Search Results (19)

Fractional multi-delay differential systems, as an important mathematical model, can effectively describe viscoelastic materials and non-local delay responses in ecosystem population dynamics. It has a wide and profound application background in interdisciplinary fields such as physics, biomedicine, and intelligent control. Through literature review and classification, it is evident that for the fractional multi-delay differential system, the existence and uniqueness of the solution and Ulam-Hyers stability (UHS), Ulam-Hyers-Rassias stability (UHRS) of the fractional multi-delay differential system are rarely studied by using the multi-delayed perturbation of two parameter Mittag-Leffler typematrix function. In this paper, we first establish the existence and uniqueness of the solution for the Riemann-Liouville fractional multi-delay differential system on finite intervals using the Banach and Schauder fixed point theorems. Second, we demonstrate the existence and uniqueness of the solution for the system on the unbounded intervals in the weighted function space. Furthermore, we investigate UHS and UHRS for the nonlinear fractional multi-delay differential system in unbounded intervals. Finally, numerical examples are provided to validate the key theoretical results. Full article

In this work, we develop and analyze a novel fractional-order framework to investigate the interactions among oxygen, phytoplankton, and zooplankton under changing climatic conditions. Unlike standard integer-order formulations, our model incorporates a Proportional–Caputo $\left(\mathbb{PC}\right)$ fractional derivative, allowing the system dynamics to capture non-local influences and memory effects over time. Initially, we rigorously verify that a unique solution exists by suitable fixed-point theorems, demonstrating that the proposed fractional system is both well-defined and robust. We then derive stability criteria to ensure Ulam–Hyers stability (UHS), confirming that small perturbations in initial states lead to bounded variations in long-term behavior. Additionally, we explore extended UHS to assess sensitivity against time-varying parameters. Numerical simulations illustrate the role of fractional-order parameters in shaping oxygen availability and plankton populations, highlighting critical shifts in system trajectories as the order of differentiation approaches unity. Full article

This paper investigates a general class of variable-kernel discrete delay differential equations (DDDEs) with integral boundary conditions and impulsive effects, analyzed using Caputo piecewise derivatives. We establish results for the existence and uniqueness of solutions, as well as their stability. The existence of at least one solution is proven using Schaefer’s fixed-point theorem, while uniqueness is established via Banach’s fixed-point theorem. Stability is examined through the lens of Ulam–Hyers (U-H) stability. Finally, we illustrate the application of our theoretical findings with a numerical example. Full article

This paper focuses on the existence and uniqueness of solutions for $\vartheta$-fractional stochastic integral equations ($\vartheta$-FSIEs) using the Banach fixed point theorem (BFPT). We explore the Ulam–Hyers stability (UHS) of $\vartheta$-FSIEs through traditional methods of stochastic calculus and the BFPT. Moreover, the continuous dependence of solutions on initial conditions is proven. Additionally, we provide three examples to demonstrate our findings. Full article

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples. Full article

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation. Full article

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics. Full article

This manuscript is related to undertaking a mathematical model (susceptible, vaccinated, infected, and recovered) of rotavirus. Some qualitative results are established for the mentioned challenging childhood disease epidemic model of rotavirus as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractal calculus. We compute the boundedness positivity of the solution of the proposed model. Additionally, the basic reproduction ratio and its sensitivity analysis are also performed. The global stability of the endemic equilibrium point is also confirmed graphically using some available values of initial conditions and parameters. Sufficient conditions are deduced for the existence theory, the Ulam–Hyers ($\mathbb{UH}$) stability. Specifically, the numerical approximate solution of the rotavirus model is investigated using efficient numerical methods. Graphical presentations are presented corresponding to a different fractional order to understand the transmission dynamics of the mentioned disease. Furthermore, researchers have examined the impact of lowering the risk of infection on populations that are susceptible and have received vaccinations, producing some intriguing results. We also present a numerical illustration taking the stochastic derivative of the proposed model graphically. Researchers may find this research helpful as it offers insightful information about using numerical techniques to model infectious diseases. Full article

In this article, we investigate the existence and uniqueness Theorem of Pantograph Hadamard fractional stochastic differential equations (PHFSDE) using the fixed-point Theorem of Banach (BFPT). According to the generalized Gronwall inequalities, we prove the stability in the sense of Ulam–Hyers (UHS) of PHFSDE. We give some examples to show the effectiveness of our results. Full article

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 may be less equipped to handle them. It is recognized that the hepatitis B virus (HBV) infection remains a critical global public health issue. In this study, we develop a comprehensive model for HBV infection that includes vaccination and hospitalization through a fractional framework. It has been shown that the solutions of the recommended system of HBV infection are positive and bounded. We examine the steady states of the model and determine the basic reproduction number; denoted by ${\mathcal{R}}_0$. The qualitative and quantitative behavior of the model is demonstrated using mathematical skills and numerical techniques. It has been proved that the infection-free steady state of the system is locally asymptotically stable if ${\mathcal{R}}_0<1$ and unstable otherwise. Furthermore, the Ulam–Hyers stability (UHS) of the recommended fractional models is investigated and the significant conditions are provided. We present an iterative technique to visualize the dynamical behavior of the system. We perform different simulations to illustrate the effect of different input factors on the solution pathways of the system of HBV infection to conceptualize the role of parameters in the control and prevention of the infection. Full article

In this paper, we study a coupled fully hybrid system of $(\mathbf{k},\mathsf{\Phi })$–Hilfer fractional differential equations equipped with non-symmetric $(\mathbf{k},\mathsf{\Phi })$–Riemann-Liouville ($\mathcal{RL}$) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space ($\mathcal{GBS}$). Moreover, the Ulam–Hyers ($\mathcal{UH}$) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results. Full article

We apply Mittag–Leffler-type functions to introduce a class of matrix-valued fuzzy controllers which help us to propose the notion of multi-stability (MS) and to obtain fuzzy approximate solutions of matrix-valued fractional differential equations in fuzzy spaces. The concept of multi stability allows us to obtain different approximations depending on the different special functions that are initially chosen. Additionally, using various properties of a function of Mittag–Leffler type, we study the Ulam–Hyers stability (UHS) of the models. Full article

The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel $psi$-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the literature. The required results are proven using Banach’s contraction mapping and Krasnoselskii’s fixed-point theorem. Additionally, results pertaining to UH stability are obtained using traditional procedures of nonlinear functional analysis. Additionally, in light of our current findings, a more general challenge for the pantograph system is presented that includes problems similar to the one considered. We provide a pertinent example as an application to support the theoretical findings. Full article

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs. Full article

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or Riemann–Liouville (RL) derivatives, the system with Mittag–Leffler (ML)-type fractional derivatives can eliminate singularity such that the solution of the system has better analytical properties. Therefore, we concentrate on a nonlinear Langevin system of ML-type fractional derivatives affected by time-varying delays and differential feedback control in the manuscript. We first utilize two fixed-point theorems proposed by Krasnoselskii and Schauder to investigate the existence of a solution. Next, we employ the contraction mapping principle and nonlinear analysis to establish the stability of types such as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) as well as generalized UH and UHR. Lastly, the theoretical analysis and numerical simulation of some interesting examples are carried out by using our main results and the DDESD toolbox of MATLAB. Full article