Fractals

The concept of superhydrophobicity has been widely used after years of theoretical and experimental exploration. Researchers have obtained materials with excellent surface superhydrophobicity for numerous areas (e.g. textiles, paints, and coatings industries), through different design and synthesis methods using low surface energy components (LSECs) and micro/nanohierarchy composite structures. However, the durability of superhydrophobic material surfaces has gradually become a prominent obstacle to restricting applications. In this paper, advances in improving the durability of superhydrophobic surfaces in the past decade are reviewed, covering different schemes for recovering fractal surfaces based on the LSECs and micro/nanocomposite structures. It also presents a balanced review on wet chemical methods, chemical vapor deposition (CVD), layer-by-layer (LbL), microarc oxidation (MAO), and self-healing of the fractal surfaces, with the aim at developing sustainable superhydrophobicity. In addition, it innovatively summarized the research on the synthesis of self-healing superhydrophobic materials by machine learning methods. By the end it discusses perspectives on future development in this emerging research area.

The smart wearable devices that can track the fitness activities are getting famous these days due to their easy-to-use features. The fitness trackers can work for an individual in a promising manner, provided that the user is well familiar with the device and is committed with the timelines. Several reports have provided evidence that these smart wearable devices have not showed promising results and in most of the cases, people have stopped using them, few weeks after the purchase. There are several reasons linked with this response. During this research, we have worked on the correlations of weight loss via smart device with the age, gender, body mass index (BMI) and ideal body weight (IBW), with the aid of gradient boosted decision trees (XGBoost) and support vector machine (SVM) learning tools. XGBoost and SVM are capable of dealing with complex datasets, with higher frequencies, and for data emerging from multiple sources. These machine learning tools use kernel functions for the clustering and other classification measures, and are thus better as compared to the logistic methods. Next, the time series forecasting tools are discussed with the Bayesian hyperparametric optimization. The time series of the weight loss monitoring of each individual, depicted in this manner, provided complex fractal patterns, with reduction in amplitude, with the passage of time.

The Kozeny–Carman (KC) equation is a well-known semi-empirical formula, which is used to calculate the permeability of porous media in the seepage field. The KC constant is an empirical constant in the KC equation. In this paper, based on the fractal theory, the Fractal-Monte Carlo technique is used to simulate the KC constant of the roughened fibrous porous media (RFPM) with micro-scale effects. There is no empirical constants in the proposed model, and each parameter has its physical meaning. The KC constant model of RFPM can be expressed as a function of structural parameters, including relative roughness ($𝜀$), porosity ($\varphi$), pore area fractal dimension ($D_f$), tortuosity fractal dimension ($D_T$), capillary diameter ($\lambda$) and Knudsen number ($\mathrm{\text{Kn}}$). The result shows that the KC constant increases with increases in $\varphi$, $𝜀$, $D_f$ and $D_T$. On the contrary, with increases in $\lambda$ and $\mathrm{\text{Kn}}$, the KC constant decreases. In addition, the KC constant model constructed in the paper agrees well with the existing experimental data and the model. According to the proposed Fractal-Monte Carlo technique, it is possible to better clarify the transmission physical mechanism in RFPM with micro-scale effects.

In short-fiber composites (SFCs), fiber length distribution (FLD) is complicated and has a considerable impact on the mechanical properties of SFCs. This work proposes a fractal FLD in SFCs on the basis of the fractal theory, and develops a multi-step mean-field homogenization (MSMFH) method to accurately and efficiently predict the mechanical properties of SFCs with fractal FLD. In the developed MSMFH method, SFCs are first decomposed into virtual pseudo-grains (PGs) according to fiber orientation distribution (FOD), followed the further division of the PGs into virtual sub-pseudo-grains (SPGs) according to FLD. The Mori–Tanaka or Double-Inclusion model is adopted to homogenize the mechanical properties of each SPG in the first step, and the Voigt model is implemented to homogenize the mechanical properties of all the SPGs and the PGs, respectively, in the sequential steps. Fiber length and orientation averaging algorithms for the developed MSMFH method are detailed. The developed MSMFH method and the proposed fractal FLD are validated to accurately predict the mechanical properties of SFCs by the means of the comparison with the FE method and the available experimental tests.

The coefficient of vertical permeability is a common filter design criterion for woven slit-film geotextiles. According to the pipe flow theory, a permeability coefficient model has been proposed for predicting the permeability coefficient of woven geotextiles, in which the pore characteristics were described by the Sierpiński carpet fractal theory. The verification of models is performed in 12 woven geotextiles of two types using a digital image method and a vertical water permeability test of the geotextile. The influence of the pore characteristics on the permeability coefficient is further explored using univariate and multivariate analyses. The results show that the porosity model can accurately predict the total percent pore area (POA) and the POA distribution of the woven geotextiles, and the permeability model can accurately predict the permeability coefficient of the geotextiles. The permeability coefficient of geotextiles decreases with the pore area fractal dimension and the minimum pore size under the action of a single factor. The permeability coefficient increases as the maximum pore size increases. POA and maximum pore size are the most significant independent variables affecting the permeability coefficients of geotextile under the action of multifactor.

The effective thermal conductivity of unsaturated fibrous media is simulated by Fractal-Monte Carlo simulations in this work. Based on the probability model and thermal-electrical analogy method, the proposed model of the effective thermal conductivity can be expressed as an explicit function of fractal dimensions, $D_t$, $D_f$, intrinsic thermal conductivities, $k_s$, $k_w$, $k_g$, the straight capillary length of a unit cell $L_0$, liquid saturation $S$, pore size ${\lambda }_i$ as well as porosity $\varphi$. Each parameter of the proposed model has physical meaning and the function has no empirical constant. The results calculated by Fractal-Monte Carlo simulations have been compared with the experimental data available, and an excellent agreement has been shown for an appropriate range of porosity except for the condition at a lower liquid saturation. It is found that, when $k_s/k_g>1$ and $k_w/k_g>1$, the dimensionless effective thermal conductivity of unsaturated fibrous media decreases with increment in fractal tortuosity dimension and porosity and decrement in liquid saturation. On the other hand, when $k_s/k_g<1$ and $k_w/k_g<1$, the correlations are opposite. The present Fractal-Monte Carlo technique may have its own advantages in predicting other transports in unsaturated fibrous media.

The pore geometry of porous rocks is fundamental for accurate description of stress dependence of effective permeability, which is an important parameter of mass transfer in porous rocks. An important physical assumption that porous rocks contain numerous elliptical or spherical pores has been shown to be successfully applied to many aspects of hydromechanical coupling properties of porous rocks. To investigate the detailed description of pore structures on the degree of effect on the coupled hydromechanical process, in this work, a generalized stress-dependent model for permeability of porous rocks has been proposed based on fractal geometry theory and mechanics of porous rock. The proposed model is expressed as a nonlinear function of pore structure parameters, such as aspect ratio (${\gamma }_{\sigma }$), the fractal dimensions ($D_{T,\sigma }$ and $D_{T,av}$) for tortuosity, the initial fractal dimension ($D_{f,0}$), and initial porosity (${\varphi }_0$) as well as matrix elastic constants ($E$ and $\nu$) of porous rocks without any empirical parameter. The validity of the proposed models is verified by the good agreements between available experimental data and theoretical predictions of stress-dependent permeabilities of porous rocks. Detailed discussions of the essential effects of pore structures parameters and material elastic constants of porous rocks on the dimensionless stress-dependent permeabilities are performed. It is found that the stress parameters ($E$ and ${\gamma }_0$) have remarkable effects on the dimensionless stress-dependent permeabilities compared with other parameters (${\varphi }_0$ and $\nu$). The proposed model may contribute to a better quantitative understanding of the coupled hydromechanical properties of porous rocks.

The price and return time series are two distinct features of any financial asset. Hence, examining the evolution of multiscale characteristics of price and returns sequential data in time domain would be helpful in gaining a better understanding of the dynamical evolution mechanism of the financial asset as a complex system. In fact, this is important to understand their respective dynamics and to design their appropriate predictive models. The main purpose of the current work is to investigate the multiscale fractals of price and return high frequency data in Turkish stock market. In this regard, the wavelet leaders computational method is applied to each high frequency data to reveal its multi-fractal behavior. In particular, the method is applied to a large set of Turkish stocks and statistical results are performed to check for (i) presence of multi-fractals in price and return series and (ii) differences between prices and returns in terms of multi-fractals. Our statistical results show strong evidence that high frequency price and return data exhibit multi-fractal dynamics. In addition, they show evidence of distinct fractal characteristics on different scales between price and return series. Furthermore, our statistical results show evidence of differences in local fluctuation characteristics of price and return time series. Therefore, differences in local characteristics are useful to build specific predictive models for each type of data for better modeling and prediction to generate profits. Besides, we found evidence that both long-range correlations and fat-tail distributions contribute to the multifractality in Turkish stocks. This finding can be attributed to the major role played by international investors in increasing the volatility of Turkish stocks.

This paper investigates the variation of fractal dimensions of continuous functions under operations. It is found that the Box dimension of the logarithm of positive continuous functions remains constant, and any nonzero real power of a positive continuous function can keep fractal dimensions variation closed. The study also discusses the fractal dimension of logarithmic function with bases of fractal functions, and the corresponding fractal dimension is determined by the function with the larger fractal dimension.

Data-driven learning of the fractional discrete-time unified system is studied in this paper. A neural network method is suggested in the parameter estimation of fractional discrete-time chaotic systems. An optimization problem is obtained and the famous Adam algorithm is employed to train the neural network’s weights and parameters. The parameter estimation result is compared with that of the stepwise response sensitivity approach (SRSA). This paper provides a high accuracy method for parameter inverse problems. The method also can be applied to data-driven learning of other fractional chaotic systems.

Many scholars have studied the interporosity flow function (IFF) and shape factor. However, the existing research does not consider the complex microstructure of dual media and the influence of fluid types on interporosity flow. In this paper, the anisotropic IFF and shape factor model of power-law fluid in smooth and rough fracture dual media are established by using the rough capillary bundle model and considering the influence of the microstructure of dual media and the type of fluid. The relationship between IFF, shape factor, and relative roughness in dual media is studied. It is found that the relative roughness is inversely proportional to the IFF and inversely proportional to the shape factor. In addition, the model is compared with the existing research and experimental data to verify the rationality of the model.

In this paper, we investigate the equivalence of connectedness for the Sierpinski-like sponge and skeleton networks, and find out the relation between the geodesic distance on the sponge and renormalized shortest path distance on the skeleton networks. Furthermore, under some assumption on the IFS, we obtain the comparability of the Manhattan distance and the geodesic distance on the sponge.

Based on the fractal geometry science and the model of flow friction resistance when fluid flows through rough microcapillaries, the formula of diffusion layer thickness and the spatial charge density distribution of the pipe were derived. The relationship between the amount of space charge and the structural parameters of rough elements, relative roughness, liquid conductivity, pipeline diameter, and other physical quantities was discussed. It was found that the height of rough elements is directly proportional to the thickness of the diffusion layer, and also positively proportional to the amount of current. The predicted results of the model are in good agreement with existing models and various published experimental data, verifying the accuracy and reliability of the model.

Scientists worldwide have always been interested in the study of networks that resemble trees and porous materials. Therefore, based on fractal theory, this paper systematically studies the heat transfer problem of the damaged tree-like networks under different saturations in multi-medium composite materials and derives their dimensionless thermal conductivity (DTC). The research has shown that the dimensionless thermal conductivity (DTC) decreases with an increase in damaged channels. The saturation significantly impacts the heat transfer of the damaged tree networks, and it exhibits linear changes with the ratio of the gas phase, liquid phase, and solid phase. Additionally, it can be observed that the fractal dimension of length distribution and diameter distribution, bifurcation numbers, bifurcation series, porosity, and other parameters in the networks are related to thermal conductivity. The comparison between the data derived from this model and experimental data shows that the proposed model can effectively deepen our understanding of the heat transfer mechanism of the damaged networks in composite materials under different saturation levels. Additionally, the model in this paper does not have an empirical constant, which avoids the influence of potential factors.

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.

Student confusion is a problem that is experienced daily in school classrooms. Teachers transfer knowledge to students without knowing if they are receiving it correctly. Being aware of whether the student correctly grasps the knowledge would be of great help since different actions could be carried out to correct this problem. The proposal of this work focuses on carrying out the diagnosis of student confusion through Machine Learning algorithms. In particular, the following algorithms were applied: Logistic Regression (LR), $K$-Nearest Neighbor (K-NN), Random Forest (RF) and Multi-Layer Perceptron (MLP). The metric was accuracy. The results obtained from the accuracy of each algorithm with the 5-Fold-Cross Validation validation method are 55.03% (LR), 52.98% (7-NN), 58.86% (RF) and 75.40% (MLP). An improvement in accuracy was achieved with respect to already published papers.

The main goal of this work is to focus on using He’s two-scale fractal dimension transform, the Caputo–Fabrizio fractional-order derivative, and the harmonic balance and the homotopy methods are applied for deriving the approximate solution of the fractal, damped cubic–quintic Duffing’s equation when the fractional derivative order of the inertia term is not twice of that of the damping term. Numerical results obtained from the derived expressions and the numerical integration solution show good agreement, especially at small values of the nonlinear parameters. Furthermore, when the fractal order of the damping term decreases, the damping oscillation frequency values increase with a decrease in the system wavelength values, which indicates a slower decay in the system oscillation amplitudes. Our solution procedures elucidate the applicability of He’s two-scale fractal dimension transform for solving nonlinear dynamic systems with inertia and damping fractal terms.

Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.

In this paper, we construct a new kind of weighted recursive iteration function system (IFS) and prove the existence of the unique attractor for the kind of IFS based on the Matkowski fixed point theorem. We confirm that the attractor is a bivariate fractal interpolation surface (FIS), which interpolates a given set of data. In addition, we also provide an upper error estimate of such FISs caused by changes of weights. Finally, we give their box dimension estimates for a specific type of the FISs.

Studying the activity of organs during aging is a very important research area. On the other hand, simultaneous analysis of the activities of various organs is important to understand how their activities are correlated. For the first time, this research analyzes the brain-heart correlation in younger and older subjects. We analyzed the sample entropy (SampEn) and approximate entropy (ApEn) of EEG and R-R signals (as heart rate variability (HRV)) of younger and older participants while they sat comfortably in an armchair with their eyes open. The results indicated that older subjects’ EEG and R-R signals have greater values of sample and approximate entropies than younger subjects. Therefore, as subjects age, their EEG and R-R signals become more complex. This analysis can be extended to investigate the correlation between other physiological signals among different age groups.

This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect’s anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions.

The laws governing life should be as simple as possible; however, theoretical investigations into allometric laws have become increasingly complex, with the long-standing debate over the scaling exponent in allometric laws persisting. This paper re-examines the same biological phenomenon using two different scales. On a macroscopic scale, a cell surface appears smooth, but on a smaller scale, it exhibits a fractal-like porous structure. To elaborate, a few examples are given. Employing the two-scale fractal theory, we theoretically predict and experimentally verify the scaling exponent values for basal, active, and maximal metabolic rates. This paper concludes that Rubner’s 2/3 law and Kleiber’s 3/4 law are two facets of the same truth, manifested across different scale approximations.

Recurrence lacunarity has been recently proposed to detect dynamical state transitions over various temporal scales. In this paper, we combine suggested distribution moments and introduce multifractal recurrence lacunarity to unearth rich information of trajectories in phase space. By considering generalized moments, it provides an enhanced measurement to account for differences of black pixels in the recurrence plot at various scales. Numerical simulations have proved that the proposed method is able to differentiate varying types of time series and provide further insights of inherent features including stochastic series, chaotic maps and series contaminated interference components. In real-world applications, it performs well on quantifying the subtle structural changes of financial time series. In addition, it is intriguing to confirm that corrugation signals possess much more vivid information of heterogeneity in terms of recurrence plots than normal ones.

Air transportation constitutes a significant advancement in enhancing transportation efficiency. Nonetheless, when this modality is employed for the transit of large-scale armaments and equipment, the vibrational properties of these items within the aircraft’s cabin, coupled with potential deviations from their designated installation positions, emerge as critical factors that could compromise the safety of such transportation endeavors. To accommodate the unique environmental conditions of low-temperature and low-pressure prevalent in high-altitude air transportation, this model employs a fractal frequency formula for an expedited and accurate characterization of vibrational properties, while also providing a detailed analysis of the errors attributable to positional deviations in these vibration assessments. The findings of this research demonstrate that the computational accuracy achieved herein surpasses that of the variational iteration method (VIM) and the homotopy perturbation method (HPM). Moreover, the investigation into the damping effects of inertial forces within fractal dimensions unveils innovative prospects for optimizing nonlinear vibration systems under the challenging conditions of low-temperature and low-pressure environments.

This paper sheds light on the changes suffered in cryptocurrencies due to the COVID-19 shock through a nonlinear cross-correlations and similarity perspective. We have collected daily price and volume data for the seven largest cryptocurrencies considering trade volume and market capitalization. For both attributes (price and volume), we calculate their volatility and compute the Multifractal Detrended Cross-Correlations (MF-DCCA) to estimate the complexity parameters that describe the degree of multifractality of the underlying process. We detect (before and during COVID-19) a standard multifractal behavior for these volatility time series pairs and an overall persistent long-term correlation. However, multifractality for price volatility time series pairs displays more persistent behavior than the volume volatility time series pairs. From a financial perspective, it reveals that the volatility time series pairs for the price are marked by an increase in the nonlinear cross-correlations excluding the pair Bitcoin versus Dogecoin $({\alpha }_{xy}(0)=-1.14\%)$. At the same time, all volatility time series pairs considering the volume attribute are marked by a decrease in the nonlinear cross-correlations. The K-means technique indicates that these volatility time series for the price attribute were resilient to the shock of COVID-19. While for these volatility time series for the volume attribute, we find that the COVID-19 shock drove changes in cryptocurrency groups.

Fractured porous media is of great significance to the exploration and development of unconventional reservoirs. In this paper, a fractal model for permeability through micro-fractured porous media with consideration of the electric double layer (EDL) effect is proposed based on the fractal theory. The present model indicates that the permeability is a function of the electrokinetic parameters and micro-structural parameters of fractured porous media, and each parameter has a clear physical meaning, the results from the proposed model are found to be in good agreement with experimental data. Moreover, factors influencing the permeability are also analyzed in detail. The results indicate that the more obvious EDL effects will lead to permeability becoming lower. The present fractal model for the permeability with the EDL effect can provide guidance for tight reservoir development or other micro-porous media transportation.

The hyper-Wiener index plays an important role in chemical graph theory. In this paper, using the technique named finite pattern, we discuss the hyper-Wiener index on level-3 Sierpinski gasket which is a self-similar fractal.

Fractal integro-differential equations (IDEs) can describe the effect of local microstructure on a complex physical problem, however, the traditional numerical methods are not suitable for solving the new-born models with the fractal integral and fractal derivative. Here we show that deep learning can be used to solve the bottleneck. By the two-scale transformation, the fractal IDE is first approximately converted to its traditional integro-differential partner, which is further converted to a differential equation system by introducing an auxiliary variable to remove the integral operation. Moreover, a flexible adaptive technology is adopted to deal with the loss weights of a deep learning neural network. A fractal Volterra IDE is used to show the effectiveness and simplicity of this new physics-informed deep AI simulation model. All results indicate the AI simulation model has good robustness and convergence, and the fractal Volterra IDE might explore the different properties of viscoelasticity for a porous medium.

In ocean engineering, the long- and short-wave interaction system represents a crucial nonlinear evolution equation that elucidates the resonant interaction phenomenon between ocean waves. In this study, we describe the fractional long and short-wave interaction (FLSWI) system employing the M-truncated derivative. Subsequently, we employ the extended fractional ${tanh}_{\chi }-{coth}_{\chi }$ and the fractional $csc{h}_{\chi }$ methods to address the FLSWI system. These two approaches yields novel and intriguing soliton solutions. To further elucidate the derived soliton solutions, three-dimensional visualizations are constructed and analyzed.

The aim of this study is to explore some semi-domain soliton solutions for the fractal (3+1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation (GKPBe) within He’s fractal derivative. First, the fractal soliton molecules are plumbed by combining the Hirota equation and fractal two-scale transform. Second, the Bernoulli sub-equation function approach together with the fractal two-scale transform is employed to investigate the other soliton solutions, which include the kink soliton and the rough wave soliton solutions. The impact of the different fractal orders on the physical behaviors of the semi-domain soliton solutions is also discussed graphically. The methods mentioned in this research are expected to provide some new viewpoints on the behaviors of the fractal PDEs.

Local fractional calculus theory and parameterized method have greatly assisted in the advancement of the field of inequalities. To continue its enrichment, this study investigates the multi-parameter fractal–fractional integral inequalities containing the fractal $(P,m)$-convex functions. Initially, we formulate the new conception of the fractal $(P,m)$-convex functions and work on a variety of properties. Through the assistance of the fractal–fractional integrals, the $2\ell$-fractal identity with multiple parameters is established, and from that, integral inequalities are inferred regarding twice fractal differentiable functions which are fractal $(P,m)$-convex. Furthermore, a few typical and novel outcomes are discussed and visualized for specific parameter values, separately. It concludes with some applications in respect of the special means, the quadrature formulas and random variable moments, respectively.

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the ${\mathrm{\text{LT}}}_{\delta }({\mathrm{\Xi }}^{\delta })$ and ${\mathrm{\text{LC}}}_{\delta }({\mathrm{\Xi }}^{\delta })$ functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang’s non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for $\delta =ln2/ln3$ are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for $\delta =1$ are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs.

This paper focuses on the numerical investigation of the fractal modification of the (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation. A variational approach based on the two-scale fractal complex transformation and the variational principle is presented for solving this fractal equation. The fractal potential-YTSF equation can be transformed as the original potential-YTSF equation by means of the fractal complex transformation. Some fractal soliton-type solutions and fractal periodic wave solutions are provided by using the variational principle proposed by He, which are not touched in the existing literature. Some remarks about the variational formulation and the wave solutions for the original potential-YTSF equation by Manafian et al. [East Asian J. Appl. Math. 10(3) (2020) 549–565] are also given. Numerical results of the fractal wave solutions with different fractal dimensions and amplitudes are presented to show the propagation behavior.

The edge-Wiener index, an invariant index representing the summation of the distances between every pair of edges in the graph, has monumental influence on the study of chemistry and materials science. In this paper, drawing inspiration from Gromov’s idea, we use the finite pattern method proposed by Wang et al. [Average geodesic distance of Sierpinski gasket and Sierpinski networks, Fractals 25(5) (2017) 1750044] to figure out the exact formula of edge-Wiener index of the Sierpinski fractal networks.

Consider orbits ${𝒪}(z,\kappa )$ of the fractal iterator $f_{\kappa }(z):=z^2+\kappa$, $\kappa \in ℂ$, that start at initial points $z\in {\widehat{K}}_{\kappa }^{(m)}\subset \widehat{ℂ}$, where $\widehat{ℂ}$ is the set of all rational complex numbers (their real and imaginary parts are rational) and ${\widehat{K}}_{\kappa }^{(m)}$ consists of all such $z$ whose complexity does not exceed some complexity parameter value $m$ (the complexity of $z$ is defined as the number of bits that suffice to describe the real and imaginary parts of $z$ in lowest form). The set ${\widehat{K}}_{\kappa }^{(m)}$ is a bounded-complexity approximation of the filled Julia set $K_{\kappa }$. We present a new perspective on fractals based on an analogy with Chaitin’s algorithmic information theory, where a rational complex number $z$ is the analog of a program $p$, an iterator $f_{\kappa }$ is analogous to a universal Turing machine $U$ which executes program $p$, and an unbounded orbit ${𝒪}(z,\kappa )$ is analogous to an execution of a program $p$ on $U$ that halts. We define a real number ${\mathrm{{\rm Y}}}_{\kappa }$ which resembles Chaitin’s $\mathrm{\Omega }$ number, where, instead of being based on all programs $p$ whose execution on $U$ halts, it is based on all rational complex numbers $z$ whose orbits under $f_{\kappa }$ are unbounded. Hence, similar to Chaitin’s $\mathrm{\Omega }$ number, ${\mathrm{{\rm Y}}}_{\kappa }$ acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set $K_{\kappa }$. We present a procedure that, when given $m$ and $\kappa$, it uses ${\mathrm{{\rm Y}}}_{\kappa }$ to generate ${\widehat{K}}_{\kappa }^{(m)}$. Several numerical examples of sets that estimate ${\widehat{K}}_{\kappa }^{(m)}$ are presented.

In this paper, we have done some research studies on the fractal dimension of the sum of two continuous functions with different $K$-dimensions and approximation of $s$-dimensional fractal functions. We first investigate the $K$-dimension of the linear combination of fractal function whose $K$-dimension is $s$ and the function satisfying Lipschitz condition is still $s$-dimensional. Then, based on the research of fractal term and the Weierstrass approximation theorem, an approximation of the $s$-dimensional continuous function is given, which is composed of finite triangular series and partial Weierstrass function. In addition, some preliminary results on the approximation of one-dimensional and two-dimensional fractal continuous functions have been given.

This paper presents an advanced control strategy based on Fractional-Order Sliding Mode Control (FO-SMC), which introduces a robust solution to significantly improve the reliability of robotic manipulator systems and increase its control performance. The proposed FO-SMC strategy includes a two-key term-based Fractional Sliding Function (FSF) that presents the main contribution of this work. Additionally, a fractional-order-based Lyapunov stability analysis is developed for a class of nonlinear systems to guarantee the asymptotic stability of the closed loop system. Four FSF-based versions of the designed FO-SMC are studied and discussed. Various scenarios of the proposed control strategy are tested on a 3-degree-of-freedom SCARA robotic arm and compared to recent FO-SMC works, demonstrating the effectiveness of the new proposed control strategy to (i) ensure the asymptotic stability, (ii) achieve a smooth start-up, (iii) cancel the static error, giving a good tracking trajectory, and (iv) reduce the control torques, yielding a consumed energy minimization.