Fractal Version Research Articles

Fractal version of average Fermat distance on some small-world hierarchical networks

Fermat–Wiener index based on topological Wiener index is the total sum of Fermat distance over all the triplets for vertices. In this paper, we construct a class of hierarchical graphs based on hierarchical product generalized from Cartesian product. We study some critical properties of the hierarchical networks by investigating its topological indices. Applying the finite pattern method, we analytically deduce the dominant term of average Fermat distance and obtain its asymptotic formula, which implies small-world property. We finally exhibit a close connection between Fermat–Wiener index and related graph invariants like average geodesic distance, Wiener index and eigenvalues of Laplacian matrix.

Some basic theorems and formulas for building fractal nonlinear wave models

As a tool to deal with the phenomenon in layered or porous media, many theoretical problems need to be improved urgently. In this paper, the definitions of two kinds of fractal derivatives based on scale transformation are developed from the perspective of mathematics, and the four operation rules, chain rules, Rolle theorem, mean value theorem, Taylor formula, variable upper limit integral derivation rule, definite integral basic principle, partial integral formula and double partial integral formula of the fractal version are given. Better still, we derive the fractal Euler-Lagrangian equation by using the variational method, and demonstrate the steps and effectiveness of deriving the fractal nonlinear wave equation based on the fractal Euler-Lagrangian equation through the example of the functionally graded beam.

FRACTAL VERSION OF ZAGREB ECCENTRICITY INDEX

In this paper, we introduce the fractal version of Zagreb eccentricity index which is a singular integral of the eccentricity squared in terms of self-similar measure. Based on the geometric characteristics of the farthest point, we calculate the fractal version of Zagreb eccentricity index on the (3,1) flower fractal to be [Formula: see text] by using the technique of finite pattern.

Fractal version of hyper-Wiener index

We propose the Fractal version of the hyper-Wiener index and we calculate the hyper-Wiener index on the Sierpiński gasket, a classical self-similar fractal, by developing the notion of pattern introduced by S. Wang et al (2017).

Motion of Poly-vectors in Fractal Clifford Spaces

Fractal Clifford Spaces (FCS) may be considered as a challenging approach to the unification of micro-physics and macro-physics. Trajectories of these manifolds are described by different poly-vectors describing paths and their deviation equations for not only to test particles but also for charged, rotating and rotating charged objects. Due to relaxing the condition of differentiability for all extended objects in the manifold. In this approach, a specialized Bazanski Lagrangian will be discussed to describe the fractal version of extended objects in a fractal space-time.

Solving fractal differential equations via fractal Laplace transforms

AbstractThe intention of this study is to investigate the fractal version of both one-term and three-term fractal differential equations. The fractal Laplace transform of the local derivative and the non-local fractal Caputo derivative is applied to investigate the given models. The analogues of both the Wright function with its related definitions in fractal calculus and the convolution theorem in fractal calculus are proposed. All results in this paper have been obtained by applying certain tools such as the general Wright and Mittag-Leffler functions of three parameters and the convolution theorem in the sense of the fractal calculus. Moreover, a comparative analysis is conducted by solving the governing equation in the senses of the standard version and fractal calculus. It is obvious that whenα=γ=β=1{\alpha=\gamma=\beta=1}, we obtain the same results as in the standard version.

The fractal version of the bond‐based peridynamics method

AbstractThis article presents a new formulation of bond‐based peridynamic theory (PD) encompassing material texture heterogeneity through fractal geometry concepts and transforming all relevant variables from Euclidean space to fractal space. The heterogeneity incorporation is realized through the introduction of a linear transformation fractal coefficient (LTFC). By considering a mean value for all LTFCs of all spatial coordinates. For example, x, y, z, and using the relationship between fractal dimension and the Hurst exponent . We present a fractal version of the bond‐based PD method that the equation of motion is conformed to fractal space. Validation of the formulation is accomplished in two steps: first, the effect of irregular mass distribution is characterized by the fractal dimension of the fracture surface roughness through experimental results of a series of single‐edge notched tensile tests of a ductile material. In the next step, the proposed formulation is calibrated with the fractal dimension obtained experimentally. The results are compared with the classical bond‐based PD formulation and the finite element analysis results. The obtained results in this article indicate that the irregular mass distribution of material can be introduced into particle‐based numerical methods using an adequate fractal exponent. The fractal version provides a good approximation of the fracture behavior of materials.

A Fractal Version of a Bivariate Hermite Polynomial Interpolation

A Fractal Version of a Bivariate Hermite Polynomial Interpolation

Shape preserving rational cubic trigonometric fractal interpolation functions

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C1-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form pi(θ)/qi(θ), where pi(θ) and qi(θ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C1-RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

Approximation by shape preserving fractal functions with variable scalings

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions $$f^{\varvec{\alpha }}\in {\mathcal {C}}(I)$$ , where $$\varvec{\alpha }$$ is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the $$\varvec{\alpha }$$ -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the $$\varvec{\alpha }$$ -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the $$\varvec{\alpha }$$ -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Muntz theorem in $${\mathcal {C}}[0,1]$$ .

Fractal Higher-order Dispersions Model and Its Fractal Variational Principle Arising in the Field of Physcial Process

This paper introduces the fractal version of the higher-order dispersion model for the construction of novel soliton solutions through fractal variational technology. Higher-order dispersion model theoretical study of the soliton propagation dynamics is known in the absence of self-phase modulation. In the context of negligibly small group velocity dispersion, this model involves higher-order spatio-temporal dispersion and can be a core component of the telecommunications industry. Using the variational approach, the model effectively produces bright and dark soliton solutions. Essential novel conditions guaranteeing the existence of suitable solitons have been developed. The 3D, 2D and contour graphs of the computed effects are seen in the collection of the relevant parameter values. This study shows the significance and immense latency of variational technologies to the derivative nonlinear Schrödinger equation (DNLSE).

Fractals and discrete dynamics associated to prime numbers

Abstract On this work, it is shown that fractal theory can be applied in number theory to analyze the distribution of prime numbers. Based on the similarities between the distribution of prime numbers’ powers for each natural number and generalized Cantor’s sets, some symmetric properties are analyzed and used to propose a set of discrete dynamics that allow visualizing the recursive and symmetric properties of prime numbers, which leads to the proposition of an approximation of a fractal version of the distribution function of prime numbers. Then, with this set of discrete dynamics, it is proposed a prime numbers’ sieve algorithm that seems to describe the growth of a fractal and its fractal dimension is computed, suggesting that the fractal and chaotic behavior on the distribution of prime numbers emerges from: 1) the increase of the memory or dimension of the discrete dynamics, that introduces new frequencies of oscillation each stage and 2) the symmetry break due to the exponential growth of the length of the symmetries with respect to the domain of applicability of each discrete dynamics. Therefore, although chaotic, the distribution of prime numbers has a deterministic chaotic behavior, with symmetries and harmonics, that follows the classical path to chaos, i.e. from periodic to quasi periodic to chaos.

Parameter Identification for a Class of Bivariate Fractal Interpolation Functions and Constrained Approximation

The current article intends to study some elementary constrained approximation aspects of the bivariate fractal functions. To this end, firstly the construction of bivariate fractal interpolation functions available in the literature is revisited with a focus to obtain a parameterized family of fractal functions corresponding to a prescribed bivariate continuous function on a rectangular region in The parameters are chosen appropriately so that the corresponding fractal version preserves some properties inherent in the original function. We apply these results to invite the notion of bivariate fractal functions to the field of constrained approximation. Furthermore, we attempt to investigate the box dimension and Hausdorff dimension of the graph of the constructed bivariate fractal function.

Platonic Quantum Set Theory Proposal and Fractal-Cantorian Heterotic Kaluza- Klein Spacetime

The four basic building blocks of the cosmos hypothesized by the Athenian philosopher Plato and his corresponding theory are replaced by the backbone golden mean based scaling pertinent to each of the four kinds of blocks. In the course of doing this we stretch and generalize Plato’s philosophical ideas to ultimately find out that it is essentially the deep philosophical origin of the golden mean number system of E-infinity Cantorian spacetime theory of high energy physics and quantum cosmology. Subsequently we use this new platonic form of E-infinity quantum set theory to uncover a remarkably rich Kaluza-Klein fractal version of Gross et al’s ingenious Heterotic superstring theory.

More General Fractal Functions on the Sphere

In this article, a family of continuous functions on the unit sphere $$S\subseteq \mathbb {R}^{3}$$ generalizing the spherical harmonics, is considered. Using fractal methodology, the fractal version of this family of continuous functions on the sphere S is constructed. To do this, an iterated function system (IFS) and a linear bounded operator that maps classical functions to its fractal analogues is defined. Some approximation properties of fractal functions on the sphere are investigated. Restricting the scale vector involved in the IFS, a fractal Hilbert basis is established for the functions on the sphere.

Viscosity of a Melt of Polymer/Carbon Nanotube Nanocomposites. An Analogy with a Polymer Solution

A melt of polymer/carbon nanotube nanocomposites is considered as an analog of a polymer solution within the scope of fractal physical chemistry, in which a polymer matrix is simulated as a solvent and ring-like structures of carbon nanotubes are simulated as macromolecular coils of polymer. Further application of the fractal version of Mark–Kuhn–Houwink, which was developed to describe the behavior of polymer solutions, shows good agreement between the theoretical and experimental values of the viscosity of a melt of polymer/carbon nanotube nanocomposites.

Kantorovich-Bernstein α-fractal function in 𝓛P spaces

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space 𝓛p(I), 1 ≤ p ≤ ∞. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in 𝓛p(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal 𝓛p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in 𝓛p spaces is proven. Further, we derive the fractal analogues of some 𝓛p(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein α-fractal function is developed.

Bernstein Fractal Trigonometric Approximation

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.

Textural characterization of chars using fractal analysis of N2 and CO2 adsorption

The application of fractal analysis to nitrogen and carbon dioxide adsorption isotherms is used together with other textural analysis techniques with the aim of obtaining a better knowledge of the phenomena leading to char formation. This is an important point since chars are key intermediate materials during the manufacturing process of active carbons. The materials under study were series of chars obtained from a high-volatile A bituminous coal oxidized in air at 543 and 473K for different periods of time extending up to 42days. The fractal characteristics of the chars were obtained from the nitrogen and carbon dioxide adsorption isotherms by using the methods proposed by Neimark, Wang and Li, and the fractal version of the Frenkel-Halsey-Hill method. The latter approach was found suitable for the high pressure interval of N2 isotherms. Evolution of the fractal properties in the series of chars is compared with that corresponding to the series of precursor coals. Changes in pore network during carbonization depend on the severity of the air preoxidation of the coal. This happens since preoxidation conditions other properties and processes such as the texture of the precursor coals, the release of the volatile matter as well as the plastic behaviour of coals. These three points were found to be the key factors that determine the fractal characteristics of char. The role of the pore characteristics of the coal on the textural features of the obtained chars is boosted if coal preoxidation is carried out at the more elevated temperature, 543K. The role that the phenomena occurring throughout the carbonization of oxidized coals have on the textural characteristics of the chars obtained is explained considering fractal features together with other textural properties. The results of this work are compared with previous fractal studies based on mercury porosimetry data and conventional textural techniques.

H\xf6lder norm estimate for the fractal Hilbert transform in Douglis analysis

The main goal of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Douglis analysis context acting from Hölder spaces of Douglis algebra valued functions defined on h-summable curves.