Mathematics

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18 pages, 317 KiB

Open AccessArticle

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

by Abdelhamid Mohammed Djaouti, Khellaf Ould Melha and Muhammad Amer Latif

Mathematics 2024, 12(8), 1268; https://doi.org/10.3390/math12081268 - 22 Apr 2024

Viewed by 707

Abstract

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating [...] Read more.

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

13 pages, 337 KiB

Open AccessArticle

On a Generalized Wave Equation with Fractional Dissipation in Non-Local Elasticity

by Teodor M. Atanackovic, Diana Dolicanin Djekic, Ersin Gilic and Enes Kacapor

Mathematics 2023, 11(18), 3850; https://doi.org/10.3390/math11183850 - 8 Sep 2023

Cited by 1 | Viewed by 982

Abstract

We analyze wave equation for spatially one-dimensional continuum with constitutive equation of non-local type. The deformation is described by a specially selected strain measure with general fractional derivative of the Riesz type. The form of constitutive equation is assumed to be in strain-driven [...] Read more.

We analyze wave equation for spatially one-dimensional continuum with constitutive equation of non-local type. The deformation is described by a specially selected strain measure with general fractional derivative of the Riesz type. The form of constitutive equation is assumed to be in strain-driven type, often used in nano-mechanics. The resulting equations are solved in the space of tempered distributions by using the Fourier and Laplace transforms. The properties of the solution are examined and compared with the classical case. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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24 pages, 1930 KiB

Open AccessArticle

Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform

by Didar Urynbassarova and Aajaz A. Teali

Mathematics 2023, 11(9), 2201; https://doi.org/10.3390/math11092201 - 7 May 2023

Cited by 4 | Viewed by 1623

Abstract

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional [...] Read more.

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional free parameters. We explore the properties of 2D right-sided QOLCT, including inversion and Parseval formulas, besides its relationship with other transforms. We also examine the convolution and correlation theorems of 2D right-sided QOLCT, followed by several uncertainty principles. Additionally, we present an illustrative example of the proposed transform, demonstrating its graphical representation of a given signal and its transformed signal. Finally, we demonstrate an application of QOLCT, where it can be utilized to generalize the treatment of swept-frequency filters. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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20 pages, 357 KiB

Open AccessArticle

On Impulsive Implicit ψ-Caputo Hybrid Fractional Differential Equations with Retardation and Anticipation

by Abdelkrim Salim, Jehad Alzabut, Weerawat Sudsutad and Chatthai Thaiprayoon

Mathematics 2022, 10(24), 4821; https://doi.org/10.3390/math10244821 - 19 Dec 2022

Cited by 6 | Viewed by 1240

Abstract

In this paper, we investigate the existence and Ulam–Hyers–Rassias stability results for a class of boundary value problems for implicit

ψ

-Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The results are based on the Banach contraction principle [...] Read more.

In this paper, we investigate the existence and Ulam–Hyers–Rassias stability results for a class of boundary value problems for implicit

ψ

-Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The results are based on the Banach contraction principle and Krasnoselskii’s fixed point theorem. In addition, the Ulam–Hyers–Rassias stability result is proved using the nonlinear functional analysis technique. Finally, illustrative examples are given to validate our main results. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

21 pages, 1098 KiB

Open AccessArticle

Computational Scheme for the First-Order Linear Integro-Differential Equations Based on the Shifted Legendre Spectral Collocation Method

by Zhuoqian Chen, Houbao Xu and Huixia Huo

Mathematics 2022, 10(21), 4117; https://doi.org/10.3390/math10214117 - 4 Nov 2022

Cited by 1 | Viewed by 1408

Abstract

First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited [...] Read more.

First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited by the order of the differences in

L^{1}

space. Therefore, how to design a computational scheme for the first-order linear IDEs with computational efficiency becomes an urgent problem to be solved. To this end, a polynomial approximation scheme based on the shifted Legendre spectral collocation method is proposed in this paper. First, we transform the first-order linear IDEs into an Cauchy problem for consideration. Second, by decomposing the system operator, we rewrite the Cauchy problem into a more general form for approximating. Then, by using the shifted Legendre spectral collocation method, we construct a computational scheme and write it into an abstract version. The convergence of the scheme is proven in the sense of

L^{1}

-norm by employing Trotter-Kato theorem. At the end of this paper, we summarize the usage of the scheme into an algorithm and present some numerical examples to show the applications of the algorithm. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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12 pages, 460 KiB

Open AccessArticle

New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain

by Siddiqui Saima, Bingzhao Li and Samad Muhammad Adnan

Mathematics 2022, 10(8), 1217; https://doi.org/10.3390/math10081217 - 8 Apr 2022

Cited by 7 | Viewed by 1677

Abstract

The theory of quaternions has gained a firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation [...] Read more.

The theory of quaternions has gained a firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and curves appearing in images, therefore, such a sampling formula is of substantial importance for image processing, particularly in image super-resolution procedures. Moreover, the presented sampling expansion can be applied in the field of image enhancement, color image processing, image restoration and compression and filtering, etc. Finally, an illustrative example is presented to demonstrate the efficacy of the proposed technique with vivid simulations in MATLAB. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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20 pages, 329 KiB

Open AccessArticle

Hermite-Hadamard-Fejér Type Inequalities with Generalized K-Fractional Conformable Integrals and Their Applications

by Humaira Kalsoom and Zareen A. Khan

Mathematics 2022, 10(3), 483; https://doi.org/10.3390/math10030483 - 2 Feb 2022

Cited by 3 | Viewed by 1336

Abstract

In this work, we introduce new definitions of left and right-sides generalized conformable

K

-fractional derivatives and integrals. We also prove new identities associated with the left and right-sides of the Hermite-Hadamard-Fejér type inequality for

ϕ

-preinvex functions. Moreover, we use these new [...] Read more.

In this work, we introduce new definitions of left and right-sides generalized conformable

K

-fractional derivatives and integrals. We also prove new identities associated with the left and right-sides of the Hermite-Hadamard-Fejér type inequality for

ϕ

-preinvex functions. Moreover, we use these new identities to prove some bounds for the Hermite-Hadamard-Fejér type inequality for generalized conformable

K

-fractional integrals regarding

ϕ

-preinvex functions. Finally, we also present some applications of the generalized definitions for higher moments of continuous random variables, special means, and solutions of the homogeneous linear Cauchy-Euler and homogeneous linear

K

-fractional differential equations to show our new approach. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

20 pages, 16816 KiB

Open AccessArticle

A New Active Contour Medical Image Segmentation Method Based on Fractional Varying-Order Differential

by Yanshan Zhang and Yuru Tian

Mathematics 2022, 10(2), 206; https://doi.org/10.3390/math10020206 - 10 Jan 2022

Cited by 10 | Viewed by 1747

Abstract

Image segmentation technology is dedicated to the segmentation of intensity inhomogeneous at present. In this paper, we propose a new method that incorporates fractional varying-order differential and local fitting energy to construct a new variational level set active contour model. The energy functions [...] Read more.

Image segmentation technology is dedicated to the segmentation of intensity inhomogeneous at present. In this paper, we propose a new method that incorporates fractional varying-order differential and local fitting energy to construct a new variational level set active contour model. The energy functions in this paper mainly include three parts: the local term, the regular term and the penalty term. The local term combined with fractional varying-order differential can obtain more details of the image. The regular term is used to regularize the image contour length. The penalty term is used to keep the evolution curve smooth. True positive (TP) rate, false positive (FP) rate, precision (P) rate, Jaccard similarity coefficient (JSC), and Dice similarity coefficient (DSC) are employed as the comparative measures for the segmentation results. Experimental results for both synthetic and real images show that our method has more accurate segmentation results than other models, and it is robust to intensity inhomogeneous or noises. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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13 pages, 1009 KiB

Open AccessArticle

Dynamic Analysis of Software Systems with Aperiodic Impulse Rejuvenation

by Huixia Huo, Houbao Xu and Zhuoqian Chen

Mathematics 2022, 10(2), 197; https://doi.org/10.3390/math10020197 - 9 Jan 2022

Viewed by 1380

Abstract

This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt [...] Read more.

This paper aims to obtain the dynamical solution and instantaneous availability of software systems with aperiodic impulse rejuvenation. Firstly, we formulate the generic system with a group of coupled impulsive differential equations and transform it into an abstract Cauchy problem. Then we adopt a difference scheme and establish the convergence of this scheme by applying the Trotter–Kato theorem to obtain the system’s dynamical solution. Moreover, the instantaneous availability as an important evaluation index for software systems is derived, and its range is also estimated. At last, numerical examples are shown to illustrate the validity of theoretical results. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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9 pages, 276 KiB

Open AccessArticle

Blow-Up and Global Existence of Solutions for the Time Fractional Reaction–Diffusion Equation

by Linfei Shi, Wenguang Cheng, Jinjin Mao and Tianzhou Xu

Mathematics 2021, 9(24), 3248; https://doi.org/10.3390/math9243248 - 15 Dec 2021

Cited by 2 | Viewed by 2319

Abstract

In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial [...] Read more.

In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial data, we obtain two sufficient conditions for the blow-up of the solutions in finite time. Moreover, the existence of global solutions is studied when the initial data is small enough. Finally, the long-time behavior of bounded solutions is analyzed. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

13 pages, 1608 KiB

Open AccessEditor’s ChoiceArticle

Fractional Growth Model Applied to COVID-19 Data

by Fernando Alcántara-López, Carlos Fuentes, Carlos Chávez, Fernando Brambila-Paz and Antonio Quevedo

Mathematics 2021, 9(16), 1915; https://doi.org/10.3390/math9161915 - 11 Aug 2021

Cited by 4 | Viewed by 2842

Abstract

Growth models have been widely used to describe behavior in different areas of knowledge; among them the Logistics and Gompertz models, classified as models with a fixed inflection point, have been widely studied and applied. In the present work, a model is proposed [...] Read more.

Growth models have been widely used to describe behavior in different areas of knowledge; among them the Logistics and Gompertz models, classified as models with a fixed inflection point, have been widely studied and applied. In the present work, a model is proposed that contains these growth models as extreme cases; this model is generalized by including the Caputo-type fractional derivative of order

0 < β \leq 1

, resulting in a Fractional Growth Model which could be classified as a growth model with non-fixed inflection point. Moreover, the proposed model is generalized to include multiple sigmoidal behaviors and thereby multiple inflection points. The models developed are applied to describe cumulative confirmed cases of COVID-19 in Mexico, US and Russia, obtaining an excellent adjustment corroborated by a coefficient of determination

R^{2} > 0.999

. Full article

(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)

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