Fractals

The variational theory has triggered skyrocketing interest in the solitary theory, and the semi-inverse method has laid the foundation for the search for a variational formulation for a nonlinear system. This paper gives a brief review of the last development of the fractal soliton theory and discusses the variational principle for fractal Boussinesq-like $B(m,n)$ equation in the literature. The paper establishes a variational formulation for $B(m,1)$ equation to show the effectiveness of the semi-inverse method, and a general trial-Lagrange function with two free parameters is established for $B(m,n)$ equation, the identification of the unknown parameters and the unknown function involved in the trial-Lagrange function is shown step by step. This paper opens a new path for the fractal variational theory.

Zeng and Xi introduced the Hausdorff dimension of a family of networks and investigated the dimensions of touching networks. In this paper, using the self-similarity and induction we obtain the Hausdorff dimension of flower networks and Hanoi graphs, which are not touching networks.

The KdV–Zakharov–Kuznetsov equation is an important and interesting mathematical model in plasma physics, which is used to describe the effect of magnetic field on weak nonlinear ion-acoustic waves. A fractional KdV–Zakharov–Kuznetsov equation in the $M$-truncated derivative sense is investigated. By taking into account the fractional ${tanh}_{\delta }$ method and fractional $sin{e}_{\delta }$–$\mathrm{\text{cosin}}{e}_{\delta }$ method, larger numbers of a new type of solitary wave solutions are obtained. The dynamic characteristics of these new solitary wave solutions are elaborated by sketching some three-dimensional (3D) and two-dimensional (2D) figures. The study reveals that the proposed two methods are very powerful to solve fractional evolution equations.

The hyper-Wiener index on a graph is an important topological invariant that is defined as one half of the sum of the distances and square distances between all pairs of vertices of a graph. In this paper, we develop the discrete version of finite pattern to compute the accurate formulas of the hyper-Wiener indices of the Sierpiński skeleton networks.

A fast higher-order scheme is established for solving inhomogeneous time-fractional generalized Burgers’ equation. The time-fractional operator is taken as the modified operator with the Mittag-Leffler kernel. Through stability analysis, it has been demonstrated that the proposed numerical approach is unconditionally stable. The convergence of the numerical method is analyzed theoretically using von Neumann’s method. It has been proved that the proposed numerical method is fourth-order convergent in space and second-order convergent in time in the $L_2$-norm. The scheme’s proficiency and effectiveness are examined through two numerical experiments to validate the theoretical estimates. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.

This study presents the modified form of the homotopy perturbation method (HPM), and the Yang transform is adopted to simplify the solving process for the Kuramoto–Sivashinsky (KS) problem with fractal derivatives. This scheme is established by combining the two-scale fractal scheme and Yang transform, which is very helpful to evaluate the approximate solution of the fractal KS problem. Initially, we transfer the fractal problem into its partners using the two-scale fractal approach, and then we use the Yang transform (${𝒴}$T) to obtain the recurrent relation. Second, the HPM is then introduced to deal with the nonlinear elements of the fractal model. The numerical example demonstrates how the suggested technique is incredibly straightforward and precise for nonlinear fractal models. In addition, the graphical error of the proposed fractal model is compared with the calculated results of our suggested approach and the exact results. This graphical error displays the strength and authenticity of our proposed scheme.

A fractal integral identity with the parameter $\tau$ related to twice-differentiable mappings is first proposed in this paper. Based on the identity, the parameterized inequalities over the fractal domains are then derived for the mappings whose second-order derivatives in absolute value at certain powers are generalized $n$-polynomial convex, which is the main purpose of this investigation. Moreover, a series of fractal findings of some applications, involving the special mean values, the midpoint formulas, the moments of random variable and the wave equations on Cantor sets, are acquired correspondingly.

A vibration system with discontinuities has triggered off rocketing interest in various fields including mechanical engineering, physics, and mathematics because it has many striking and amazing properties which cannot be unexplained by traditional vibration theory. This paper studies the problem using the energy conservation frame in a fractal time. A variational formulation is developed, and its periodic property is elucidated by the frequency–amplitude formulation. For discontinuity, this paper aims to propose a new frequency formulation by means of the golden average. Comparison between the approximate and exact frequency values is presented to validate our method.

This paper proposes an innovative method to take advantage of Blockchain Convolutional Neural Networks (BCNNs) in Emotion Recognition (ER). Based on Artificial Intelligence, this proposal uses audio-visual emotion patterns to determine psychiatric profiles to attend to the most urgent as a priority. BCNN architectures were used to identify emergency patterns. The results indicate that the proposed method is adequate for classifying and identifying audio-visual patterns using Deep Learning (DL) with Boltzmann’s restricted machines. It is concluded that it is sufficient to consider the audio-visible critical features from the patient’s face and voice for the proposed model to recognize a psychiatric services emergency for immediate action: the emergency with no control and the Emergency under control. User personal dynamic profiles are stored in the blockchain ecosystem since they are deemed sensitive data. System security is provided by blockchain and authentication uses non-fungible tokens (NFT) technology.

In this work, we have given an analogical method for estimating the fractal dimension for three-dimensional fracture tortuosity (3D-FT). The comparison and error analysis of analogical and rigorous methods on fractal dimension for 3D-FT were carried out in this work. The fractal dimension $D_{T-R}$ for 3D-FT from the proposed analogical method is the function of 3D fracture average tortuosity (${\tau }_{\mathrm{\text{av}}})$ and average fracture length ($L_{\mathrm{\text{av}}})$. The analogical method for estimating fractal dimension ($D_{T-A})$ with high accuracy indicates good consistency with the rigorous method ($D_{T-R})$. The fractal dimension ($D_{T-R})$ from the rigorous method is the embodiment of the physical meaning of $D_{T-R}$. The fractal dimension ($D_{T-A})$ from the analogical method is relatively convenient for calculating the premise of ensuring accuracy.

Given $0<t<\infty$, we study the Assouad dimension and weak tangents of closed $t$-popcorn graphs. For all $0<t<2$, we prove that ${[0,1]}^2$ is a weak tangent of the closed $t$-popcorn graphs by using some arguments on prime numbers. For all $2\le t<\infty$, we first show that the Assouad dimension of the closed $t$-popcorn graphs is 1, and then prove that $[0,1]\times \{0\}$ is a weak tangent of them. We also discuss some specific weak tangents of closed $t$-popcorn graphs when $t=2$ and $2<t<\infty$.

We introduce a class of sets defined by digit restrictions in ${ℝ}^2$ and study its fractal dimensions. Let $E_{S,D}$ be a set defined by digit restrictions in ${ℝ}^2$. We obtain the Hausdorff and lower box dimensions of $E_{S,D}$. Under some condition, we gain the packing and upper box dimensions of $E_{S,D}$. We get the Assouad dimension of $E_{S,D}$ and show that it is 2 if and only if $E_{S,D}$ contains arbitrarily large arithmetic patches. Under some conditions, we study the upper spectrum, quasi-Assouad dimension and Assouad spectrum of $E_{S,D}$. Finally, we give an intermediate value property of fractal dimensions of the class of sets.

Electrical conductivity is an important physical property of porous media, and has great significance to rock physics and reservoir engineering. In this work, a conductivity model including pore water conductivity and surface conductivity is derived for water-saturated tree-like branching network. In addition, combined with Archie’s law, a general analytical formula for the formation factor is presented. Through the numerical simulation of the analytical formula above, we discuss the impact of some structural parameters ($\alpha$, $\beta$, $N$, $m$, $d_0$, $l_0$, $D_d$, $D_l)$ in tree-like branching network on the resistance, conductivity and formation factor. The results show that the total resistance $R$ is proportional to $l_0$, $\alpha$, and inversely proportional to $d_0$, $\beta$. The relation between conductivity and porosity in this model is contrasted with previous models and experimental data, and the results show considerable consistency at lower porosity. It is worth noting that when $\alpha =0$, the conductivity and porosity curve of this model overlap exactly with those plotted by the parallel model. The fractal conductance model proposed in this work reveals the operation of the current in the tree-like branching network more comprehensively.

Image denoising has been a fundamental problem in the field of image processing. In this paper, we tackle removing impulse noise by combining the fractal image coding and the nonlocal self-similarity priors to recover image. The model undergoes a two-stage process. In the first phase, the identification and labeling of pixels likely to be corrupted by salt-and-pepper noise are carried out. In the second phase, image denoising is performed by solving a constrained convex optimization problem that involves an objective functional composed of three terms: a data fidelity term to measure the similarity between the underlying and observed images, a regularization term to represent the low-rank property of a matrix formed by nonlocal patches of the underlying image, and a quadratic term to measure the closeness of the underlying image to a fractal image. To solve the resulting problem, a combination of proximity algorithms and the weighted singular value thresholding operator is utilized. The numerical results demonstrate an improvement in the structural similarity (SSIM) index and peak signal-to-noise ratio.

This paper studies the average trapping time of honeypots on some evolving networks. We propose a simple algorithmic framework for generating networks with Sturmian structure. From the balance property and the recurrence property of Sturmian words, we estimate the average trapping time of our proposed networks with an asymptotic expression ${〈T〉}_t\sim M_t(\alpha )2^t$, where $M_t(\alpha )$ is a bounded expression related to word $\alpha \in {\{0,1\}}^{\infty }$. We next consider networks with multi-honeypots and generalize our basic models. Additionally, we give an symmetrical method to create a series of networks with the Sturmian structure, and the average trapping time satisfies ${〈T〉}_t\sim 5\times 2^t$, which is independent of any word $\alpha$. The generalized methods may have some illuminating effects on the study of networks with randomness.

Among all probability distributions, power law distribution is an intriguing one, which has been studied by many researchers. However, the derivation of power law distribution is still an inconclusive topic. For deriving a distribution, there are various methods, among which maximum entropy principle is a special one. Entropy of random permutation set (RPS), as an uncertainty measure of RPS, is a newly proposed entropy with special features. Deriving power law distribution with maximum entropy of RPS is a promising method. In this paper, certain constraints are given to constrain the entropy of RPS. Power law distribution is able to be finally derived with maximum entropy principle. Numerical experiments are done to show characters of proposed derivation.

The key to shale gas exploration is the characterization of gas migration under the combination of multiple factors. To address the long-standing energy challenge of rapidly and accurately quantifying the behavior of natural fractures and matrix pores in shale at an engineering scale in interaction with gas migration. This study proposes an interdisciplinary model for shale gas extraction by adopting fractal theory. Five innovative microstructural parameters are developed to characterize the size and scale of natural matrix pores/fractures in shale, so as to investigate the contributions of fractal distributed pores and fractal power-law distributed fractures to shale gas extraction. The present results of the proposed model are consistent with the exploitation state of the UK Bowland Shale #114 well. The evolution of the shale microstructure will lead to changes in gas migration behavior throughout the reservoir and in turn affect shale stress, temperature and gas adsorption–desorption effect, and finally have a significant impact on permeability. It is found that in the present analysis of the entire Bowland Shale, the overall permeability changes by 10.8% with the evolution of fractal distributed pores and by 41.3% with the evolution of fractal power-law fractures. This work provides a new approach for rapidly exploring the behavior of shale fractures and matrix pores at engineering scales. This work also offers a new and practical baseline for shale gas extraction assessment and fossil energy management.

Richards’ equation is a classical differential equation describing water transport in unsaturated porous media, in which the moisture content and the soil matrix depend on the spatial derivative of hydraulic conductivity and hydraulic potential. This paper proposes a nonlocal model and the peridynamic formulation replace the temporal and spatial derivative terms. Peridynamic formulation utilizes a spatial integration to describe the path-dependency, so the fast diffusion process of water transport in unsaturated porous media can be captured, while the Caputo derivative accurately describes the sub-diffusion phenomenon caused by the fractal nature of heterogeneous media. A one-dimensional water transport problem with a constant permeability coefficient is first addressed. Convergence studies on the nonlocal parameters are carried out. The excellent agreement between the numerical and analytical solutions validates the proposed model for its accuracy and parameter stability. Subsequently, the wetting process in two porous building materials is simulated. The comparison of the numerical results with experimental observations further demonstrates the capability of the proposed model in describing water transport phenomena in unsaturated porous media.

This paper presents a novel approach for modeling Boost converters using the Caputo–Fabrizio (C-F) definition-based fractional-order model to address singular characteristics in fractional-order definitions and enhance model accuracy. A small signal modeling method is proposed to improve the accuracy of circuit parameter design and to derive state-averaged models, state-space equations, and transfer functions. The influence of capacitor and inductor orders on steady-state characteristics is analyzed and the influence of fractional-order on ripple characteristics is investigated through simulation. When the fractional-order approaches 1, the output voltage increases and the inductance current decreases, with waveform jitter mitigation. Moreover, boundary conditions for continuous conduction mode operation are established based on ripple characteristics. The numerical and circuit-oriented simulations verify the correctness of the proposed model. Finally, the orders and accurate parameters of capacitors and inductors based on the C-F definition are determined and the experiments are conducted. The comparison between the experimental and simulation results demonstrates that the proposed model can accurately describe the steady-state characteristics of the practical circuit systems, which further validates the accuracy of the proposed method.

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic $P$-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.

A fractal modification of the modified KdV–Zakharov–Kuznetsov equation is suggested and its fractal generalized variational structure is established by means of the semi-inverse method. Furthermore, the obtained fractal generalized variational structure is discussed and verified through the two-scale transform from another dimension field in detail. The obtained fractal generalized variational structure reveals the conservation laws via the energy form in the fractal space and can be employed to study the fractal solitary wave properties.

In this paper, exact traveling wave solutions of space-time fractional simplified modified Camassa–Holm (mCH) equation are investigated by the bifurcation theory. The phase portraits of the equation are obtained with different parameter conditions. By analyzing different orbits, periodic wave, kink, anti-kink, burst wave, bright and dark solitary solutions of the equation are acquired. Finally, numerical simulation and the effects of fractional order taking on the dynamic behaviors of various forms of solutions are analytically discussed.

In this paper, we investigate a deterministic mathematical model of Marburg–Monkeypox virus co-infection transmission under the Caputo fractional-order derivative. We discussed the dynamics behavior of the model and carried out qualitative and quantitative analysis, including the positivity–boundedness of solution, and the basic reproduction number ${\Re }_{\text{o}}$. In addition, the Banach and Schauder-type fixed point theorem is utilized to explore the existence–uniqueness of the solution in the suggested model and the proposed model stability under the Ulam–Hyers condition is demonstrated. In numerical simulation, the Predictor–Corrector method is used to determine the numerical solutions. According to the numerical result, increasing the rate of quarantine and detecting unknown Marburg virus, will be the most effective control intervention to reduce Marburg and Monkeypox virus transmission in the population.

A new definition of fractional differentiation of nonlocal and non-singular kernels has recently been developed to overcome the shortcomings of the traditional Riemann–Liouville and Caputo fractional derivatives. In this study, the dynamic behaviors of the fractional financial chaotic model have been investigated. Singular and non-singular kernel fractional derivatives are used to examine the proposed model. To solve the financial chaotic model with nonlocal operators, the fractional Adams–Bashforth method (ABM) is applied based on Lagrange polynomial interpolation (LPI). The existence and uniqueness of the solution of the model can be demonstrated using fixed point theory and nonlinear analysis. Further, the error analysis of the present method and Ulam–Hyers stability of the considered model have also been included. Obtained numerical simulations reveal that the model based on three different fractional derivatives shows various chaotic behaviors that may be useful in a practical sense which may not be observed in the integer case.

This paper derives a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation based on the conformable fractional derivative for the first time. Some new types of the fractal traveling wave solutions are successfully constructed by applying a novel approach which is called the fractal semi-inverse variational method. To our knowledge, the obtained results are all new and have not reported in the other literature. In addition, the dynamic characteristics of the different solutions on the fractal space are discussed and presented via the 3D plots, 2D contour and 2D curves. It can be found that: (1) The fractal order can not only affect the peak value of the fractal traveling waves, but also affect the wave structures, that is, the smaller the fractional order value is, the more curved the waveform is, and the slower waveform changes. (2) In the fractal space, the fractal wave keeps its shape unchanged in the process of the propagation and still meets the energy conservation. The methods in this paper can be used to study the other fractal PDEs in the physics, and the findings are expected to bring some new thinking and inspiration toward the fractal theory in physics.

The fractal and small-word properties are two important properties of complex networks. In this paper, we propose a new random rewiring method to transform fractal networks into small-world networks. We theoretically prove that the proposed method can retain the degree of all nodes (hence the degree distribution) and the connectivity of the network. Further, we also theoretically prove that our method also retains the tree structure of tree graphs. Our method can transform many different types of fractal networks into small-world networks while the degree distribution and connectivity of these networks remain unchanged, demonstrating the generality of small-world networks. In addition, the method also works for other types of complex networks. The rewiring method proposed in this paper can be used in a broader range of applications of network analysis.

Grains account for more than 50% of the calories consumed by people worldwide, and military conflicts, pandemics, climate change, and soaring grain prices all have vital impacts on food security. However, the complex price behavior of the global grain spot markets has not been well understood. A recent study performed multifractal moving average analysis (MF-DMA) of the Grains & Oilseeds Index (GOI) and its sub-indices of wheat, maize, soybeans, rice, and barley and it was found that only the maize and barley sub-indices exhibit an intrinsic multifractal nature with convincing evidence. Here, we utilize multifractal fluctuation analysis (MF-DFA) to investigate the same problem. Extensive statistical tests confirm the presence of intrinsic multifractality in the maize and barley sub-indices and the absence of intrinsic multifractality in the wheat and rice sub-indices. Different from the MF-DMA results, the MF-DFA results suggest that there is also intrinsic multifractality in the GOI and soybeans sub-indices. Our comparative analysis does not provide conclusive information about the GOI and soybeans and highlights the high complexity of the global grain spot markets.

Many real-world systems are connected together, in natural and man-made networks. A complex-valued laser network can simulate the working mechanism of human brain. However, amplitude control of a complex-valued laser network is seldom studied. In this paper, a ring network of complex-valued Lorenz laser systems is investigated. The ring network exhibits complex dynamics including hyper-chaos, quasi-periodic orbits, and coexisting hyper-chaos. Three kinds of single-parameter oriented amplitude controls are realized with varying or unvarying Lyapunov exponents in the ring network. Meanwhile, a simple knob can realize the amplitude rescaling of hyper-chaotic signals, which reduces the cost of circuit implementation. Moreover, a criterion of chaotic complete synchronization among all the nodes is established for a network with strong coupling. For relatively weak coupling, quasi-periodic complete synchronization is found, and the performance of chaotic synchronization is studied in terms of the cross-correlation coefficient. It is moreover revealed that the improvement and trend of synchronization performance are robust to the parity of the number of nodes for the same-scale laser networks.

In this paper, two weighted parameterized fractal identities are first proposed, wherein the mappings involved are second-order local fractional differentiable. Based upon these equalities, a series of the weighted parameterized inequalities, which are related to the fractal convex mappings, are then deduced. Moreover, making use of boundedness and $(\phi ,{\rho }^{\widehat{𝜗}})$-Lipschitzian mappings, some error estimates are attained as well. Finally, certain fractal outcomes in accordance to random variable and the weighted formula, respectively, are presented as applications.

This paper elucidates how the two-scale fractal dimension transform, and a transformation method can be applied to replace the Warner–Gent equation that models the fractal dynamic response of porous viscohyperelastic materials with an equivalent power-form equation. Furthermore, this research work elucidates the advantages of modeling viscohyperlastic materials using the fractal Warner–Gent’s model since the values of the fractal dimension parameter unveil how the global molecular structure of viscohyperelastic materials varies as a function of the vibration frequency wavelength. Compared to the original one, the accuracy attained from the Warner–Gent power-form equivalent equation is examined by plotting the frequency–amplitude and time–amplitude curves obtained from the corresponding numerical integration solutions. It is found that both numerical integration solutions agree well since the root-mean-square-error (RMSE) values remain small.

The Klein–Gordon–Zakharov equation is an important and interesting model in physics. A fractional Klein–Gordon–Zakharov model is described by employing beta-derivative. Some new solitary wave solutions are acquired by utilizing the fractional rational ${sinh}_{\sigma }$–${cosh}_{\sigma }$ method and fractional ${\mathrm{\text{sech}}}_{\sigma }$ method. Some 3D graphs are depicted to elaborate these new solitary wave solutions. The work is very helpful to study other related types of fractional evolution equations.

Accurate classification of the medical signals is urgently needed in clinical medicine. This paper aims to create a classifier to shorten the time of the classification and ensure the sorting accuracy, which assists physicians in saving diagnostic time and formulating the treatment plans. We create the classifier based on Kolmogorov complexity, Shannon entropy, Higuchi’s Hurst exponent and multifractal features. We obtain a feature value from Kolmogorov complexity, Shannon entropy and Higuchi’s Hurst exponent, and three feature values based on multifractal features to compose a vector and analyze it. Furthermore, we study a vector composed of six multifractal features as a control group. Electrocardiogram (ECG) and electroencephalogram (EEG) signals are applied to examine the performance of the classifier by support vector machine (SVM). The accuracy of ECG signals based on mixed classification (MC–ECG–SVM) reaches 94.17%, which is approximately 15% higher than that of ECG signals only based on multifractal features classification (UC–ECG–SVM). The sensitivities of MC–ECG–SVM and UC–ECG–SVM are 86.09% and 64.54%, respectively. The specificities of MC–ECG–SVM and UC–ECG–SVM are 98.26% and 93.65%, respectively. Analogously, the accuracy, sensitivity, and specificity of EEG signals based on mixed classification (MC–EEG–SVM) reach 95.29%, 96.28%, and 94.55%, respectively. The accuracy, sensitivity, and specificity of EEG signals based on multifractal features classification (UC–EEG–SVM) are 87.40%, 89.28%, and 88.11%, respectively. Therefore, the mixed classification method is more accurate than the classification method only based on multifractal features.

The alignment or mapping of Deoxyribonucleic Acid (DNA) reads produced by the new massively parallel sequencing machines is a fundamental initial step in the DNA analysis process. DNA alignment consists of ordering millions of short nucleotide sequences called reads, using a previously sequenced genome as a reference, to reconstruct the genetic code of a species. Even with the efforts made in the development of new multi-stage alignment programs, based on sophisticated algorithms and new filtering heuristics, the execution times remain limiting for the development of various applications such as epigenetics and genomic medicine. This paper presents an overview of recent developments in the acceleration of DNA alignment programs, with special emphasis on those based on hardware, in particular Graphics Processing Units (GPUs), Field Programmable Gate Arrays (FPGAs), and Processing-in-Memory (PIM) devices. Unlike most of the works found in the literature, which review only the proposals that gradually emerged in some specific acceleration technology, this work analyzes the contemporary state of the subject in a more comprehensive way, covering from the conception of the problem, the modern sequencing technologies and the analysis of the structure of the new alignment programs, to the most innovative software and hardware acceleration techniques. The foregoing allows to clearly define, at the end of the paper, the trends, challenges and opportunities that still prevail in the field. We hope that this work will serve as a guide for the development of new and more sophisticated DNA alignment systems.