Fractals

The foremost purpose of this paper is to present a valuable numerical procedure constructed on Elzaki transform and He’s Homotopy perturbation method (HPM) for nonlinear partial differential equation arising in spatial flow characterizing the general biological population model for animals. The actions are made usually by mature animals driven out by intruders or by young animals just accomplished maturity moving out of their parental region to initiate breeding region of their own. He–Elzaki method is a blend of Elzaki transform and He’s HPM. The results attained are compared with Sumudu decomposition method (SDM). The numerical results attained by suggested method specify that the procedure is easy to implement and precise. These outcomes reveal that the proposed method is computationally very striking.

In order to research roughness of rock fracture surfaces and interior damaged status of rock, some Brazil discs of white marble and sandstone are loaded to fracture by Brazil Test and the mathematical model of fractal variation is used to distinguish the morphology characterization of fracture surface of white marble from that of sandstone. Through computing the acquired scanned data of rock fracture surfaces, the following three results are obtained. First, the roughness of fracture surface increases with increase of loading angles, furthermore, the anisotropy of white marble fracture surface is more obvious than that of sandstone one. Second, under the same loading mode, by analyzing the mean fractal variation, the fracture surface of white marble is rougher than that of sandstone. Finally, rock disc specimens loaded to 1/2 or 3/4 of mechanics of mean fracture threshold still do not fracture at the first time loading, but their interiors have been badly damaged.

In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given.

To study the damage evolution mechanism of gas-bearing coal and formation causes of acoustic emission signals during this process, the loaded experiments of gas-bearing coal were performed, and acoustic emission (AE) data radiated in this process were collected. Based on the multifractal theory, the causes of AE were explored in various loaded phases. The results showed that at the low stress stage, the fractures close and the friction/slip could cause low-energy acoustic emission events, and the multifractal spectrum had a smaller width. By contrast, at the high stress stage, the cracks expand, penetrate, and rupture, which would lead to AE events with the release of high energy, reflecting an increase in the width of the multifractal spectrum. At the initial loading stage, the time-varying multifractal spectrum was characterized by a chaotic behavior, but as the loading progressed, it gradually became orderly. In the elastic stage, coal experienced elastic deformation without damage, the ratio of strong and weak AE signals was almost the same, and both $\mathrm{\Delta }{f}_m$ and $\mathrm{\Delta }{\alpha }_m$ were close to 0. In the plastic fracture stage, coal body consumed huge amounts of energy and suffered fracture. This also caused the coal body to radiate a large amount of AE signals. An analysis of these signals indicated that strong signals dominated and showed an increasing trend, and $\mathrm{\Delta }{f}_m$ was less than 0 and continued to decrease. The time-varying multifractal characteristics reveal the formation mechanism of AE signals from gas-bearing coal, which contributes to improve our understanding of the mechanism of gas-bearing coal damage.

In this paper, we present examples of one-parametric iterated function systems converging to the standard middle-third Cantor set. The main goal is the study of the continuous growth of the attractor of the corresponding parametrization.

In recent years, weighted networks have been extensively studied in various fields. This paper studies characteristic polynomial of adjacency or Laplacian matrix for weighted treelike networks. First, a class of weighted treelike networks with a weight factor is introduced. Then, the relationships of adjacency or the Laplacian matrix at two successive generations are obtained. Finally, according to the operation of the block matrix, we obtain the analytic expression of the characteristic polynomial of the adjacency or the Laplacian matrix. The obtained results lay the foundation for the future study of adjacency spectrum or Laplacian spectrum.

This paper presents a new methodology of wear state recognition by using fractal parameters, multifractal parameters and recurrence parameters. The relationship between these nonlinear parameters is analyzed. A nonlinear state point of worn surface is established by fractal dimension, average diagonal length and spectrum width. Further, a steady state sphere is obtained by the nonlinear state point and $K$-means clustering algorithm. Results show that fractal, multifractal and recurrence parameters characterize the worn surface from different perspectives. They should be used simultaneously to comprehensively characterize the integral structures, partial structures and internal structures of worn surface. The proposed nonlinear state point shows a variation process of concentration–stabilization–separation during the wear process. The wear states can be identified effectively by the relationship between nonlinear state points and steady state sphere.

Acquiring the desired surface quality is one of the major efforts in machining of materials. Milling operation is a widely used machining operation to shape the material in different forms. Machining parameters and conditions are two major factors that affect the surface quality of machined workpiece in milling operation. In this paper, we analyze the surface finish of machined workpiece under the variations of machining parameters and conditions (wet and dry conditions) in milling operation. For our analysis, we use fractal dimension as the indicator of complexity of structure. Based on the obtained results, in the case of wet machining condition, by increasing the depth of cut, feed rate and spindle speed in separate experiments, the fractal dimension of machined surface increases. However, the obtained results in the case of dry machining condition are not consistent with the variations of different machining parameters. The obtained results will be discussed in terms of complex structure of machined surface. The method of analysis employed in this research can be investigated with other machining operations to check how the machining parameters and conditions affect the surface quality of machined surface.

In this paper, we investigate the average geodesic distance on the Sierpinski hexagon in terms of finite patterns on integrals. Applying this result, we also obtain the asymptotic formula for average distances of Sierpinski hexagon networks.

Supersingular integrals arise in many areas of applied mathematics: Fluid dynamics and fracture mechanics are among the most important ones. This paper is devoted to investigating the relationship between Riesz, Riesz–Caputo, Hilfer fractional derivatives and the corresponding finite part integrals in Hadamard sense. We prove that Riesz, Riesz–Caputo, and Hilfer derivatives of a given function can be expressed by the finite part integrals of a supersingular integrals which do not exist.

To maximize the advantages of both spectral and spatial information, we introduce a new spectral–spatial jointed hyperspectral image classification approach based on fractal dimension (FD) analysis of spectral response curve (SRC) in spectral domain and extended morphological processing in spatial domain. This approach first calculates the FD image based on the whole SRC of the hyperspectral image and decomposes the SRC into segments to derive the FD images with each SRC segment. These FD images based on the segmented SRC are composited into a multidimensional FD image set in spectral domain. Then, the extended morphological profiles (EMPs) are derived from the image set through morphological open and close operations in spatial domain. Finally, all these EMPs and FD features are combined into one feature vector for a probabilistic support vector machine (SVM) classification. This approach was demonstrated using three hyperspectral images in urban areas of the university campus and downtown area of Pavia, Italy, and the Washington DC Mall area in the USA, respectively. We assessed the potential and performance of this approach by comparing with PCA-based method in hyperspectral image classification. Our results indicate that the classification accuracy of our proposed method is much higher than the accuracies of the classification methods based on the spectral or spatial domain alone, and similar to or slightly higher than the classification accuracy of PCA-based spectral–spatial jointed classification method. The proposed FD approach also provides a new self-similarity measure of land class in spectral domain, a unique property to represent hyperspectral self-similarity of SRC in hyperspectral imagery.

A molecular dynamics simulation is performed to study the liquid film evaporation on rough surfaces constructed by the Cantor set fractal. The liquid film evaporation on the rough surfaces with different roughness and fractal dimensions is presented and is compared with that on smooth surface. The results indicate that the response speed for evaporation on the rough surface is faster than that on the smooth surface and is accelerated with the increase of the roughness height and the fractal dimensions. Protrusions on the rough surface are conducive to heat transfer enhancement, and thus, the evaporation on a rough surface can reach equilibrium earlier than that on a smooth surface. With increasing roughness and fractal dimension, the evaporation rate and the thickness of the non-evaporating liquid layer increase.

In this paper, we discuss the lower type dimensions for some Moran sets. On one hand, for Moran set $E$ with $c_{\ast }>0$, we prove that ${dim}_{qL}E={sup}_f{dim}_{L}f(E)$, where the supremum is taken over all quasi-Lipschitz mappings $f$. On the other hand, we obtain the lower spectrum formula for homogeneous Moran sets. In the proof a lower spectrum formula for a large class of fractal sets is established.

This paper proposes a general time-space metric by an extension of the power-function-based fractal concept to the structural function fabric. The structural function can be an arbitrary-function to describe complex metric underlying physical systems. We call such a metric “structal”, and the fractal metric is its special case. This work is inspired by the recent work on the structural derivative, which takes into account the significant influence of time-space fabric of a complex system on its physical behaviors, in particular, the ultraslow diffusion. Based on the structal concept, this study gives the structural time-space transformation and introduces the general diffusion models. In addition, the fundamental solutions of structural derivative equations and statistics implication of the structal metric are also briefly discussed.

In this paper, fractal dimensions of fractional calculus of continuous functions defined on $[0,1]$ have been explored. Continuous functions with Box dimension one have been divided into five categories. They are continuous functions with bounded variation, continuous functions with at most finite unbounded variation points, one-dimensional continuous functions with infinite but countable unbounded variation points, one-dimensional continuous functions with uncountable but zero measure unbounded variation points and one-dimensional continuous functions with uncountable and non-zero measure unbounded variation points. Box dimension of Riemann–Liouville fractional integral of any one-dimensional continuous functions has been proved to be with Box dimension one. Continuous functions on $[0,1]$ are divided as local fractal functions and fractal functions. According to local structure and fractal dimensions, fractal functions are composed of regular fractal functions, irregular fractal functions and singular fractal functions. Based on previous work, upper Box dimension of any continuous functions has been proved to be no less than upper Box dimension of their Riemann–Liouville fractional integral. Fractal dimensions of Riemann–Liouville fractional derivative of certain continuous functions have been investigated elementary.

We present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on triangular domains. Then we introduce affine RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of affine RFISs under certain constraints.

There are several methods given for constructing the Sierpinski Triangle (ST). In particular, the ST can be created as a countably infinite intersection of contained figures, or as the countably infinite union of triangles. At first glance they both appear to create the same object. In this paper, we show that they are not the same. In addition, we analyze and compare with the previous two methods, a third method of constructing the ST which entails generating a countable number of points using an algorithm.