This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite

In this paper, we propose a new fractional operator which is based on the weight function for Atangana–Baleanu (𝒜ℬ)-fractional operators. A motivating characteristic is the generalization of classical variants within the weighted 𝒜ℬ-fractional integral. We aim to establish Minkowski and reverse Hölder inequalities by employing weighted 𝒜ℬ-fractional integral. The consequences demonstrate that the

In this research, the analytical and numerical solutions of the fractional nonlinear space-time Phi-four model are investigated by employing two systematic schemes and the B-spline schemes. A new fractional operator definition is applied to this model to convert the model from its fractional formula to an integer-order nonlinear ordinary differential equation. The considered model is of major interest

In this paper, a novel fractional-order control strategy for the SCARA robot is developed. The proposed control is composed of PI𝜗 and a fractional-order passivity-based adaptive controller, based on the Caputo–Fabrizio and Atangana–Baleanu derivatives, respectively; both controls are robust to external disturbances and change in the desired trajectory and effectively enhance the performance of robot

Let A=n00m, where n,m≥3 are integers and 𝒟={0,1,…,n−1}×{0,1,…,m−1} be a digit set. Then the pair (A,𝒟) generates a fractal set T:=T(A,𝒟) satisfying AT=T+𝒟 which is a unit square. However, if we remove one digit from 𝒟, then the structure of T will become very interesting. A well-known example is the Sierpinski carpet. In this paper, we study the resulting self-affine sets of moving a digit in

Permeability is one of the most important parameters for accurately predicting water flow in reservoirs and quantifying underground water inrush into coal mines. This study developed a predictive permeability model by considering the microstructural parameters and tortuosity effects of low-permeability sandstone. The model incorporates the fractal geometry theory, Darcy’s law, and Poiseuille equation

The paper deals with the existence of three different weak solutions of p -Laplacian fractional for an overdetermined nonlinear fractional partial Fredholm–Volterra integro-differential system by using variational methods combined with a critical point theorem due to Bonanno and Marano.

In this paper, our main objective is to develop the conditions that assure the existence of solution to a system of boundary value problems (BVPs) of sequential hybrid fractional differential equations (SHFDEs). The problem is considered under the nonlinear boundary conditions. Nonlinear functions involved in the considered system of SHFDEs are continuous and satisfy the growth conditions. We convert

In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense. The

In this paper, we prove the existence, uniqueness and various kinds of Ulam stability for fractional order coupled systems with fractional order boundary conditions involving Riemann–Liouville fractional derivatives. The standard fixed point theorem like Leray–Schauder alternative and Banach contraction are applied to establish the existence theory and uniqueness. Furthermore, we build sufficient conditions

Supply chain finance is a new financing model tailored for small and medium-sized enterprises, which integrates capital flow into supply chain management, providing commercial trade capital services for enterprises in all aspects of the supply chain and providing new loan financing services for vulnerable enterprises in the supply chain. Fractal originally is a general term for a graph, structure or

Traditional interpolation algorithms often blur the edges of the target image due to low-pass filtering effects, making it difficult to obtain satisfactory visual effects. Especially when the reduction ratio becomes small, the phenomenon of jagged edges and partial information loss will occur. In order to obtain better image scaling quality, an adaptive image scaling algorithm based on continuous fraction

Users use the network more and more frequently, and more and more data is published on the network. Therefore, how to find, organize, and use the useful information behind these massive data through effective means, and analyze user intentions is a huge challenge. There are many time series problems in user intentions. Time series have complex characteristics such as randomness and multi-scale variability

The contradiction between the supply and demand of water resources is becoming increasingly prominent, whose main reason is the eutrophication of rivers and lakes. However, limited and inaccurate data makes it impossible to establish a precise model to successfully predict eutrophication levels. Moreover, it is incompetent to distinguish the degree of eutrophication status of lakes by manual calculation

For a long time, the mismatch between material flow and capital flow in the supply chain operation management practice is very prominent, which has led to the inefficiency of supply chain operation and hindered the exertion of supply chain’s advantages. In the field of supply chain management theory research, the information completeness in capital market has long been a hypothetical premise, which

The fractal dimension of fractional Brownian motion can effectively describe random sets, reflecting the regularity implicit in complex random sets. Data mining algorithms based on fractal theory usually follow the calculation of the fractal dimension of fractional Brownian motion. However, the existing fractal dimension calculation methods of fractal Brownian motion have high time complexity and space

Fractal analysis of time series is an important tool for describing complex systems and solving nonlinear problems based on time series datebases, of which the core is believed that the geometric bodies composed of various parts of the system have self-similarity and scale invariance. Supply chain management uses integrated thinking, from the perspective of system, to design, plan and control the logistics

The RL and RC circuits are analyzed in this research paper. The classical model of these circuits is generalized using the modern concept of fractional derivative with Mittag-Leffler function in its kernel. The fractional differential equations are solved for exact solutions using the Laplace transform technique and the inverse transformation. The obtained solutions are plotted and presented in tables

It has become vital to effectively characterize the self-similar and regular patterns in time series marked by short-term and long-term memory in various fields in the ever-changing and complex global landscape. Within this framework, attempting to find solutions with adaptive mathematical models emerges as a major endeavor in economics whose complex systems and structures are generally volatile, vulnerable

In this paper, stability results of central concern for control theory are given for finite-dimensional linear and nonlinear local fractional or fractal differential systems. The main purpose of this paper is to provide some results on stability and asymptotic stability of conformable order systems, together with some illustrating examples.

In this research work, we discuss an approximation techniques for boundary value problems (BVPs) of differential equations having fractional order (FODE). We avoid the method from discretization of data by applying polynomials of Laguerre and developed some matrices of operational types for the obtained numerical solution. By applying the operational matrices, the given problem is converted to some

In the modeling of dynamical problems the fractional order integro-differential equations (IDEs) are very common in science and engineering. The scientists are developing different aspects of these models. The existence of solutions, stability analysis and numerical simulations are the most commonly studied aspects. There is no paper in literature describing the Hyers–Ulam stability (HU-stability)

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique

In this paper, the nonlinear space–time fractal–fractional advection–diffusion–reaction equation is introduced and a highly accurate methodology is presented for its numerical solution. In the time direction, the fractal–fractional derivative in the Atangana–Riemann–Liouville concept is utilized whereas the fractional derivatives in the Caputo and Atangana–Baleanu–Caputo senses are mutually used in

In this research work, we investigate the existence of solutions for a class of nonlinear boundary value problems for fractional-order differential inclusion with respect to another function. Endpoint theorem for φ-weak contractive maps is the main tool in determining our results. An example is presented in aim to illustrate the results.

In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and anti-periodic-type boundary conditions. Also, we investigate the Hyres–Ulam (HU) stability of solutions. The existence result is derived by applying Krasnoselskii’s fixed point theorem and

The primary goal of this study is to define the weighted fractional operators on some spaces. We first prove that the weighted integrals are bounded in certain spaces. Afterwards, we discuss the weighted fractional derivatives defined on absolute continuous-like spaces. At the end, we present a modified Laplace transform that can be applied perfectly to such operators.

The aim of the present paper is to state a simplified nonlinear mathematical model to describe the dynamics of the novel coronavirus (COVID-19). The design of the mathematical model is described in terms of four categories susceptible (S), infected (I), treatment (T) and recovered (R), i.e. SITR model with fractals parameters. These days there are big controversy on if is needed to apply confinement

In this paper, the newly proposed concept of the generalized proportional fractional integral operator with respect to another function Φ has been utilized to generate integral inequalities using convex function. This new concept will have the option to reduce self-similitudes in the fractional attractors under investigation. We discuss the implications and other consequences of the integral inequalities

We conducted a longitudinal study involving 240 patients grouped according to the classification of periodontal diseases agreed in the World Workshop by the different groups of specialists gathered there. We proceed to select images of Cone Beam Computed Tomography (CBCT) that were used to perform a study of bone density through a precise algorithm allowing an accurate calculation of the fractal dimensions

In this paper, we use the generalized fractional derivative in order to study the fractional differential equation associated with a fractional Gaussian model. Moreover, we propose new properties of generalized differential and integral operators. As a practical application, we estimate the order of the derivative of the fractional Gaussian models by solving an inverse problem involving real data on

In this paper, we discussed the longitudinal harmonic waves reflection from a solid elastic half-space with electromagnetic and gravity fields influence, considering a fractional order via fractional exponential function method. The clarifications are required for the reflection amplitudes ratios (i.e. the ratios between the reflected waves amplitude and the incident waves amplitude). The results obtained

Due to the complexity of digital imaging targets and imaging conditions, fractal theory techniques in existing digital imaging systems still have various shortcomings. In this paper, a digital imaging processing method based on fractal theory is proposed for the first time. For X-ray images, the rapid calculation method of H-parameters is derived based on the fractional Brownian random field model

The role of modern logistics in economic development is being highlighted and regional logistics has an effective role in the development of various industries such as regional trade, tourism and finance and has important value for the reduction of enterprise material consumption and the improvement of comprehensive service quality. The main research content of multifractal theory is fractal systems’

By using the notion of endpoints for set-valued functions and some classical fixed point techniques, we investigate the existence of solutions for two fractional q-differential inclusions under some integral boundary value conditions. By providing an example, we illustrate our main result about endpoint. Also, we give some related algorithms and numerical results.

We construct new examples of parametric iterated function systems converging to some fractal shapes. The main goal is the study of the continuous growth and the rate of change of the attractor of the corresponding parametrization.

Recently there are several works devoted to the study of self-similar subsets of a given self-similar set, which turns out to be a difficult problem. Let L≥2 be an integer and let α∈(0,1/L). Let Cα,L be the uniform Cantor set defined by the following set equation: Cα,L=⋃j=0L−1α(Cα,L+j). We show that for any α,β∈(0,1/L2), Cα,L and Cβ,L essentially have the same self-similar subsets. Precisely, E is

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 construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on

This paper presents a preliminary study about a kind of chain coupling system, which we hope could have some enlightened effect for the research on the spatial fractal set of more strongly coupled systems. By analyzing the trajectories of system variables and applying the magnifying or reducing method, the upper bounds of the original and controlled Julia sets from the proposed system are given. Numerical

In this paper, we investigate the exact solutions of the generalized time fractional foam drainage equation. The Lie-group scaling transformation method and improved (G′/G)-expansion method are adopted here. The equation describes the evolution of the vertical density profile of a foam under gravity. New exact solutions and maple diagrams of the generalized time fractional foam drainage equation can

The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second

For any x∈(0,1], let x=1d1+1d1(d1−1)d2+⋯+1d1(d1−1)⋯dn−1(dn−1−1)dn+⋯ be its Lüroth expansion with digits {dj≥2,∀j≥1}. This paper is concerned with the growth rate of the digits in the Lüroth expansions. Let ψ:ℕ→ℝ+ be a function satisfying ψ(n+1)−ψ(n)→∞ as n→∞ and limn→∞ψ(n+1)ψ(n)=b≥1. In this paper, we consider the set E(ψ):=x∈(0,1]:limn→∞1ψ(n)∑j=1nlogdj(x)=1, and we quantify the size of E(ψ) in the

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized

Because of the self-similarity properties of nature, fractals are widely adopted as generators of natural object multimedia contents. Unfortunately, fractals are difficult to control due to their iterated function systems, and traditional researches on fractal generating visual objects focus on mathematical manipulations. In Generative Adversarial Nets (GANs), visual object generators can be automatically

A generalized KdV equation with fractal derivatives is suggested, and a special function is introduced to establish a fractal variational principle. A detailed derivation is elucidated step by step by the semi-inverse method, and some special cases are discussed.

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions

This study aims at combining the machine learning technique with the Hausdorff derivative to solve one-dimensional Hausdorff derivative diffusion equations. In the proposed artificial neural network method, the multilayer feed-forward neural network is chosen and improved by using the Hausdorff derivative to the activation function of hidden layers. A trial solution is a combination of the boundary

In the process of gas extraction, fracture-pore structure significantly influences the macroscopic permeability of coal seam. However, under the multi-field coupling, the mechanism of coal seam fracture-pore evolution remains to be clarified. In this paper, considering the effect of adsorption expansion, the fractal theory for porous media coupled with the multi-field model for coal seam is considered

In this paper, in order to calculate the average path length for unweighted and weighted hierarchical networks, we define the sum of the distances from each node to the vertex and the bottom nodes. For the unweighted network, we show that the average path length grows with the size of Nt as ln(Nt). For the weighted network, we prove its weighted path length approaches a constant related to the weighting

In this work, we present a robotic arm assisted by a visual system to decide whether an object with different colors, parallel flat surfaces and other types of surfaces would be subject to be manipulated without a drop risk. This robotic arm is assisted with sensors such as temperature, humidity, artificial vision, etc. and monitored with a Blockchain Internet of Things (BIoT) expert system assistance

During coal and rock loading, a significantly large number of electromagnetic signals are generated as a result of fracture appearance and crack expansion. The generation of electromagnetic signal is the comprehensive embodiment of the coal rock failure behavior. Therefore, the generated signals contain complex and rich messages that can reflect the damage process and degree of coal and rock. In this

We present evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene. We resolve the problem of the "non-stationary" feature of the sequence of base pairs by