Fractal and Fractional

Research

15 pages, 3852 KiB

Open AccessArticle

In Vivo HIV Dynamics, Modeling the Interaction of HIV and Immune System via Non-Integer Derivatives

by Asif Jan, Hari Mohan Srivastava, Amin Khan, Pshtiwan Othman Mohammed, Rashid Jan and Y. S. Hamed

Fractal Fract. 2023, 7(5), 361; https://doi.org/10.3390/fractalfract7050361 - 28 Apr 2023

Cited by 34 | Viewed by 2133

Abstract

The economic burden of HIV extends beyond the individual level and affects communities and countries. HIV can lead to decreased economic growth due to lost productivity and increased healthcare costs. In some countries, the HIV epidemic has led to a reduction in life [...] Read more.

The economic burden of HIV extends beyond the individual level and affects communities and countries. HIV can lead to decreased economic growth due to lost productivity and increased healthcare costs. In some countries, the HIV epidemic has led to a reduction in life expectancy, which can impact the overall quality of life and economic prosperity. Therefore, it is significant to investigate the intricate dynamics of this viral infection to know how the virus interacts with the immune system. In the current research, we will formulate the dynamics of HIV infection in the host body to conceptualize the interaction of T-cells and the immune system. The recommended model of HIV infection is presented with the help of fractional calculus for more precious outcomes. We introduce numerical methods to demonstrate how the input parameters affect the output of the system. The dynamical behavior and chaotic nature of the system are visualized with the variation of different input factors. The system’s tracking path has been numerically depicted and the impact of the viruses on T-cells has been demonstrated. In addition to this, the key factors of the system has been predicted through numerical findings. Our results predict that the strong non-linearity of the system is responsible for the chaos and oscillation, which are so closely related. The chaotic parameters of the system are highlighted and are recommended for the control of the chaos of the system. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

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19 pages, 378 KiB

Open AccessArticle

On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓ_p-Classes

by Ravi P. Agarwal, Safi S. Rabie and Samir H. Saker

Fractal Fract. 2023, 7(3), 261; https://doi.org/10.3390/fractalfract7030261 - 14 Mar 2023

Viewed by 1334

Abstract

In this paper, we study some relations between different weights in the classes

B_{p},

B_{p}^{*},

M_{p}

and

M_{p}^{*}

that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that [...] Read more.

In this paper, we study some relations between different weights in the classes

B_{p},

B_{p}^{*},

M_{p}

and

M_{p}^{*}

that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space

Λ_{p}

. These results will be proven by using the properties of classes

B_{p},

B_{p}^{*},

M_{p}

and

M_{p}^{*}

, including the self-improving properties and also the properties of the generalized Hardy operator

H_{p}

, the adjoint operator

S_{q}

and some fundamental relations between them connecting their composition to their sum. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

17 pages, 580 KiB

Open AccessArticle

Mittag–Leffler Functions in Discrete Time

by Ferhan M. Atıcı, Samuel Chang and Jagan Mohan Jonnalagadda

Fractal Fract. 2023, 7(3), 254; https://doi.org/10.3390/fractalfract7030254 - 10 Mar 2023

Cited by 2 | Viewed by 1573

Abstract

In this paper, we give an efficient way to calculate the values of the Mittag–Leffler (h-ML) function defined in discrete time

h N

, where

h > 0

is a real number. We construct a matrix equation that represents an iteration [...] Read more.

In this paper, we give an efficient way to calculate the values of the Mittag–Leffler (h-ML) function defined in discrete time

h N

, where

h > 0

is a real number. We construct a matrix equation that represents an iteration scheme obtained from a fractional h-difference equation with an initial condition. Fractional h-discrete operators are defined according to the Nabla operator and the Riemann–Liouville definition. Some figures and examples are given to illustrate this new calculation technique for the h-ML function in discrete time. The h-ML function with a square matrix variable in a square matrix form is also given after proving the Putzer algorithm. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

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16 pages, 321 KiB

Open AccessArticle

Abstract Univariate Neural Network Approximation Using a q-Deformed and λ-Parametrized Hyperbolic Tangent Activation Function

by George A. Anastassiou

Fractal Fract. 2023, 7(3), 208; https://doi.org/10.3390/fractalfract7030208 - 21 Feb 2023

Cited by 1 | Viewed by 1349

Abstract

In this work, we perform univariate approximation with rates, basic and fractional, of continuous functions that take values into an arbitrary Banach space with domain on a closed interval or all reals, by quasi-interpolation neural network operators. These approximations are achieved by deriving [...] Read more.

In this work, we perform univariate approximation with rates, basic and fractional, of continuous functions that take values into an arbitrary Banach space with domain on a closed interval or all reals, by quasi-interpolation neural network operators. These approximations are achieved by deriving Jackson-type inequalities via the first modulus of continuity of the on hand function or its abstract integer derivative or Caputo fractional derivatives. Our operators are expressed via a density function based on a q-deformed and

λ

-parameterized hyperbolic tangent activation sigmoid function. The convergences are pointwise and uniform. The associated feed-forward neural networks are with one hidden layer. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

10 pages, 284 KiB

Open AccessArticle

Some Local Fractional Hilbert-Type Inequalities

by Predrag Vuković

Fractal Fract. 2023, 7(2), 205; https://doi.org/10.3390/fractalfract7020205 - 19 Feb 2023

Cited by 1 | Viewed by 1131

Abstract

The main purpose of this paper is to prove some new local fractional Hilbert-type inequalities. Our general results are applicable to homogeneous kernels. Furthermore, the best possible constants in terms of local fractional hypergeometric function are obtained. The obtained results prove that the [...] Read more.

The main purpose of this paper is to prove some new local fractional Hilbert-type inequalities. Our general results are applicable to homogeneous kernels. Furthermore, the best possible constants in terms of local fractional hypergeometric function are obtained. The obtained results prove that the employed method is very simple and effective for treating various kinds of local fractional Hilbert-type inequalities. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

15 pages, 307 KiB

Open AccessArticle

Generalized Hermite-Hadamard Inequalities on Discrete Time Scales

by Qiushuang Wang and Run Xu

Fractal Fract. 2022, 6(10), 563; https://doi.org/10.3390/fractalfract6100563 - 4 Oct 2022

Viewed by 1372

Abstract

This paper is concerned with some new Hermite-Hadamard inequalities on two types of time scales,

Z

and

N_{c, h}

. Based on the substitution rules, we first prove the discrete Hermite-Hadamard inequalities on

Z

relating to the midpoint [...] Read more.

This paper is concerned with some new Hermite-Hadamard inequalities on two types of time scales,

Z

and

N_{c, h}

. Based on the substitution rules, we first prove the discrete Hermite-Hadamard inequalities on

Z

relating to the midpoint

\frac{a + b}{2}

and extend them to discrete fractional forms. In addition, by using traditional methods, we prove discrete Hermite-Hadamard inequalities on

N_{c, h}

and explore the corresponding fractional inequalities involving the nabla h-fractional sums. Finally, two examples are given to illustrate the obtained results. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

17 pages, 335 KiB

Open AccessArticle

Diamond Alpha Hilbert-Type Inequalities on Time Scales

by Ahmed A. El-Deeb, Dumitru Baleanu, Sameh S. Askar, Clemente Cesarano and Ahmed Abdeldaim

Fractal Fract. 2022, 6(7), 384; https://doi.org/10.3390/fractalfract6070384 - 6 Jul 2022

Cited by 1 | Viewed by 1435

Abstract

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 scales, and unify and extend [...] Read more.

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 scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen’s inequality on time scales. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

15 pages, 333 KiB

Open AccessArticle

Existence and U-H Stability Results for Nonlinear Coupled Fractional Differential Equations with Boundary Conditions Involving Riemann–Liouville and Erdélyi–Kober Integrals

by Muthaiah Subramanian, P. Duraisamy, C. Kamaleshwari, Bundit Unyong and R. Vadivel

Fractal Fract. 2022, 6(5), 266; https://doi.org/10.3390/fractalfract6050266 - 13 May 2022

Cited by 4 | Viewed by 3479

Abstract

The purpose of this article is to discuss the existence, uniqueness, and Ulam–Hyers stability of solutions to a coupled system of fractional differential equations with Erdélyi–Kober and Riemann–Liouville integral boundary conditions. The Banach fixed point theorem is used to prove the uniqueness of [...] Read more.

The purpose of this article is to discuss the existence, uniqueness, and Ulam–Hyers stability of solutions to a coupled system of fractional differential equations with Erdélyi–Kober and Riemann–Liouville integral boundary conditions. The Banach fixed point theorem is used to prove the uniqueness of solutions, while the Leray–Schauder alternative is used to prove the existence of solutions. Furthermore, we conclude that the solution to the discussed problem is Hyers–Ulam stable. The results are illustrated with examples. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

14 pages, 321 KiB

Open AccessArticle

A Numerical Approach to Solve the q-Fractional Boundary Value Problems

by Ying Sheng and Tie Zhang

Fractal Fract. 2022, 6(4), 200; https://doi.org/10.3390/fractalfract6040200 - 2 Apr 2022

Cited by 8 | Viewed by 1989

Abstract

In this present paper, we study the difference method for solving a boundary value problem of the Caputo type q-fractional differential equation. This method is based on the numerical quadrature of the q-fractional derivative and the q-Taylor expansion of related [...] Read more.

In this present paper, we study the difference method for solving a boundary value problem of the Caputo type q-fractional differential equation. This method is based on the numerical quadrature of the q-fractional derivative and the q-Taylor expansion of related function. We first derive the truncation error boundness of

O (▵ x_{n}^{2})

-order and prove the existence and uniqueness of the numerical solution. Then, we prove the stability of the numerical solution and give the error estimation. Numerical experiments finally verify the validity of the theoretical analysis. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

16 pages, 318 KiB

Open AccessArticle

Local Fractional Integral Hölder-Type Inequalities and Some Related Results

by Guangsheng Chen, Jiansuo Liang, Hari M. Srivastava and Chao Lv

Fractal Fract. 2022, 6(4), 195; https://doi.org/10.3390/fractalfract6040195 - 31 Mar 2022

Cited by 14 | Viewed by 2014

Abstract

This paper is devoted to establishing some functional generalizations of Hölder and reverse Hölder’s inequalities with local fractional integral introduced by Yang. Then, based on the obtained results, we derive some related inequalities including local fractional integral Minkowski-type and Dresher-type inequalities, which are [...] Read more.

This paper is devoted to establishing some functional generalizations of Hölder and reverse Hölder’s inequalities with local fractional integral introduced by Yang. Then, based on the obtained results, we derive some related inequalities including local fractional integral Minkowski-type and Dresher-type inequalities, which are some extensions of several existing local fractional integral inequalities. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

17 pages, 1035 KiB

Open AccessArticle

A Local Fractional Elzaki Transform Decomposition Method for the Nonlinear System of Local Fractional Partial Differential Equations

by Halil Anac

Fractal Fract. 2022, 6(3), 167; https://doi.org/10.3390/fractalfract6030167 - 18 Mar 2022

Cited by 7 | Viewed by 2610

Abstract

In this paper, the nonlinear system of local fractional partial differential equations is solved via local fractional Elzaki transform decomposition method. The local fractional Elzaki decomposition transform method combines a local fractional Elzaki transform and the Adomian decomposition method. Applications related to the [...] Read more.

In this paper, the nonlinear system of local fractional partial differential equations is solved via local fractional Elzaki transform decomposition method. The local fractional Elzaki decomposition transform method combines a local fractional Elzaki transform and the Adomian decomposition method. Applications related to the nonlinear system of local fractional partial differential equations are presented. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

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13 pages, 333 KiB

Open AccessArticle

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

by Noureddine Djenina, Adel Ouannas, Taki-Eddine Oussaeif, Giuseppe Grassi, Iqbal M. Batiha, Shaher Momani and Ramzi B. Albadarneh

Fractal Fract. 2022, 6(3), 158; https://doi.org/10.3390/fractalfract6030158 - 14 Mar 2022

Cited by 11 | Viewed by 2038

Abstract

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor [...] Read more.

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

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