Narrow Results

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.

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.

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.

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.

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.

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.

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.

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 ${ℝ}^{\alpha }$ ($0<\alpha \le 1)$. Further, we also demonstrate some generalized Jensen–Mercer-type inequalities by employing local fractional calculus. Lastly, some applications related to Jensen–Mercer inequality and $\alpha$-type special means are given. The present approach is efficient, reliable, and may motivate further research in this area.

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 ${ℝ}^{\alpha }$ ($0<\alpha \le 1)$ 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.

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.

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, $𝜀$-type special means, as well as moments of random variable, are acquired, respectively.

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.

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.

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.

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.

Compact derivation of mathematical equations similar to those of quantum and classical mechanics is given on the base of fractal decomposition of a three-dimensional space. In physical units the equations become Shrödinger and Hamilton-Jacobi equations, the wave function of a free particle associated with a virtual ring. Locally uniform motion of the ring in the physical space provides an original helix (or regular cylindrical spring) model of a relativistic theory equivalent in results with special relativity, the free particle's relativistic Lagrangian emerging automatically. Irregular spring model generates theory similar to general relativity.