Fractal and Fractional, Volume 7, Issue 5

This Special Issue of the MDPI journal, Fractal and Fractional, on the subject area of “Operators of Fractional Calculus and Their Multidisciplinary Applications” consists of 19 peer-reviewed papers, including some invited feature article...

We consider a general metric Steiner problem, which involves finding a set $\mathcal{S}$ with the minimal length, such that $\mathcal{S}\cup A$ is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini...

In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some C...

Advances in our knowledge of nonlinear dynamical networks, systems and processes (as well as their unified repercussions) currently allow us to study many typical complex phenomena taking place in nature, from the nanoscale to the extra-galactic scal...

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications...

In this paper, we analyze the Bohr $\rho$-almost periodic type sequences and the generalized $\rho$-almost periodic type sequences of the form $F:I\times X\to Y$, where $\varnothing \ne I\subseteq {\mathbb{Z}}^n$, X and Y are complex Banach spaces and $\rho$ is a general bina...

In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weight...

This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furtherm...

In this paper, we introduce a SIVR model using the Laplace Adomian decomposition. This model focuses on a new trend in mathematical epidemiology dedicated to studying the characteristics of vaccination of infected communities. We analyze the epidemio...

Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemis...

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous f...

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems ar...

Fractional differ-integral operators are used to obtain the equation of state for a substance that can be seen as fractal. Two equations of state have been obtained, the first of which depends on two parameters that characterize the fractal dimension...

A new fractional accumulation technique based on discrete sequence convolution transform was developed. The accumulation system, whose unit impulse response is the accumulation convolution sequence, was constructed; then, the order was extended to fr...

Aiming to accurately predict the leakage rate of the sealing interface, this work proposes a two-dimensional finite element model of a proton exchange membrane fuel cell, which includes the microscopic surface morphology and the asperity contact proc...

Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems ma...

Let $\partial \mathsf{T}$ be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of $\partial \mathsf{T}$ along which the sequence $S_n\mathsf{X}\left(t\right)/S_n\stackrel{\&ti}{\mathsf{X}}$...

In the paper, the authors find a sufficient and necessary condition for the power-exponential function ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ and ${(1}^{}$...

Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested a...

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The...

The present paper introduces a new class of generalized differential and integral operators. This class includes and generalizes a large number of definitions of fractal-fractional derivatives and integral operators used to model the complex dynamics...

Depth image enhancement techniques can help to improve image quality and facilitate computer vision tasks. Traditional image-enhancement methods, which are typically based on integer-order calculus, cannot exploit the textural information of an image...

Remanence is an important parameter of magnetic property for Nd-Fe-B magnets, and high remanent magnetization is a prerequisite for high-performance magnets. In this paper, the surface morphology perpendicular to the texture orientation direction and...

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the afor...

The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and gre...

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial diffe...

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a...

The dynamics of cardiac signals can be studied using methods for nonlinear analysis of heart rate variability (HRV). The methods that are used in the article to investigate the fractal, multifractal and informational characteristics of the intervals...

Marine and terrestrial natural surfaces exhibit statistical scale invariance properties that are well modelled by fractional Brownian motion (fBm), two-dimensional random processes. Accordingly, for microwave remote sensing applications it is useful...

The continuous Bernoulli distribution is defined on the unit interval and has a unique property related to fractiles. A fractile is a position on a probability density function where the corresponding surface is a fixed proportion. This article prese...

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construc...

Since each rock type represents different deformation characteristics, prediction of the damage beforehand is one of the most fundamental problems of industrial activities and rock engineering studies. Previous studies have predicted the stress&ndash...

The paper is oriented on the existence of almost periodic solutions of factional-order impulsive delayed reaction-diffusion gene regulatory networks. Caputo type fractional-order derivatives and impulsive disturbances at not fixed instants of time ar...

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliabl...

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order $0<\alpha <1,$ and $1<\beta <2$. The numerical method is based on the La...

The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices a...

Based on the derivation of the equation of state for systems with a fractional power spectrum, the relationship between the van der Waals constant and the fractional derivative order has been established. The fractional model of electron–phonon inter...

In this article, we solve the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) using an accelerated form of the Adomian decomposition method (ADM). Based on $(\alpha ,\delta ,\beta )$-cut, we convert the NVPFIE to the nonlinea...

Shannon entropy, also known as information entropy or entropy, measures the uncertainty or randomness of probability distribution. Entropy is measured in bits, quantifying the average amount of information required to identify an event from the distr...

Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establ...

We introduce fractal Tsallis entropy and show that it satisfies Shannon–Khinchin axioms. Analogously to Tsallis divergence (or Tsallis relative entropy, according to some authors), fractal Tsallis divergence is defined and some properties of it are s...

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation...

Recent studies have demonstrated the benefits of using fractional derivatives to simulate a blood pressure profile. In this work we propose to combine a one-dimensional model of coronary blood flow with fractional-order Windkessel boundary conditions...

The current work is devoted to studying the dynamical behavior of the Sakovich equation with beta derivatives. We announce the conditions of problem parameters leading to the existence of periodic, solitary, and kink solutions by applying the qualita...

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the invers...

The rising tide of smoking-related diseases has irreparably damaged the health of both young and old people, according to the World Health Organization. This study explores the dynamics of the age-structure smoking model under fractal-fractional (F-F...

This paper investigates exact solutions of cosmological interest in fractional cosmology. Given $\mu$, the order of Caputo’s fractional derivative, and w, the matter equation of state, we present specific exact power-law solutions. We discuss th...

Fractals is a new branch of nonlinear science that was established in the 1970s, focusing on irregularities, haphazard phenomena and self-similarities in nature [...]

In this paper, we consider a fractional equation of order between one and two which may be looked at as an interpolation between the heat and wave equations. The problem is non-linear as it involves a power-type non-linearity. We investigate the poss...

This paper studies a fractional differential equation combined with a Liouville–Caputo fractional differential operator, namely, ${}^{LC}{\mathcal{D}}_{\eta }^{\beta ,\gamma }Q\left(t\right)=\lambda \vartheta (t,Q\left(t\right)),t\in [c,d],\beta ,\gamma \in (0,1],\eta \in [0,1],$ wher...