Fractal and Fractional, Volume 6, Issue 7

In this paper, the Carleman matrix is constructed, and based on it we found explicitly a regularized solution of the Cauchy problem for the matrix factorization of the Helmholtz equation in a multidimensional unbounded domain in ${\mathbb{R}}^m,(m=2k,k\ge 2)$. The...

In this study, we will discuss the engineering construction of a special sixth generation (6G) antenna, based on the fractal called Minkowski’s loop. The antenna has the shape of this known fractal, set at four iterations, to obtain maximum per...

Fractional cumulative residual entropy is a powerful tool for the analysis of complex systems. In this paper, we first provide some properties of fractional cumulative residual entropy (FCRE). Secondly, we generate cumulative residual entropy (CRE) t...

In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for the numerical solution of the time fractional Black–Scholes model. In this way, the operational matrix of the fractional derivative based on the tw...

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result...

Let f and g be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of $f+g$ by classifying all the subsequences tending to zero into different sets. Using this method, we...

In this research, we look at the Julia set patterns that are linked to the entire transcendental function $f\left(z\right)=a{e}^{z^n}+bz+c$, where $a,b,c\in \mathbb{C}$ and $n\ge 2$, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit st...

The robust process in memorising the Quran is expected to cause neuroplasticity changes in the brain. To date, the analysis of neuroplasticity is limited in binary images because greyscale analysis requires the usage of more robust processing techniq...

In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to...

The prediction of the stock price index is a challenge even with advanced deep-learning technology. As a result, the analysis of volatility, which has been widely studied in traditional finance, has attracted attention among researchers. This paper p...

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solv...

The paper applies fractal theory to the structure of fractal coal pores and calculates the fractal dimension and integrated fractal dimension for each pore section >100 nm, 100 nm > d > 5.25 nm, and <2 nm. In the experiment, we performed...

Abrasion damage is a typical hydraulic structure failure and considerably impacts the durability of buildings. In severe circumstances, it can even prevent hydraulic structures from being used and operated normally. Thus, it is essential to research...

Moving from the study of plasmonic materials with relaxation, in this work we propose a fractional Abraham–Lorentz-like model of the complex permittivity of conductor media. This model extends the Ciancio–Kluitenberg, based on the Mazur&n...

Fractional calculus (FC) generalizes the concepts of derivative and integral to non-integer orders [...]

Extensive research has been conducted on the scaling fractal fractor using various structures. The development of high-resolution emulator circuits to achieve a variable-order scaling fractal fractor with high resolution is a major area of interest....

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established...

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta me...

This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the...

Many physical aspects emerging from the local structure and micromotions of liquid particles can be studied by utilizing the governing model of micropolar liquid. It has the ability to explain the behavior of a wide range of real fluids, including po...

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scale...

Breakage of particles has a great influence on the particle size distribution (PSD) and the associated mechanical behavior of ballast under train loads. A discrete element method (DEM) simulation of triaxial testing under monotonic loading was carrie...

In this paper, based on the $L2$-$1_{\sigma }$ scheme and nonconforming $E{Q}_1^{rot}$ finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel an...

This brief investigates the Mittag–Leffler formation bounded control problem for second-order fractional multi-agent systems (FMASs), where the dynamical nodes of followers are modeled to satisfy quadratic (QUAD) condition. Firstly, under the u...

Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a...

This paper reports the new advances in biological fractal dynamics. The following contents are included: (1) physical (or functional) fractal spaces abstracted from biological materials, biological structures and biological motions; (2) fractal opera...

This paper presents a numerical technique to approximate the Rayleigh–Stokes model for a generalised Maxwell fluid formulated in the Riemann–Liouville sense. The proposed method consists of two stages. First, the time discretization of th...

We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as $(p,h)$ fractional type inequality and many others, as the $(p,h)$-c...

In this paper, we study the passive problem of uncertain fractional-order neural networks (UFONNs) with time-varying delays. First, we give a sufficient condition for the asymptotic stability of UFONNs with bounded time-varying delays by using the fr...

The integer-order LCL (IOLCL) filter has excellent high-frequency harmonic attenuation capability but suffers from resonance, which causes system instability in grid-connected inverter applications. This paper studied a class of resonance-free fracti...

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solut...

The article considers the fractional Poisson process as a mathematical model of deformation activity in a seismically active region. The dislocation approach is used to describe five modes of the deformation process. The change in modes is determined...

In this paper, the partial topology identification of stochastic multi-group models with multiple dispersals is investigated. Based on adaptive pinning control and a graph-theoretic method, some sufficient criteria about partial topology identificati...

Fractional-order chaotic systems are widely used in the field of encryption because of its initial value sensitivity and historical memory. In this paper, the fractional-order definition of Caputo is introduced based on a nonideal flux-controlled mem...

In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy $1<\alpha <2$ and $0<\beta <1$. Numerical computing schemes of...

This study investigates the problem of finite-time boundedness of a class of neural networks of Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to en...

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties...

In this paper, we investigate the problem of sliding mode control for singular fractional-order systems that have matched uncertainties. We design an innovative integral sliding mode function and controller based on the normalizable condition. A stri...

There are many applications of optimization and nonlinear analysis in various fields of basic science, engineering, and natural phenomena [...]

In this paper, we propose a one-dimensional (1D) multifractal sign retention detrending fluctuation analysis algorithm (MF-S-DFA). The proposed method is based on conventional multifractal detrending fluctuation analysis (MF-DFA). As negative values...

A primary aim of this study is to examine and simulate a fractional Coronavirus disease model by providing an efficient method for solving numerically this important model. In the Liouville-Caputo sense, the examined model consists of five fractional...

In this paper, we establish the existence and uniqueness of a strong solution to a fractional mean field games system with non-separable Hamiltonians, where the fractional exponent $\sigma \in (\frac{1}{2},1)$. Our result is new for fractional mean field game...

By using Lucas polynomials, we define a new subclass of analytic bi-univalent functions, class $\mathsf{\Sigma }$, in the open unit disc with respect to symmetric conjugate points connected with the combination Binomial series and Babalola operator. The bounds on the...

Discrimination of various microseismic (MS) events induced by blasting and mining in coal mines is significant for the evaluation and forecasting of rock bursts. In this paper, multifractal and moment tensor inversion methods were used to investigate...

In this current manuscript, we study a fractional-order modificatory hybrid optical model (FOMHO model). Experiments manifest that under appropriate parameter conditions, the fractional-order modificatory hybrid optical model will generate chaotic be...

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space ${\mathbb{R}}^m$. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of...

The functional implications of substances, such as retardation and relaxation, can be studied for magnetized diffusion coefficient based on the relative increase throughout magnetization is a well-known realization. In this context, we have explored...

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-o...

A framework of distributed interval observers is introduced for fractional-order multiagent systems in the presence of nonlinearity. First, a frame was designed to construct the upper and lower bounds of the system state. By using monotone system the...

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this c...