Fractals

This paper intends to study the analytical properties of the Riemann–Liouville fractional integral and fractal dimensions of its graph on ${ℝ}^n$. We show that the Riemann–Liouville fractional integral preserves some analytical properties such as boundedness, continuity and bounded variation in the Arzelá sense. We also deduce the upper bound of the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of Hölder continuous functions. Furthermore, we prove that the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of a function, which is continuous and of bounded variation in Arzelá sense, are $n$.

In this paper, we prove the existence of the bivariate fractal interpolation function using the Rakotch contraction theory and iterated function system for a countable data set. We also give the existence of the invariant Borel probability measure supported on the graph of the bivariate fractal interpolation function. In particular, we highlight that our theory encompasses the bivariate fractal interpolation theory in both finite and countably infinite settings available in literature.

There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This paper studies the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures.

Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the “subset of positive and finite measure” property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures.

This paper establishes a linear relationship between the order of the Riemann–Liouville fractional calculus and the exponent of the Hölder condition, whether the Hölder condition is global, local, or at a single point. We propose and prove a control inequality between the Hölder derivative ($H_f(x,\alpha )$ as defined in Proposition 12) of a continuous function and the Hölder derivative of the Riemann–Liouville fractional calculus of this function. In addition, this paper provides a more accurate estimation of the Box dimension of the graph of the Riemann–Liouville fractional integral of an arbitrary continuous function. More specifically, it establishes the result that whenever there is a continuous function whose graph has the upper Box dimension $s$ with $1<s\le 2$, the graph of its Riemann–Liouville fractional integral of order $\nu$, with $0<\nu <1$, has the upper Box dimension not greater than $s-(s-1)\nu$.

In this paper, we research on the dimension preserving monotonous approximation by using fractal interpolation techniques. A constructive result of the approximating sequence of self-affine continuous functions has been given, which can converge to the object continuous function of bounded variation on $[0,1]$ monotonously and unanimously, meanwhile their graphs can be any value of the Hausdorff and the Box dimension between one and two. Further, such approximation for continuous functions of unbounded variation or even general continuous functions with non-integer fractal dimension has also been discussed elementarily.

In this paper, we consider a function with only one unbounded variation point and study the box dimension of its graph. We prove that the function is continuous and differentiable on a certain interval. Moreover, we show that the function is of unbounded variation on the domain of definition. Using our techniques, we also estimate the box dimension of the graph of the function.

In this paper, new results on the $\alpha$-fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an $\alpha$-fractal function with variable parameters is examined by imposing certain conditions on the scaling factors. Following the investigation of fractional derivative, the definite integral of the $\alpha$-fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the $\alpha$-fractal function is provided using the base $q$ representation of numbers.

The purpose of this paper is to deal with a system governed by a system of semilinear nonlocal fractional evolution inclusions with partial Clarke subdifferential and its optimal control. First, we establish an existence theorem of the mild solution for the presented control system by applying the measure of noncompactness, a fixed point theorem of a condensing multivalued map and some properties of partial Clarke subdifferential. Moreover, under some mild conditions, we obtain a result on the existence of an optimal control to the presented control system. Finally, an example is provided to demonstrate the main results. The results presented in this paper improve, extend and develop the corresponding results in the earlier and recent literature.

A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

In an emergency, fear can spread among crowds through one-on-one encounters, with negative societal consequences. The purpose of this research is to create a novel theoretical model of fear (panic) spread in the context of epidemiology during an emergency using the fractal fractional operator. For quantitative analysis, the system’s boundedness and positivity are checked. According to the Arzela Ascoli theorem, the model is completely continuous. As a result of the discovery of Schauder’s fixed point, it has at least one solution. The existence and uniqueness of the concerned solution have been examined using the fixed point theory technique. Numerical simulations are used to demonstrate the accuracy of the proposed techniques using a generalized form of Mittag-Leffler kernel with a fractal fractional operator. Finally, simulations are utilized to represent the spread of group emotional contagion (spontaneous spread of emotions and related behaviors) dynamically.

Currently, immunotherapy is seen to be the most effective cancer treatment. This is especially true while treating chronic lymphocytic leukemia (CLL), a slow-growing B-lymphocyte neoplasm that gradually compromises the immune system. Mathematical modeling is acknowledged as a key technique for analyzing theoretical and practical challenges in this field of cancer research and others. We were inspired to develop a mathematical model because of its dearth in investigations of chemotherapy-induced immunotherapy for CLL. This study effort formulates the dynamics of lymphocytic leukemia utilizing fractional calculus to conceptualize the complex processes of this viral illness. The basic idea of fractional calculus has been shown using the Atangana–Baleanu framework. For the suggested model’s chaotic and dynamic behavior, a new numerical approach is described. We inspect the stability and convergence of the suggested numerical technique in our work. The fluctuation of various system input elements has demonstrated the oscillatory and chaotic behavior of the system. Moreover, we have demonstrated how the suggested mechanism of lymphocytic leukemia infection is affected by fractional order. Through numerical simulations, the most important input parameters are emphasized, and the policymakers are given control intervention suggestions.

In the Caputo sense, the goal of this paper is to develop a thorough analysis for the Nipah virus sickness under piecewise equations with fractional order derivative. Here, we take into account a SIRD-type model with classes for susceptible, infected, recovered, and dead individuals. We evaluate the solution’s viability using the notion of fractional order derivative. The fundamental reproduction number is also calculated. Here, we offer sufficient requirements for the Ulam–Hyers (UH) stability and the existence theory of the solution. We also give some numerical simulations for several compartments using the RK4 approach. In order to deepen our investigation, we therefore investigate the aforementioned disease’s transmission process using some novel fractional calculus notions.

The social life, economic issues, and health issues resulting from various diseases will be impacted by the use of the epidemiological model to address the negative effects of drinking in society. The paper aims to investigate a nonlinear drinking epidemic fractional SHTR model in the sense of a Constant Proportional Caputo (CPC) operator. For the CPC operator, a stability study of the fractional order model and the presence of a solution have been made. A nonlinear system of the suggested system has an approximative solution provided by the Laplace Adomian Decomposition technique. A convergence analysis of the system is also treated to show the effect and efficiency of the scheme. Additionally, we offer some numerical outcomes to demonstrate the efficiency of this operator. A comparative study for different values of $\alpha$ shows that these methods are effective for giving better results in fractional order as compared to classical order derivatives.

Rapid emissions of green-house gases (GHGs) are causing global warming, which is wreaking havoc on the earth’s climate system. As a result, the coastal ecosystems of the world are on the verge of becoming endangered. We develop a fractal-fractional hybrid model to estimate the influence of rapid emissions of GHGs on the coastal ecosystems and climate changes. The fractal-fractional climate change model is considered for the theoretical and numerical aspects. The theoretical aspect includes the existence and Hyers–Ulam (HU) stability of the solutions while the numerical aspect is based on Lagrange’s interpolation polynomial. Moreover, unique solutions are proved by the Banach contraction principle. Different fractal-fractional orders are considered for the numerical study of dynamics of solutions which prove the importance and accuracy of these hybrid operators.

In this work, we prove a parameterized fractional integral identity involving differentiable functions. Then, we use the newly established identity to establish some new parameterized fractional Hermite–Hadamard–Mercer-type inequalities for differentiable function. The main benefit of the newly established inequalities is that these inequalities can be converted into some new Mercer inequalities of midpoint type, trapezoidal type, and Simpson’s type for differentiable functions. Finally, we show the validation of the results with the help of some mathematical examples and their graphs.

In this paper, we consider the networks modeled by several self-affine sets based on the Bedford–Mcmullen carpet. We calculate three properties of the networks, including the cumulative degree distribution, the average clustering coefficient and the average path length. We show that such networks have scale-free and small-world effects.

Motivated by previous authors’ work, where Shannon entropy, box covering and information dimension were applied to quantify pulmonary lesions, this paper extends such a contribution in two fashions: (i) Following the approach to quantify pulmonary lesions with Deng entropy and Deng information dimension obtained through box covering method; (ii) exploiting the Shannon and Deng lesion quantification for pulmonary illnesses classification with a bidirectional Long Short Term Memory (bLSTM). The referred pulmonary illnesses are Common Pneumonia (CP) and COVID-19. Shannon entropy and information dimension are performed here and called the Shannon sequence. Then, Deng entropy and Deng information dimension are computed for chest Computed Tomography (CT) images to obtain and combine two data sequences to quantify the pulmonary lesions. The data sequence resulting from the data combination is called the Deng sequence. Both Shannon and Deng sequences are independently used as input for the bLSTM. CT lung scans of 531 healthy subjects, 497 confirmed COVID-19 diagnoses and 516 with CP were analyzed to obtain the Shannon and Deng sequences. The results demonstrate that Deng entropy and Deng information dimension of CT images can differentiate similar lung lesions between COVID-19 and CP. Besides, a statistical analysis shows that: (a) Classification by the bLSTM is better when using the Deng sequence than the Shannon sequence; (b) Deng sequences plus bLSTM significantly outperform DenseNet-201, GoogLeNet and MobileNet-v2 in classifying COVID-19, CP and Normal CT (healthy subjects) in time and accuracy. Hence, the Deng sequence and bLSTM are fast and accurate tools for helping in diagnosing CP and COVID-19.

In this paper, the fractional Schrödinger equation is described with beta derivative, which is used to elucidate the dynamic interaction of ultra-short pulses with quantum properties in optical fibers. This work is to study the solitary wave and periodic solutions of the fractional Schrödinger equation by employing three powerful and simple mathematical approaches like fractional Kudryashov method, fractional cosine–sine method and fractional tanh function method. The acquired outcomes illustrate that the proposed three computational approaches are simple, efficient, concise and can be adopted to study more complex phenomena. Finally, the dynamical behavior of these acquired solitary wave solutions is illustrated by sketching some 3D figures with proper parameters.

The fracture network in fractured coal is the main channel of coal seam gas flow. Not only the geometric topology properties (such as fractal characteristics) of a single fracture but also the connection topology properties (interconnection characteristics between fractures) of the fracture network have an important impact on the fluid flow in fracture networks. In this study, the connection topology properties of the fracture network in the fractured coal are explored based on the complex network theory for the first time. The property parameters such as the fracture node degree, the clustering coefficient, and the average path length are analyzed. It shows that the average clustering coefficient of the fracture network in fractured coal is larger, and the average path length is smaller. The connection property of the fracture network in the fractured coal presents a typical “small-world” clustering model. Further, by considering the fractal characteristics of the single fracture and the clustering characteristics of the fracture network, an improved clustered fractal discrete fracture network (DFN) model is developed. Then, based on the lattice Boltzmann method, the permeability properties of the generated clustered fractal DFNs are analyzed. The results show that the permeability of DFNs is positively correlated with the average clustering coefficient of fracture network, and negatively correlated with the fractal dimension of fracture. Therefore, the topological clustering characteristics of fracture networks and the fractal characteristics of fractures cannot be ignored in describing the fluid flow in the fracture network, and our clustered fractal DFN model provides a new idea for guiding the optimization design in DFN engineering.

Underground coal mining in China has gradually moved into deeper seams in recent years, which results in a higher ambient temperature in the mining space and significantly affects the mechanical behavior of coal. In this study, dehydrated coal samples were obtained at different temperatures ranging from 30${}^{\circ }$ to 70${}^{\circ }$, and the mechanical behavior of the dehydrated coal was investigated through compressive loading tests. The digital image correlation (DIC) method was used to acquire the strain field of coal, and a multifractal analysis was conducted to characterize the strain evolution of coal. The findings suggest that the increasing temperatures result in higher moisture desorption rates and greater volumetric contraction strain in coal. Furthermore, coal with higher moisture desorption exhibits higher peak stress and peak strains when subjected to compressive loading. The multifractal analysis of the inhomogeneous strain evolution indicates a gradual decrease in the parameter $\mathrm{\Delta }\alpha$ under compressive loading, followed by a sudden increase before reaching the failure point due to strain localization. The multifractal mechanism was further investigated, revealing that the inhomogeneous strain field of coal is inherently affected by the microstructure of coal. In addition, a mathematical model was proposed to elucidate the relationship between the inhomogeneous coal strain and the microstructure of coal. The result indicates that the inhomogeneity of the coal strain is directly associated with the multifractal singularity of the coal microstructure. Finally, the feasibility of using the multifractal parameter $\mathrm{\Delta }\alpha$ to identify coal strain localization has been demonstrated, indicating its potential value in aiding engineers to determine the SLZ in deep coal mines.

In this work, a new fractal pseudo-parabolic equation is derived by means of He’s fractal derivative. The semi-inverse method (SIM) is employed to develop the generalized fractal variational principle (GFVP), which can reveal the energy conservation law in the fractal space and provide some new insights on the study of the variational method.

Finger seal is a new flexible dynamic sealing technology, and its heat transfer characteristics and seepage characteristics are one of the main research hotspots. In this paper, based on the fractal theory, a fractal model of the total thermal conductance of the finger seal considering the heat transfer effect of the contact gap of the rough surface is established, a fractal model of the effective gas permeability of the adjacent finger seals annulus considering the gas slip effect and the temperature change is established, and a finite element calculation method of the two-way thermo-mechanical coupling for the finger seal is proposed. The results show that the solid-phase thermal conductance decreases with the increase of the scale coefficient. When the axial pressure difference is greater than 0.4MPa, the rotor speed is greater than 11,000r/min, the radial displacement excitation is [0.03mm, 0.09mm], and the temperature is less than 600K, the gas-phase thermal conductance between the finger seal and the rotor shows an increasing trend. The gas-phase thermal conductance of the finger seal and the rotor is always the main position under different working conditions. Under different fractal dimensions, the solid-phase thermal conductance gradually occupies the dominant position. Temperature has a certain effect on the effective gas permeability, and fractal dimension, scale coefficient, and axial pressure difference have less effect on the effective gas permeability. At an axial pressure difference of 0.08MPa, the numerical calculation results of the two-way thermo-mechanical coupling calculation method for finger seal are closer to the experimental results, with a maximum error rate of 1.96%. The above results further improve the theoretical research system of the heat transfer characteristics of the finger seal.

The permeability of shale controls gas transport in shale gas reservoirs. The shale has a complex pore structure at the nanoscale and its permeability is affected by multiple transport and action mechanisms. In this study, a 3D fractal model for predicting the apparent gas permeability of shale matrix is presented, accounting for the effects of the transport mechanisms (bulk gas transport and adsorption gas diffusion) and action mechanisms (gas adsorption, real gas properties, water film, stress dependence, and total organic carbon (TOC) content). The proposed model is validated with the published experimental data. A series of sensitivity analyses are performed to investigate the influence of fractal characteristics and action mechanisms on the apparent permeability caused by each transport mechanism. The results show that the real gas properties, water film, and stress dependence cause different effects on the total apparent permeability of shale under different fractal characteristics. The maximum pore diameter is inversely proportional to the effects of these action mechanisms, and the porosity is positively proportional to the effects of real gas properties and water film but inversely proportional to the effects of stress dependence. An increase in TOC content can cause an improvement in the total apparent permeability. Furthermore, the effects of action mechanisms on the apparent permeability caused by different transport mechanisms are differently affected by the fractal characteristics. Changes in fractal characteristics mainly affect the apparent permeability caused by slip flow in the real gas effect, slip flow and Knudsen diffusion in the water film effect, and all transport mechanisms in the stress dependence effect.

Every fluid problem is greatly affected by its boundary conditions, especially the near-shore seabed could produce an irrevocable harm when a tsunami wave is approaching, and a real-life mathematical model could stave off the worst effect. This paper assumes that the unsmooth seabed is a fractal surface, and fractal-fractional governing equations are established according to physical laws in the fractal space. The geometrical potential theory is used to explain the force produced by the wave surface, and Kong-He friction law is applied to further figuring out the local and memory properties of the friction along the fractal boundary. This paper aims at studying tsunami waves in a fractal space, rendering a reliable mathematical model for both prediction of the tsunami motion and the coastal protection.

Since the discovery of momentum effect, people have started the journey of using the momentum effect to construct momentum strategies. As a result of coupling cross-sectional and time-series momentum strategy, dual momentum strategy (DM strategy) has been widely used in practice and closely followed by academics. To address the shortcoming of the classical DM strategy that has not considered the multi-fractal characteristics of the stock market, we construct the fractal dual momentum strategy (FDM strategy) from the two aspects of optimizing the ranking index of the cross-sectional momentum strategy by using fractal statistical measures and improving the timing selection of the time-series momentum strategy by using the trend entropy dimension. The empirical results show that the FDM strategy outperforms the DM strategy. Both in terms of the size and stability of the strategy returns, the FDM strategy shows an optimization effect compared with the DM strategy, which is beneficial to provide investors with better decision-making references.

Fractional $q$-calculus is considered to be the fractional analogs of $q$-calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) $q$-integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving $\hslash$-convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF $q$-integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided.

In this paper, we have extended the concept of advanced Julia function for the discovery of new type of trajectories existing inside outer and inner wings. A dynamical system based on four rotors, referred to as quadrotor unmanned aerial vehicle (QUAV), is considered for the first time to seek the generation of extra wings using fractal theory. Moreover, we have used Julia and advanced Julia function techniques for the creation of outer and inner wings, respectively. Furthermore, the trajectories achieved using both techniques are analyzed and compared with the aid of basin of attraction and bifurcation diagram. This work covers a gap in the literature, that what is the impact of global dynamics on the hidden fractals. Therefore to fill this gap, sets of initial points are plotted from each basin constructed for fractals-based QUAV systems. Finally, numerical simulations are carried out using MATLAB, whereas the data generated for the corresponding basins are attached for the convenience of readers to reproduce each figure.

Several new network information dimension definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. This paper proposes a new definition based on Deng entropy and d-summability (a concept from geometric measure theory). We will prove to what extent the new formulation will be useful in the theoretical and applied points of view.