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The current research studies a fractal Duffing oscillator in the presence of periodic force. To find an analytic solution for this oscillator, the aspects explained in the following are considered. First, we obtain an alternative unforced fractal fourthorder equation and then convert it into a continuous space. Therefore, the nonperturbative (NP) approach is used to calculate the analytic solution for the alternate equation in the second-order form after reducing its rank. It is seen that the analytical and numerical solutions agree very well. The computations reveal that for every value of the fraction parameter, the approximation and numerical solutions are identical. The present study gives reliability in the technique of reducing the order of differential equations. Furthermore, the required periodic solution is also obtained by Galerkin’s technique. In contrast to the traditional technique, which works to transform the variable and is valid only in the absence of external forces, if there is an external force, it leads to significant mathematical difficulties. The current technique works on the operator, which is simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties. [ABSTRACT FROM AUTHOR]

In recent years, there has been a considerable increase of interest in the sixth-generation (6G) communications. The advent of 6G wireless communications will facilitate a comprehensive upgrade and transformation of the industry, enabling higher data rates, lower latency, and ubiquitous connectivity. Research on microelectromechanical systems (MEMS) is a crucial element in the advancement of 6G wireless communications. In this paper, we examine the impact of geometric potential on MEMS within the context of fractal space, where a fractal geometric potential MEMS model is proposed and the critical conditions that govern the pull-in phenomenon are elucidated. Moreover, a periodic solution of the model is proposed and compared with other numerical methods. [ABSTRACT FROM AUTHOR]

Gronwall–Bellman-type inequalities provide a very effective way to investigate the qualitative and quantitative properties of solutions of nonlinear integral and differential equations. In recent years, local fractional calculus has attracted the attention of many researchers. In this paper, based on the basic knowledge of local fractional calculus and the method of proving inequality on the set of real numbers, some new Gronwall–Bellman inequalities are discussed and established on the fractal space. Our results are a powerful supplement to the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we give some further comments to the counterexample and the results of Bisht in [R.K. Bisht, Comment on: A new fixed point theorem in the fractal space, Indagationes Mathematicae, 29(2018), no.2, 819-823.]. [ABSTRACT FROM AUTHOR]

In this article, we examine the order of magnitude of the Yang-Fourier transforms for local fractional continuous functions on R and satisfiying certain Lipschitz conditions. Furthermore, using the analogue of the operator Steklov, we construct the generalized modulus of smoothness in the fractal space L 2 , α (R) and we use the Yang-Fourier transforms to prove the equivalence between K-functionals and modulus of smoothness. [ABSTRACT FROM AUTHOR]

The surface settlement during the tunneling process is becoming increasingly difficult to forecast as its surroundings become more and more erratic, and the maximal surface settlement raises risks posed suddenly by various uncertain factors. This paper proposes a novel approach to prediction of the surface settlement and analyzes the stability of tunnel construction. The dimensionless analysis and Buckingham’s π-theorem are adopted for this purpose, and some useful dimensionless quantities are found, which can be used to determine the surface settlement’s main properties. In this manner, the paper offers new ways of predicting surface settlement in various cases, and it sheds a new light on the tunnel’s design and safety monitoring. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We first obtain a new auxiliary identity by utilizing twice differentiable functions on a-type sets. Afterwards, two generalized Ostrowski type estimations for mappings whose second local fractional derivatives are bounded are derived, and special cases of these comprehensive inequalities are observed. Finally, some related applications such as inequalities including special means and generalized quadrature rules are presented. [ABSTRACT FROM AUTHOR]

The charge of a quantum particle is suggested to have a contribution from fractal potential energy which arises from the non differentiability of space-time along with imaginary mass. It has been suggested from scale relativity that quantum particle are subjected to a fractal space-time which predicts two valuedness in the velocity field. The velocity contains a divergent term which in turn depends on the fractal dimension of the particle. The divergent part has been extracted for hadron which are described as fractal objects with fractal dimension of 9/2 in the context of statistical model. A contribution from fractal potential has been suggested and an expression for the charge is extracted with imaginary fractal potential energy which is found to depend on the covariant of velocity field. It is suggested that the fractal potential manifests the charge of the quantum particles and closely related to the space-time of the quantum particle which is fractal. [ABSTRACT FROM AUTHOR]

In this paper, for the first time, the fractal–fractional short water wave model (FFSWWM) is used to simulate the propagation of seawater waves with unsmooth boundaries or in microgravity, where the conformable fractional derivative sense is adopted. A new and simple approach is presented to derive a variety of fractal solitary wave solutions of FFSWWM, which is called the sech function wave method. These obtained fractal solitary wave solutions are completely new and different from the existing literature. The fractal dynamical behavior of the gained fractal solitary wave solutions is presented by sketching some 2D and 3D graphics with different fractal dimensions and fractal parameters, which are very helpful for further study of the propagation of complex ocean waves. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function.

As a generalization of the convex mappings, the generalized s -type preinvex mappings are firstly introduced. Their meaningful properties are then investigated and the Hermite–Hadamard-type integral inequalities via the newly proposed mappings in fractal space are developed. In accordance with the newly proposed identity with three parameters, it is interesting to present certain integral inequalities with regard to the mappings whose first-order derivatives in absolute value belong to the generalized s -type preinvexity. As applications, on the basis of local fractional calculus, certain inequalities in view of numerical integration, -type special means, as well as moments of random variable, are acquired, respectively. [ABSTRACT FROM AUTHOR]

The thin film equation is very important in electroanalytical chemistry arising in diffusion–reaction process. Considering the effort of the surface morphology on the diffusion process, the thin film equation is modified by using the fractal derivative, and its fractal variational principle is successfully established according to the semi-inverse method and two-scale transform method. It is very helpful to study the structure of the approximate solution. [ABSTRACT FROM AUTHOR]

Fractal analysis is a totally new area of research based on local fractional calculus. It has interesting applications in various fields such as a complex graph, computer graphics, the music industry, picture compression and many more fields. In this paper, we present new variants of Hadamard–Mercer-type inequalities on fractal sets ℝ α (0 < α ≤ 1) by employing generalized convex function. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hadamard–Mercer-type integral inequalities for local differentiable generalized convex functions on real linear fractal space. Finally, we give applications for probability density functions and compute new generalized means. [ABSTRACT FROM AUTHOR]

The most notable inequality pertaining convex functions is Jensen's inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen's inequality called as Jensen–Mercer's inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. The purpose of this paper is to establish a generalized Jensen–Mercer inequality for a generalized convex function on a real linear fractal set ℝ α (0 < α ≤ 1). Further, we also demonstrate some generalized Jensen–Mercer-type inequalities by employing local fractional calculus. Lastly, some applications related to Jensen–Mercer inequality and α -type special means are given. The present approach is efficient, reliable, and may motivate further research in this area. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function. [ABSTRACT FROM AUTHOR]

As an important electronic device, filter is applied to all kinds of electronic products. In this paper, a new ℘-order R-L High-pass filter (HPF) modeled by the local fractional derivative (LFD) is proposed for the first time. With the help of the local fractional Laplace transform (LFLT), we obtain the non-differentiable(ND) transfer function, and present the expressions of ND amplitude-frequency characteristic (AFC) and ND phase-frequency characteristics (PFC). The corresponding parameters and properties of the ℘-order R-L HPF are also studied. What’s interesting is that the ℘-order R-L HPF becomesthe ordinary one in the exceptional case at ℘ = 1. The obtained results in this paper reveal the sufficiency of the local fractional derivative for analyzing the circuit systems in fractal space.

In this work, space-time fractal model about nonlinear KdV-Burgers–Kuramoto (NKBK) equation which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters has been put forward through coupling fractional complex transform (FCT) via local fractional derivative (LFD) for the first time. These measures are considered in the sense of local derivative operators. Analytical approximate solutions of the model are obtained by local fractional reduced differential transform method (LFRDTM). The obtained results related to physical phenomenon in Cantorian time-space reveal that the suggested project is easy to use and the calculation is more precise. The graphical representation of special solution of LFNKBK yields interesting and better physical consequences of NKBK with LFD. [ABSTRACT FROM AUTHOR]

Pull-in instability occurs in a micro-electromechanical system, and it greatly hinders its normal operation. A fractal modification is suggested to make the system stable in all operation period. A fractal model is established using a fractal derivative, and the results show that by suitable fabrication of the micro-electromechanical system device, the pull-in instability can be converted into a novel state of pull-in stability. [ABSTRACT FROM AUTHOR]

Pull-in instability was an important phenomenon in microelectromechanical systems (MEMS). In the past, MEMS were usually assumed to work in an ideal environment. But in the real circumstances, MEMS often work in dust-filled air, which is equivalent to working in porous media, that's mean fractal space. In this paper, we studied MEMS in fractal space and established the corresponding model. At the same time, we can control the occurrence time and stable time of pull-in by adjusting the value of the fractal index, and obtain a stable pull-in phenomenon.

He's fractal calculus is a powerful and effective tool to dealing with natural phenomena in a fractal space. In this paper, we study the fractal Korteweg-de Vries equation with He's fractal derivative. We first adopt the two-scale transform method to convert the fractal Korteweg-de Vries equation into its traditional partner in a continuous space. Finally, we successfully use He's variational iteration method to obtain its approximate analytical solution. [ABSTRACT FROM AUTHOR]

Fractal calculus is an excellent tool to explaining natural phenomena in porous media. In this paper, we first give a simple introduction on He's fractal derivative, and then it is used to establish a model for thermal conduction of silkworm cocoon reveal its biomechanism. The theoretical results obtained in this paper are helpful for the biomimetic design. [ABSTRACT FROM AUTHOR]

The fractal derivative is adopted to describe the non-linear fractional wave equation in a fractal space. A variational principle is successfully established by the semi-inverse method. The two-scale method and He's exp-function are used to solve the equation, and a good result is obtained. [ABSTRACT FROM AUTHOR]

Pull-in instability, as an inherent nonlinear problem, continues to become an increasingly important and interesting topic in the design of electrostatic Nano/Micro-electromechanical systems (N/MEMS) devices. Generally, the pull-in instability was studied in a continuous space, but when the electronic devices work in a porous medium, they need to be analyzed in a fractal partner. In this paper, we establish a fractal model for N/MEMS, and find a pull-in stability plateau, which can be controlled by the porous structure, and the pull-in instability can be finally converted to a stable condition. As a result, the pull-in instability can be completely eliminated, realizing the transformation of pull-in instability into pull-in stability. [ABSTRACT FROM AUTHOR]

We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space. [ABSTRACT FROM AUTHOR]

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method. [ABSTRACT FROM AUTHOR]

As an important electronic device, filter is applied to all kinds of electronic products. In this paper, a new ℘ -order R-L High-pass filter (HPF) modeled by the local fractional derivative (LFD) is proposed for the first time. With the help of the local fractional Laplace transform (LFLT), we obtain the non-differentiable(ND) transfer function, and present the expressions of ND amplitude-frequency characteristic (AFC) and ND phase-frequency characteristics (PFC). The corresponding parameters and properties of the ℘ -order R-L HPF are also studied. What's interesting is that the ℘ -order R-L HPF becomesthe ordinary one in the exceptional case at ℘ = 1. The obtained results in this paper reveal the sufficiency of the local fractional derivative for analyzing the circuit systems in fractal space. [ABSTRACT FROM AUTHOR]

In this paper, two inequalities of Ostrowski type are established for generalized strongly m-convex functions on the fractal sets Rα (0 < α ≤ 1) of real line numbers. Special cases of our main results are discussed and some applications in numerical integration and for generalized means are given. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain a generalized integral identity for twice local fractional differentiable mappings on fractal sets Rα (0 < α ≤ 1) of real line numbers. Then, using twice local fractional differentiable mappings that are in absolute value at certain powers generalized strongly convex, we obtain some new estimates on generalization of trapezium-like inequalities. We also discuss some new special cases which can be deduced from our main results. [ABSTRACT FROM AUTHOR]

A fractal modification of the telegraph equation with fractal derivatives is given, and its variational principle is established by the semi-inverse method. The two-scale transform method and He's homotopy perturbation method are successfully adopted to solve the fractal equation. [ABSTRACT FROM AUTHOR]

The local fractional derivative (LFD) has gained much interest recently in the field of electrical circuits. This paper proposes a non-differentiable (ND) model of high-pass filter described by the LFD, where the ND transfer function is obtained with the help of the local fractional Laplace transform, and its parameters and properties are studied. The obtained results reveal the sufficiency of the LFD for analyzing circuit systems in fractal space. [ABSTRACT FROM AUTHOR]

Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed. [ABSTRACT FROM AUTHOR]

The Hutchinson-Barnsley (HB) operators are studied for multi iterated function system in a general setting. Some existence and uniqueness results are obtained in fuzzy fractal and multi fuzzy fractal spaces. From them, we obtain our results, which generalize some recent results in the literature.

We offer in this review a description of the vacuum energy of self-similar systems. We describe two views of setting self-similar structures and point out the main differences. A review of the authors’ work on the subject is presented, where they treat the self-similar system as a many-object problem embedded in a regular smooth manifold. Focused on Dirichlet boundary conditions, we report a systematic way of calculating the Casimir energy of self-similar bodies where the knowledge of the quantum vacuum energy of the single building block element is assumed and in fact already known. A fundamental property that allows us to proceed with our method is the dependence of the energy on a geometrical parameter that makes it possible to establish the scaling property of self-similar systems. Several examples are given. We also describe the situation, shown by other authors, where the embedded space is a fractal space itself, having fractal dimension. A fractal space does not hold properties that are rather common in regular spaces like the tangent space. We refer to other authors who explain how some self-similar configurations “do not have any smooth structures and one cannot define differential operators on them directly”. This gives rise to important differences in the behavior of the vacuum.

Local fractional calculus is used to derive fractional partner of the KdV–Burgers–Kuramoto equation, its physical understanding is elucidated and its solution properties are revealed. Adomian's decomposition method (ADM) and the variational iteration method (VIM) are applied to obtain its approximate analytical solution, the results have both theoretical importance and practical applications. [ABSTRACT FROM AUTHOR]

A same phenomenon can lead to debating laws when observed on different scales, two-scale observation is needed for a discontinuous problem, the large scale matches with the continuum assumption, while the smaller one is to reveal the discontinuous properties. The basic properties of two-scale fractal are discussed in a layman way. [ABSTRACT FROM AUTHOR]

In this study, we establish new integral inequalities of the Hermite−Hadamard type for s-convexity via the Katugampola fractional integral. This generalizes the Hadamard fractional integrals and Riemann−Liouville into a single form. We show that the new integral inequalities of Hermite−Hadamard type can be obtained via the Riemann−Liouville fractional integral. Finally, we give some applications to special means.

Wang et al. established successfully a variational principle in a fractal space by the semi-inverse method. This paper argues that in the fractal space, time should be also considered as a fractal, and Wang's variational principle is extended to fractal space and fractal time. [ABSTRACT FROM AUTHOR]

While cyberpunk is often described as a dystopian genre, the paper argues that it should be seen rather as a post-utopian one. The crucial difference between the two resides in the nature of the historical imagination reflected in their respective narrative and thematic conventions. While dystopia and utopia (structurally the same genre) reflect a teleological vision of history, in which the future is radically different from the present, post-utopia corresponds to what many scholars, from Fredric Jameson and Francis Fukuyama to David Bell, have diagnosed as the "end of history" or rather, the end of historical teleology. Post-utopia reflects the vision of the "broad present", in which the future and the past bleed into, and contaminate, the experience of "now". From its emergence in the 1980s and until today, cyberpunk has progressively succumbed to the post-utopian sensibility, as its earlier utopian/dystopian potential has been diluted by nostalgia, repetition and recycling. By analyzing the chronotope of cyberpunk, the paper argues that the genre's articulation of time and space is inflected by the general post-utopian mood of global capitalism. The texts addressed include both novels (William Gibson's Neuromancer, Neal Stephenson's Snow Crash and Matthew Mather's Atopia) and movies (Blade Runner, Blade Runner 2049 and Ex Machina). [ABSTRACT FROM AUTHOR]

This paper proposes a new method called the local fractional Fourier sine transform to solve fractional differential equations on a fractal space. The method takes full advantages of the Yang-Fourier transform, the local fractional Fourier cosine, and sine transforms. A 1-D local fractional heat transfer equation is used as an example to reveal the merits of the new technology, and the example can be used as a paradigm for other applications. [ABSTRACT FROM AUTHOR]

In this article, we establish new Hermite−Hadamard-type inequalities via Riemann−Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a generalized s-convexity property of ψ . We also prove certain inequalities involving Riemann−Liouville integrals of a function ψ provided that the absolute value of the first or second order derivative of ψ possesses an appropriate fractal s-convexity property.

In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results obtained by Chifu et al. (2014) and N.A. Secelean (2015) and generalize the results of Nazir et al. (2016) by using the assumptions imposed by Dung et al. (2017) to the case of ciric type generalized multi-iterated function system (CGMIFS) composed of ciric type generalized multivalued G-contractions defined on multifractal space C ( U ) in the framework of a Hausdorff b-metric space, where U = U 1 × U 2 × ⋯ × U N , N being a finite natural number. As an application of our study, we derive collage theorem which can be used to construct general fractals and to solve inverse problem in Hausdorff b-metric spaces which are more general spaces than Hausdorff metric spaces.