the fractal geometry theory and the two-part Hooke’s model. The proposed model for the effective thermal conductivity is a function of the effective stress and structure-parameters and physical parameters of fractured rock, such as porosity (${\varphi }_{\sigma })$, fracture orientations ($𝜃$ and $\phi )$ as well as $k_s$ and $k_f$. The results show that thermal conductivity increases nonlinearly with increasing stress at low stress range due to the closure of fractures, then, it exhibits an almost linear behavior of pressure dependence at high stress range for most fractures closing completely. The proposed thermal conductivity model coupled with stress is evaluated by comparison with the available experimental data. The model predictions show a good agreement with the available experimental data. A more detailed investigation of the essential influence of structures parameters of a set of fractures on the dimensionless stress-dependent thermal conductivity is performed. The proposed model may provide a good quantitative understanding of the stress effect on thermal conductivity of fractured rock.

In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, adds the angles and multiplies the radii of the spherical coordinates. We show that the algebraic structure of pure quaternions with the spherical product is a commutative unital magma. Then, we present several generalizations of the Mandelbrot set. Among them, we present a set that is visually identical to the so-called Mandelbulb. We show that this set is bounded and that it can be generated by an escape time algorithm. We also define another generalization, the bulbic Mandelbrot set. We show that one of its 2D cuts has the same dynamics as the Mandelbrot set and that we can generate this set only with a quaternionic product, without using the spherical product.

In this paper, we investigate the generalized Milne-type integral inequalities via the framework of generalized m-convex mappings on fractal sets. To accomplish this, we propose a new generalized integral identity that involves differentiable generalized $m$-convex mappings. Based on the latest identity we drive a number of the latest fractal Milne-type integral inequalities. Also, we provide fractal Milne-type inequalities for bounded mappings. Some illustrative examples and applications to additional inequalities for the generalized special means and various error estimates for the generalized Milne-type quadrature formula are obtained to further support our results. The findings presented in this research offer important generalizations and extensions of previous work in the field.

The hydro-mechanical coupling behavior of the fractured rock is constitutive in accurately modeling the seepage properties such as permeability and porosity. In this work, the stress-dependent models for permeability and porosity of the fractured rock are proposed by employing the fractal geometry theory and the two-part Hooke’s model (TPHM). The proposed models for the permeability and porosity are related to the effective stress and the microstructures of fractures, such as the fractal dimension ($D_{0,f}$ and $D_{\sigma ,f})$, the maximum fracture length $l_{\sigma ,max}$, and fracture orientations ($𝜃$ and $\phi )$ as well as porosities (${\varphi }_0$ and ${\varphi }_{\sigma })$. The validity of the proposed permeability model coupled with stress is validated by comparison with the available experimental data. The model predictions agree well with the experimental data. The effects of microstructures of fracture networks in fractured rock on the dimensionless stress-dependent permeability and porosity as well as the fractal dimension for fracture areas are also discussed in detail. The proposed models may shed much light on the fundamental behavior of coupled hydro-mechanical properties of fractured rock.