Fractal and Fractional

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15 pages, 1210 KiB

Open AccessArticle

Difference Equations and Julia Sets of Several Functions for Degenerate q-Sigmoid Polynomials

by Jung-Yoog Kang and Cheon-Seoung Ryoo

Fractal Fract. 2023, 7(11), 791; https://doi.org/10.3390/fractalfract7110791 - 30 Oct 2023

Cited by 1 | Viewed by 1066

Abstract

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) [...] Read more.

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

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41 pages, 2342 KiB

Open AccessArticle

To the Theory of Decaying Turbulence

by Alexander Migdal

Fractal Fract. 2023, 7(10), 754; https://doi.org/10.3390/fractalfract7100754 - 12 Oct 2023

Cited by 1 | Viewed by 1759

Abstract

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension

d > 2

. This solution family is equivalent to a fractal curve in complex space

C^{d}

[...] Read more.

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension

d > 2

. This solution family is equivalent to a fractal curve in complex space

C^{d}

with random steps parametrized by N Ising variables

σ_{i} = \pm 1

, in addition to a rational number

\frac{p}{q}

and an integer winding number r, related by

\sum σ_{i} = q r

. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in

R_{d}

space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size

N \to \infty

or its chemical potential

μ \to 0

. The system with fixed N has different asymptotics at odd and even

N \to \infty

, but the limit

μ \to 0

is well defined. The energy dissipation rate is analytically calculated as a function of

μ

using methods of number theory. It grows as

ν / μ^{2}

in the continuum limit

μ \to 0

, leading to anomalous dissipation at

μ \propto \sqrt{ν} \to 0

. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as

t^{- λ}

, with a continuous spectrum of indexes

λ

in the local limit

μ \to 0

. The spectrum is determined by a resolvent, which is represented as an infinite product of

3 \otimes 3

matrices depending of the element of the Euler ensemble. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

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23 pages, 427 KiB

Open AccessArticle

Continuous Dependence on the Initial Functions and Stability Properties in Hyers–Ulam–Rassias Sense for Neutral Fractional Systems with Distributed Delays

by Hristo Kiskinov, Mariyan Milev, Magdalena Veselinova and Andrey Zahariev

Fractal Fract. 2023, 7(10), 742; https://doi.org/10.3390/fractalfract7100742 - 8 Oct 2023

Cited by 2 | Viewed by 1114

Abstract

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with [...] Read more.

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

16 pages, 1862 KiB

Open AccessArticle

Derivative-Free Conformable Iterative Methods for Solving Nonlinear Equations

by Giro Candelario, Alicia Cordero, Juan R. Torregrosa and María P. Vassileva

Fractal Fract. 2023, 7(8), 578; https://doi.org/10.3390/fractalfract7080578 - 27 Jul 2023

Viewed by 1136

Abstract

In this manuscript, we use approximations of conformable derivatives for designing iterative methods to solve nonlinear algebraic or trascendental equations. We adapt the approximation of conformable derivatives in order to design conformable derivative-free iterative schemes to solve nonlinear equations: Steffensen and Secant-type methods. [...] Read more.

In this manuscript, we use approximations of conformable derivatives for designing iterative methods to solve nonlinear algebraic or trascendental equations. We adapt the approximation of conformable derivatives in order to design conformable derivative-free iterative schemes to solve nonlinear equations: Steffensen and Secant-type methods. To our knowledge, these are the first conformable derivative-free schemes in the literature, where the Steffensen conformable method is also optimal; moreover, the Secant conformable scheme is also a procedure with memory. A convergence analysis is made, preserving the order of classical cases, and the numerical performance is studied in order to confirm the theoretical results. It is shown that these methods can present some numerical advantages versus their classical partners, with wide sets of converging initial estimations. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

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18 pages, 340 KiB

Open AccessArticle

Boundary Value Problem for Impulsive Delay Fractional Differential Equations with Several Generalized Proportional Caputo Fractional Derivatives

by Ravi P. Agarwal and Snezhana Hristova

Fractal Fract. 2023, 7(5), 396; https://doi.org/10.3390/fractalfract7050396 - 12 May 2023

Cited by 1 | Viewed by 1499

Abstract

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 explicit form of the unique solution of BVP in [...] Read more.

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 explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

19 pages, 362 KiB

Open AccessArticle

Fractional Scale Calculus: Hadamard vs. Liouville

by Manuel D. Ortigueira and Gary W. Bohannan

Fractal Fract. 2023, 7(4), 296; https://doi.org/10.3390/fractalfract7040296 - 29 Mar 2023

Cited by 10 | Viewed by 1831

Abstract

A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding [...] Read more.

A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

22 pages, 892 KiB

Open AccessArticle

Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order

by Hari M. Srivastava and Mohammad Izadi

Fractal Fract. 2023, 7(1), 94; https://doi.org/10.3390/fractalfract7010094 - 14 Jan 2023

Cited by 15 | Viewed by 1524

Abstract

In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and [...] Read more.

In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and we obtain a family of linearized equations. By making use of the generalized shifted airfoil polynomials of the second kind (SAPSK) together with some appropriate collocation points as the roots of SAPSK, we arrive at an algebraic system of linear equations to be solved in an iterative manner. The error analysis and convergence properties of the SAPSK are established in the

L_{2}

and

L_{\infty}

norms. Through numerical simulations, it is shown that the proposed hybrid QLM-SAPSK approach is not only capable of tackling the inherit singularity at the origin, but also produces effective numerical solutions to the model problem with different nonlinearity parameters and two fractional order derivatives. The accuracy of the present technique is checked via the technique of residual error functions. The QLM-SAPSK technique is simple and efficient for solving the underlying electrohydrodynamic flow model. The computational outcomes are accurate in comparison with those of numerical values reported in the literature. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

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20 pages, 392 KiB

Open AccessArticle

Abstract Impulsive Volterra Integro-Differential Inclusions

by Wei-Shih Du, Marko Kostić and Daniel Velinov

Fractal Fract. 2023, 7(1), 73; https://doi.org/10.3390/fractalfract7010073 - 9 Jan 2023

Cited by 6 | Viewed by 1243

Abstract

In this work, we provide several applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The [...] Read more.

In this work, we provide several applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The use of regularizing operator C seems to be completely new within the theory of the abstract impulsive Volterra integro-differential equations. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

10 pages, 277 KiB

Open AccessArticle

A New Contraction with an Application for Integral Equations

by Müzeyyen Sangurlu Sezen

Fractal Fract. 2023, 7(1), 58; https://doi.org/10.3390/fractalfract7010058 - 3 Jan 2023

Viewed by 1202

Abstract

In this study, we introduce a new definition called

δ

-contraction. However, we prove some theorems for mappings satisfying the

δ

-contraction and touch upon fixed points. In the obtained theorems, we also show the existence and uniqueness of fixed points. In order [...] Read more.

In this study, we introduce a new definition called

δ

-contraction. However, we prove some theorems for mappings satisfying the

δ

-contraction and touch upon fixed points. In the obtained theorems, we also show the existence and uniqueness of fixed points. In order to prove the validity of the results of our main theorems, we give a few examples as well as an application that reveals the solution of some integral equations. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

Review

Jump to: Research

31 pages, 568 KiB

Open AccessReview

Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results

by Gani Stamov and Ivanka Stamova

Fractal Fract. 2023, 7(4), 289; https://doi.org/10.3390/fractalfract7040289 - 27 Mar 2023

Cited by 6 | Viewed by 2113

Abstract

In recent years, cellular neural networks (CNNs) have become a popular apparatus for simulations in neuroscience, biology, medicine, computer sciences and engineering. In order to create more adequate models, researchers have considered memory effects, reaction–diffusion structures, impulsive perturbations, uncertain terms and fractional-order dynamics. [...] Read more.

In recent years, cellular neural networks (CNNs) have become a popular apparatus for simulations in neuroscience, biology, medicine, computer sciences and engineering. In order to create more adequate models, researchers have considered memory effects, reaction–diffusion structures, impulsive perturbations, uncertain terms and fractional-order dynamics. The design, cellular aspects, functioning and behavioral aspects of such CNN models depend on efficient stability and control strategies. In many practical cases, the classical stability approaches are useless. Recently, in a series of papers, we have proposed several extended stability and control concepts that are more appropriate from the applied point of view. This paper is an overview of our main results and focuses on extended stability and control notions including practical stability, stability with respect to sets and manifolds and Lipschitz stability. We outline the recent progress in the stability and control methods and provide diverse mechanisms that can be used by the researchers in the field. The proposed stability techniques are presented through several types of impulsive and fractional-order CNN models. Examples are elaborated to demonstrate the feasibility of different technologies. Full article

(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)

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