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In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and a prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate f...
Article
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In this study, a generalization of the Estevez–Mansfield–Clarkson equation which considers the presence of conformable time-fractional derivatives is investigated analytically. The integer-order model finds applications in mathematical physics, optics and the investigation of shape developing in liquid drops. In the present manuscript, the Sardar s...
Article
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In this paper, a numerical method based on Haar wavelet with Ca-puto derivative is developed for the solution of a system of fractional integro-differential equations (FIDEs). The solution of these equations is difficult due to the non-local nature of fractional derivatives and integrals. Different numerical and analytical methods have been develop...
Article
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Many years ago, A. Zygmund extensively studied the symmetric derivative [5, p.1001]. Over the last decade, significant properties of the conformable fractional integral have emerged. This publication bridges these two concepts, presenting results that extend the conformable fractional derivative to the symmetric derivative.
Article
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In this work, we study a class of quantum fractional nonlinear difference equations with neutral terms. The results are developed in the sense of the q-analogue of the R-L fractional difference operator. New sufficient criteria for oscillation of the considered q-fractional equations with R-L type fractional derivatives are established using the in...
Article
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The isothermal gas sphere model may be beneficial for understanding certain features of astrophysical objects like stars, but it has severe limits when used to compact stars. The paper expands the Tolman-Oppenheimer-Volkoff (TOV) equation of the fractional relativistic gas sphere to contain fractional derivatives, resulting in a more general fracti...
Article
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This article examines the resilient base containment control (CC) for fractional order multiple agent system with disturbance term in the dynamical system. In order to deal with resilient base CC in Riemann–Liouville sense with time delays, a simple and effective method is proposed where followers have weighted digraph topology among them. A distur...
Article
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A classical system of generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt equations (gWBKBKEs), describing long shallow-water waves in a dispersive medium, has been extensively studied. This paper introduces a novel fractional derivative, the truncated modified Mittag-Leffler function derivative (TMMD), and investigates its impact on behaviors of...
Article
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In the present investigation, we targeted an inverse problem of finding an unknown time moment for the mixed wave-diffusion equation involving the Riemann-Liuoville fractional derivative. We have used a solution of the first boundary problem for the time-fractional diffusion-wave equation. The explicit form of the solution was obtained using the Gr...
Article
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In this work, we consider the Ujlayan-Dixit (UD) fractional derivative and integral to present the fractional probability density function for the Continuous Uniform Distribution (CUD). Some applications for this distribution using the UD approach are developed by introduce new fractional notions on the probability theory, which involve cumulative...
Article
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A piecewise fractional differential equation (deterministic–stochastic differential equation or vice versa) has appeared in recent literature. Piecewise operators are used to study crossover real data effectively. By using stochastic–deterministic piecewise hybrid fractional derivatives, with hybrid fractional‐order and variable‐order fractional op...
Article
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Mass estimates of a spiral galaxy derived from its rotation curve must account for the galaxy’s past accretion history. There are several lines of evidence indicating that M31 experienced a major merger 2 to 3 Gyr ago. In this work, we generated a dynamical model of M31 as a merger remnant that reproduces most of its properties, including from the...
Article
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Most of the earthquakes occur when there is a movement along faults within the earth. It is more often observed that more than one fault is situated in seismically active regions, and a movement across any one of them affects the nature of the stress accumulation near the neighboring fault. To investigate the stress–strain accumulation in neighbori...
Article
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This study aims to examine the SIA ICR model with chronic infection therapy and data analysis worldwide and its actual implications. The boundedness and uniqueness solution of such an HBV model are confirmed, and Banach space is used to look for bounded discoveries. The developed system’s uniqueness is investigated to verify if it has a unique solu...
Article
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For a diffusion equation with the Djrbashian–Nersesian–Caputo fractional derivative with respect to time, we establish sufficient conditions for the unique solvability of an inverse problem of determination of m unknown functions from the Schwarz-type space of smooth functions rapidly decreasing at infinity with m time-integral overdetermination co...
Article
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This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD wi...
Article
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This paper presents the so-called shifted Jacobi method, an efficient numerical technique to solve second-order periodic boundary value problems with finitely many singularities involving nonlinear systems of two points. The method relies on the Jacobi polynomials used as natural basis functions in the conformable sense of fractional derivative. A...
Article
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This paper proposes a numerical technique to solve the time-fractional generalized Kawahara differential equation (TFGKDE). Certain shifted Lucas polynomials are utilized as basis functions. We first establish some new formulas concerned with the introduced polynomials and then tackle the equation using a suitable collocation procedure. The integer...
Preprint
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One of the most often used methods of summing divergent series in physics is the Borel-type summation with control parameters improving convergence, which are defined by some optimization conditions. The well known annoying problem in this procedure is the occurrence of multiple solutions for control parameters. We suggest a method for resolving th...
Article
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The most complex steady-state behaviour known in dynamical systems is that which is characterised as "chaos". The three-dimensional Lorenz system, which is linear and non-periodic, is a chaotic system that is used to study the properties of a two-dimensional liquid layer that is homogeneously heated from below and cooled from above. In this study,...
Article
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This paper examines the properties of finite-time stability (FTS) in the sense of Hadamard fractional-order systems. The investigation utilizes the Hadamard fractional derivative to formulate and analyze these systems, establishing FTS criteria based on the Lyapunov theory. Additionally, this paper presents a comprehensive exploration of methodolog...
Article
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This paper explores the fluid flow and heat transfer characteristics of non-Newtonian fluids within a capillary tube, employing a generalized fractional modeling framework. The analysis utilizes the Maxwell fluid model combined with the Atangana-Baleanu fractional time-fractional derivative. Analytical solutions for velocity and temperature distrib...
Article
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In this current study, first we establish the modified power Atangana-Baleanu fractional derivative operators (MPC) in both the Caputo and Riemann-Liouville (MPRL) senses. Using the convolution approach and Laplace transformation, the so-called modified power fractional Caputo and R-L derivative operators with non-singular kernels are introduced. W...
Article
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Hepatitis B is a viral infection that primarily targets the liver, potentially leading to acute or chronic liver diseases with severe complications, such as cirrhosis and liver cancer. Its persistent prevalence underscores its status as a significant global health issue. This research constructs a mathematical model for the progression of Hepatitis...
Article
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Africa still faces significant challenges due to malaria, which calls for creative methods to disease modeling and control measures. In order to capture the non-local and memory effects that are inherent in the disease transmission process, we present a novel mathematical framework in this study that integrates treatment and vaccine as control meth...
Article
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In this manuscript, we introduce a novel system of fractional differential equations incorporating both Caputo and conformable derivatives. We delve into the existence and uniqueness of solutions for this system, employing fixed-point techniques under appropriate conditions. To illustrate the practical applications of our theoretical findings, we i...
Article
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This study introduces a novel class of fractional differential equations characterized by antiperiodic parametric boundary conditions of order µ 2 (2; 3]. The parameters θ and ξ play a crucial role in shaping the boundary conditions by defining specific values and functional behavior. By employing fixed point theorems, we establish existence result...
Article
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This paper focuses on the qualitative and quantitative description of solutions for linear systems of differential equations with fractional order. The Laplace transform and the Mittag-Leffler function are the main tools used to find such solutions. As in the ordinary case, the eigenvalues determine the stability of the system via a condition depen...
Article
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Accurately modeling seepage flow dynamics in porous media is critical in environmental science, hydrology, and engineering, especially in high-dimensional spaces with fractional derivatives. These flows present significant analytical challenges due to their inherent nonlinearity and complexity. Traditional solution methods often rely on simplificat...
Article
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Although the theory of fractional operators has numerous definitions in the literature, it is not easy to know which operator is best used for it. One way to try to get around this problem is to propose more general operators where, based on the choice of parameters involved in this new operator, it is possible to obtain the maximum number of defin...
Conference Paper
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The article introduces the concept of the Caputo-Hukuhara fractional derivative for set-valued mappings, which generalizes the Caputo fractional derivative to the set-valued case. Its main properties are also proven. Also the Cauchy problem for a linear homogeneous set-valued differential equation with a Caputo-Hukuhara derivative of order is consi...
Article
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This paper presents the development of a fractional hybrid function composed of block-pulse functions and Fibonacci polynomials (FHBPF) for the numerical solution of multiterm variable-order fractional differential equations. By replacing x → x α in FHBPF and utilizing incomplete beta functions, we construct the method with a focus on fractional de...
Article
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This paper studies the initial-boundary value problem of a class of nonlinear time-fractional parabolic equations, where the fractional derivative used is in the sense of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \...
Article
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In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a...
Preprint
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This paper investigates the multivariate, stochastic, and fractional generalizations of neural network operators with convergence guarantees. The multivariate neural network operator is introduced using symmetrized activation functions, and its convergence to the target function is shown with an error bound that involves derivatives up to the N-th...
Article
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Cancer refers to a group of diseases characterized by the uncontrolled growth of abnormal cells. Conventional cancer therapies, such as chemotherapy and radiation therapy, often encounter issues like toxicity and resistance, primarily due to their inability to effectively differentiate between cancerous and normal cells. As a result, these treatmen...
Article
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Analyzing the Michaelis-Menten kinetics biochemical reaction model is essential in understanding enzyme-catalyzed reactions. Traditional models frequently fail to incorporate the complexities of fractional derivatives. This study employs the homotopy perturbation method (HPM) and the homotopy analysis method (HAM) to derive analytical expressions f...
Article
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Here we present the univariate quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. We perform also the related Banach space valued fractional approximation. These approximations are derived by establishing Jackson type...
Article
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This research explores the epidemiology of widespread monkeypox virus (MPV) transmission using a modified Atanagana-Baleanu-Caputo (mABC) fractional derivative mathematical model in the sense of a generalized Mittag-Leffler kernel. The stability of the disease-free equilibrium in the fractional MPV model is provided, demonstrating the result for ℝ...
Article
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The main objective of this work is to study the mathematical model that combines stem cell therapy and chemotherapy for cancer cells. We study the model using the fractal fractional derivative with the Mittag-Leffler kernel. In the analytical part, we study the existence of the solution and its uniqueness, which was studied based on the fixed point...
Article
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This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the sol...
Article
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This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than t...
Preprint
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We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^{\alpha}u(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<\alpha<1, \end{equation*} where $^{C}\partial_{t}^{\alpha}$ is the Djrbashian-Caputo fractional derivative, $X$ is a complex Banach space and $\mathscr{L}:\mathcal{D}(\mathscr...
Article
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In this paper, we consider the optimal control of nonlinear systems of fractional-order with control delay, where the fractional derivatives are expressed in the Caputo sense. For this problem, we first obtained the necessary first-order optimality condition in the form of the Pontryagin maximum principle. For a control that is singular in the sens...
Article
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This study introduces a novel space-time fractional diffusion equation with the capacity to model a diverse spectrum of diffusion processes. The equation incorporates Caputo-type time derivatives with an arbitrary order \(\beta\) and introduces a spatial-fractional operator known as the fractional Bessel operator. The fundamental solution of this e...
Article
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This study explores the controllability and stability criteria for the fractional integro-differential stochastical systems with control delay employing the Ψ-Caputo-type fractional derivative (Ψ-CTFD) of order σ ∈ (0, 1). The necessary and sufficient conditions for the controllability of linear stochastical systems are obtained by utilizing the po...
Article
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Osgood functions in the source term are used to produce results for non-existence of local solutions into the framework of non-Gaussian diffusion equations. The critical exponent for non-existence of local solutions is found to depend on the fractional derivative, the non-Gaussian diffusion and the non-linear term. The instantaneous blow-up phenome...
Article
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In this article, an accurate optimization algorithm based on new polynomials namely generalized shifted Vieta-Fibonacci polynomials (GSVFPs) is employed to solve the nonlinear variable order time-space fractional reaction diffusion equation (NVOTSFRDE). The algorithm combines GSVFPs, new variable order fractional operational matrices in the Caputo...
Article
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In this paper, we aim to establish a set of sufficient conditions for the existence of an integral form mild solution and approximate controllability for a class of Sobolev-type Hilfer fractional stochastic differential systems driven by the Rosenblatt process and Poisson jumps. In the proposed control problem, we deal with a system that is under n...
Article
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In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a semigroup of contractions. Our method...
Article
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This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecew...
Article
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This research aims to determine the approximate analytical solution of a one-dimensional time fractional-order cancer model using the homotopy perturbation method (HPM). Initially, the fractional derivative component which is in the Caputo sense converted into an integer order derivative by using the Laplace transform method, followed by the techni...
Article
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In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differentia...
Article
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In this work, we explore non-commutative effects in fractional classical and quantum schemes using the anisotropic Bianchi type I cosmological model coupled to a scalar field in the K-essence formalism. We introduce non-commutative variables considering that all minisuperspace variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage...
Article
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The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains two fractional derivatives with unique fract...
Article
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An experimental study and numerical simulation of short- and long-term shear stress relaxation behaviors of nonaligned and aligned magnetorheological elastomers (MREs) were investigated. The aligned MRE was created by aligning micro-size carbonyl iron particles in chains in silicon rubber using an external magnetic field during the curing process,...
Article
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The sealing efficiency of pressure grouting in coal seam boreholes is pivotal for enhancing gas extraction and mitigating environmental risks. Therefore, this study integrated theoretical and modeling approaches to investigate the development of fractures around coal seam boreholes, elucidate the migration behavior of non-Newtonian sealing slurries...
Research Proposal
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Dear Colleagues, This Special Issue, titled "New Trends in Fractional Differential Equations with Applications", aims to explore the latest advancements in the theory and applications of fractional differential equations (FDEs). It will focus on emerging methodologies for solving FDEs, including innovative analytical, numerical, and computational...
Article
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The fractional derivative computation of piecewise continuous functions is treated with generality. It is shown why some applications give incorrect results and why Caputo derivative give strange results. Some examples are described.
Article
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Acute diarrhea poses a significant global health challenge, especially in settings with poor sanitation. This study develops a mathematical model of diarrhea, employing a piecewise modified ABC (pmABC) fractional derivative to capture the disease’s transmission dynamics, including crossover effects between classical and fractional behaviors. We ana...
Article
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In this work, we will study a time fractional order spatio-temporal SIR model with therapy and vaccination. The model is described by a system of reaction-diffusion equations incorporating a fractional derivative. The therapy will be added to the model in order to describe the effect of treatment on the population dynamics. The existence, boundedne...
Article
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This article uses an approach based on the triad model–algorithm–program. The model is a nonlinear dynamic Selkov system with non-constant coefficients and fractional derivatives of the Gerasimov–Caputo type. The Adams–Bashforth–Multon numerical method from the predictor–corrector family of methods is selected as an algorithm for studying this syst...
Preprint
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Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with $q$-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising...
Article
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In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary conditio...
Preprint
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In this paper, we present the Residual Integral Solver Network (RISN), a novel neural network architecture designed to solve a wide range of integral and integro-differential equations, including one-dimensional, multi-dimensional, ordinary and partial integro-differential, systems, and fractional types. RISN integrates residual connections with hi...
Article
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Introduction: Fractional approaches have emerged as powerful tools for modeling a wide range of phenomena in engineering and science. This study focuses on a chaotic behavior for numerical method and stability analysis to investigate the nonlinear fractional-order autonomous systems using fractional derivative operators, specifically the Atangana-B...
Article
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The main aim of this study is to examine the Hyers-Ulam stability of fractional derivatives in Volterra-Fredholm integro-differential equations using Caputo fractional derivatives. We explore the existence and uniqueness of solutions for the proposed integro-differential equation using Banach and Krasnoselskii's fixed-point techniques. Furthermore,...
Article
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In the present paper, several viscoelastic models are studied for cases when time-dependent viscoelastic operators of Lamé’s parameters are represented in terms of the fractional derivative Kelvin–Voigt, Scott-Blair, Maxwell, and standard linear solid models. This is practically important since precisely these parameters define the velocities of lo...
Article
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The regularized ψ-Hilfer derivative within the sense of Caputo is an improved version of the ψ-Hilfer fractional derivative, primarily because it addresses the issue where the initial conditions of problems involving the ψ-Hilfer fractional derivative lack clear physical significance unless p=1. This article’s main contribution is the use of the ψ-...
Article
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Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equ...
Article
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In this paper, we give sufficient conditions to guarantee the asymptotic stability of the zero solution to a kind of nonlinear fractional differential equations with the Riemann Liouville fractional derivative of order α∈(n-1,n) by using Krasnoselskii's fixed point theorem and the Banach contraction mapping principle in a weighted Banach space. The...
Article
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In the present task, the dynamics of Cryosphere is represented based on the modified Caputo-Fabrizio fractional derivative. A numerical scheme based on the fifth-order fractional Adams-Bashforth approach is applied to the dynamical system of the proposed model. The nonlinear system of differential equations of arbitrary order in the model is descri...
Article
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This paper is devoted to the solution and stability of a one‐dimensional model depicting Rao–Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fract...