Fractal Function Research Articles

What is the variant of fractal dimension under addition of functions with same dimension and related discussions

<abstract><p>This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let $ g(x) $ and $ h(x) $ have the same box dimension $ t\; (1 < t\leq 2) $, the box dimension of $ g(x)+h(x) $ may or may not exist. If it exists, it can take an arbitrary real number $ \gamma $ satisfying $ 1 < \gamma\leq t $. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers $ t_1\; {\rm and}\; t_2 $ that satisfy $ 1 < t_1 < t_2 < t\leq 2 $. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function $ W_t(x) $, we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals.</p></abstract>

RESEARCH ON THE K-DIMENSION OF THE SUM OF TWO CONTINUOUS FUNCTIONS AND ITS APPLICATION

In this paper, we have done some research studies on the fractal dimension of the sum of two continuous functions with different [Formula: see text]-dimensions and approximation of [Formula: see text]-dimensional fractal functions. We first investigate the [Formula: see text]-dimension of the linear combination of fractal function whose [Formula: see text]-dimension is [Formula: see text] and the function satisfying Lipschitz condition is still [Formula: see text]-dimensional. Then, based on the research of fractal term and the Weierstrass approximation theorem, an approximation of the [Formula: see text]-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.

The uniformly continuous theorem of fractal interpolation surface function and its proof

<abstract> <p>In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area.</p> </abstract>

α-FRACTAL FUNCTION WITH VARIABLE PARAMETERS: AN EXPLICIT REPRESENTATION

In this paper, new results on the [Formula: see text]-fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an [Formula: see text]-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 [Formula: see text]-fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the [Formula: see text]-fractal function is provided using the base [Formula: see text] representation of numbers.

FRACTAL SURFACES INVOLVING RAKOTCH CONTRACTION FOR COUNTABLE DATA SETS

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.

Design of Miniaturized Microwave Filtering-Balun Component for Wireless Applications

This research paper introduces a new design being Minkowski-like fractal filtering-balun MLFFB based on the dual-mode ring resonator. The structure has three ports, converting the signal from one unbalanced port to two balanced ports (i.e., 180° phase shift). The proposed design can act as a filter and balun simultaneously using one dual-mode ring resonator. The fractal curves are applied to have compact designs. The 0th, 1st, and 2nd iterations of the Minkowski-like fractal curves are applied to demonstrate the miniaturization ratio obtained in the proposed work. The miniaturization ratios are 47.8% and 73% of the 1st iteration and 2nd iteration of the Minkowski balun-filtering components, respectively, compared to the 0th iteration. Also, the filtering balun has a distinctive feature, which is the control of the phase error and magnitude imbalance of the two balanced ports, depending on the size of the perturbation part. Several structures have been designed, modeled, and analyzed utilizing the Advanced Design System (ADS) software. Each design iteration is evaluated and optimized to attain the best performance. The operating frequency is 2.4 GHz, and the realized transmission coefficients are S21 = -7 dB and S31 = -6.5 dB, which are less than expected ideal values because the substrate is a lossy type, being FR4. The S11 is less than -10 dB. The proposed design is a good candidate for narrow-band wireless applications.

Study on Dynamic Responses of Unit Slab Track Considering the Influence of Cement Asphalt Mortar Rough Surface

Due to the particularity of the construction technology of the CRTS I slab track bagging method, there is a certain roughness between the track slab and the mortar layer in the actual project. In this work, the W–M fractal function was used to simulate the cement asphalt mortar layer in unit slab track with roughness rough surface morphology. To investigate the effects of interlayer rough contact surfaces, an interactive modeling approach using Python–Abaqus was developed, namely, the track structure performance, interlayer contact behavior, and vertical dynamic responses. The fastener pivot point reaction of the vehicle–track coupled model interacts with the track structure to examine the impact of different fractal dimensions and scale coefficients under the dynamic train load. The numerical results demonstrate that as the fractal dimension D and characteristic scale factor G increase, the dynamic response of the track slab gradually increases, and the contact area between layers decreases. The stress of rough contact surface is pointing distribution. Among the components of the track structure, the displacement of the track slab is most affected by the roughness. The rough surface affects the fatigue life of the mortar and consequently affects the service performance of the track structure. This study explores the effect of rough surfaces on the performance of CRTS I slab track structures under dynamic train load and provides a reference for improving the design of track structures and enhancing its life.

RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

On the stability of Fractal interpolation functions with variable parameters

<abstract><p>Fractal interpolation function (FIF) is a fixed point of the Read–Bajraktarević operator defined on a suitable function space and is constructed via an iterated function system (IFS). In this paper, we considered the generalized affine FIF generated through the IFS defined by the functions $ W_n(x, y) = \big(a_n(x)+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, N $. We studied the shift of the fractal interpolation curve, by computing the error estimate in response to a small perturbation on $ \alpha_n(x) $. In addition, we gave a sufficient condition on the perturbed IFS so that it satisfies the continuity condition. As an application, we computed an upper bound of the maximum range of the perturbed FIF.</p></abstract>

THE MULTI-PARAMETER FRACTAL–FRACTIONAL INEQUALITIES FOR FRACTAL (P,m)-CONVEX FUNCTIONS

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 [Formula: see text]-convex functions. Initially, we formulate the new conception of the fractal [Formula: see text]-convex functions and work on a variety of properties. Through the assistance of the fractal–fractional integrals, the [Formula: see text]-fractal identity with multiple parameters is established, and from that, integral inequalities are inferred regarding twice fractal differentiable functions which are fractal [Formula: see text]-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.

Analytic Studies of a Class of Langevin Differential Equations Dominated by a Class of Julia Fractal Functions

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory.

Fractalization of Fractional Integral and Composition of Fractal Splines

The present study perturbs the fractional integral of a continuous function $f$ defined on a real compact interval, say $(\mathcal{I}^vf)$ using a family of fractal functions $(\mathcal{I}^vf)^\alpha$ based on the scaling parameter $\alpha$. To elicit this phenomenon, a fractal operator is proposed in the space of continuous functions, an analogue to the existing fractal interpolation operator which perturbs $f$ giving rise to $\alpha$-fractal function $f^\alpha$. In addition, the composition of $\alpha$-fractal function with the linear fractal function is discussed and the composition operation on the fractal interpolation functions is extended to the case of differentiable fractal functions.

On two special classes of fractal surfaces with certain Hausdorff and Box dimensions

In this paper, using two special types of rise-dimensional operators based on existing fractal functions, we construct new fractal surfaces with any value of the Hausdorff and Box dimension between two and three. Further, we demonstrate that the lower and upper Box dimension of such fractal surfaces may be unequal to each other. This result could be useful to the research on creating various fractal surfaces with the required fractal dimensions in the future.

Effects of Macro-Pitting Fault on Dynamic Characteristics of Planetary Gear Train Considering Surface Roughness

The planetary gearbox plays a vital role in a wide range of mechanical power transmission systems, including high-speed trains, wind turbines, vehicles, and aircraft. At the same time, the planetary gear train inside the gearbox is regarded as the most susceptible to failure in the entire transmission system. To analyze the influence of surface roughness on the dynamic characteristics of the planetary gear train, a dynamic modeling method based on fractal theory is proposed. Firstly, the tooth surface contact model was established based on the W-M fractal function, and the time-varying mesh stiffness (TVMS) of the planetary gear train was calculated under healthy and tooth macro-pitting. Then, the lumped-parameter method is introduced to construct a planetary gear train translation-torsion dynamic model that comprehensively considers TVMS and tooth backlash. The vibration acceleration signals of the planetary gear train under different macro-pitting states and surface roughness are simulated and calculated, allowing a quantificative analysis of the influence of surface roughness on system vibration response. Finally, the correctness of the model for the planetary gear train is verified by experiments. The results show that compared with the planetary gear train modeling method based on Hertz contact theory, the root mean squared error of the vibration signal of this work under a macro-pitting fault state is reduced by 8.7%, further improving the reliability of the model.

FRACTAL DIMENSIONS OF THE LOGARITHM OF CONTINUOUS FUNCTIONS

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.

Assessment of reservoir heterogeneities and hydrocarbon potential zones using wavelet-based fractal and multifractal analysis of geophysical logs of Cambay basin, India

The scaling behaviour of physical properties of the earth's subsurface e.g. porosity, permeability, density, etc significantly influence reservoir characteristics. The characterization of the parameters, by analysing nonlinear geophysical logs using the application of linear and traditional empirical relations, is very tedious, complicated, and yields less accurate results. In this perspective, wavelet-based fractals and multifractals techniques provide effective insights into local and global characteristics of heterogeneity of reservoir parameters. The results from the numerical experiment on simulated synthetic data reveal that the wavelet-based fractal and multifractals are more useful to delineate the shorter fluctuations in signal and heterogeneities of the reservoir. The scaling exponent and fractal density, a component of fractal analysis, have been estimated for each sample from power law relationship that facilitates the understanding of the local characteristics of the subsurface formation. On the other hand, the multifractal spectrum, a generalization curve of the hurst exponents, provides the global nature of heterogeneities corresponding to reservoir parameters. We analyse gamma ray, density, and neutron logs of three wells (well-204, 203, and 172) of the Limbodara oil field of Cambay Basin, India. The fractal density curve of gamma ray log is found like the low-pass filtered version of the original signal, and map the lithologies by clustering their similar values while the scaling exponent curve on either side of 0.5 of scaling exponent demarcates the lithological boundaries at the intersection points. The scaling exponent curve highlights the finer variations embedded in the signal and is feasible to enhance the thin layers of the reservoir. The multifractal spectrum of gamma-ray logs across three wells encompasses three major formations emphasizing that the Kalol formation exhibits higher degree of heterogeneity than other formations (Tarapur and Olpad). The multifractal spectrum of neutron and gas corrected log for each well of Kalol formation shows the greater influence on well-203 and lowest influence on well-172, indicating the effective hydrocarbon bearing zones of the former well than the latter. This evidence that an increase in lithological heterogeneity enhances the quality of hydrocarbon reservoir as well as reduces the multifractality of porosity logs.

Fractal Mellin transform and non-local derivatives

Abstract This paper provides a comparison between the fractal calculus of fractal sets and fractal curves. There are introduced the analogues of the Riemann–Liouville and Caputo integrals and derivatives for fractal curves, which are non-local derivatives. Moreover, the concepts analogous to the fractional Laplace operator to address fractal non-local differential equations on fractal curves are defined. Additionally, in the paper it is introduced the fractal local Mellin transform and fractal non-local transform as tools for solving fractal differential equations. The results are supported with tables and examples to demonstrate the findings.

Simulation of electrical contact wear on the rough surfaces of ultra-high-voltage transmission line fittings

A novel electrical contact wear simulation method was proposed to characterize the electrical contact wear of fittings in ultra-high-voltage transmission lines by combining line fittings with the Archard wear model and oxidation loss theory. In this method, a three-dimensional (3D) rough body was generated by using the Weierstrass–Mandelbrot fractal function to simulate the contact surface. An electrical contact wear subroutine was developed, and the wear state was updated using arbitrary Lagrangian–Eulerian adaptive grid technology. Finally, finite element software was used to perform thermal stress wear coupled analysis. The results show that the wear volume, wear depth and friction temperature obtained by the rough electric contact model were 2.71 times, 4.21 times and 2.18 times of the common ideal plane model, respectively. In the rough model, the wear depth of the nodes initially accelerated, subsequently slowed down, and again accelerated with time. The friction high temperature region was distributed in a point pattern, and the temperature difference between the contact region and the non-contact region became obvious.

A traffic-fractal-element-based congestion model considering the uneven distribution of road traffic

The uneven distribution of road traffic has been proven to play a crucial role in congestion evolution within urban transportation networks. Establishing congestion models to explore congestion evolution under its impact is of great significance for predicting congestion. Current congestion models commonly describe the uneven distribution of traffic on roads through continuous differential equations or the simulation of vehicle operational rules by which to model traffic flow dynamics. However, these models fail to directly reveal the mechanism by which the uneven distribution characteristics affect congestion, which is not conducive to uncovering the fundamental factors that affect congestion evolution and general laws for predicting congestion. Therefore, this paper first constructs traffic fractal elements (TFE) based on the fractal characteristics exhibited by uneven traffic distribution. A local parameter namely the load duty ratio (LDR) is introduced to represent the uneven distribution level of TFE. By employing the fractal interpolation function (FIF), we model road traffic as the iterations of TFE. Then we define the fractal evolution rule and the load transfer rule to inscribe the influence relationship between LDR of TFE and global congestion, ultimately establishing a TFE-based congestion model. Furthermore, the proposed TFE-based model is validated using empirical data from AMAP and simulation data from a calibrated Cellular Automata (CA) model. Finally, by comparing the results with a CA-based congestion model, the LDR of TFE is confirmed to be a comprehensive and indicative parameter affecting congestion evolution. By simulating and fitting the congestion evolution data under different LDRs, our proposed model demonstrates its value in congestion prediction and analysis.

Non-stationary [formula omitted]-fractal functions and their dimensions in various function spaces

In this article, we study the novel concept of non-stationary iterated function systems (IFSs) introduced by Massopust in 2019. At first, using a sequence of different contractive operators, we construct non-stationary α-fractal functions on the space of all continuous functions. Next, we provide some elementary properties of the fractal operator associated with the non-stationary α-fractal functions. Further, we show that the proposed interpolant generalizes the existing stationary interpolant in the sense of IFS. For a class of functions defined on an interval, we derive conditions on the IFS parameters so that the corresponding non-stationary α-fractal functions are elements of some standard spaces like bounded variation space, convex Lipschitz space, and other function spaces. Finally, we discuss the dimensional analysis of the corresponding non-stationary α-fractal functions on these spaces.