Search Results (21)

Fractional stochastic differential Equations (FSDEs) with time delays and non-instantaneous impulses describe dynamical systems whose evolution relies not only on their current state but also on their historical context, random fluctuations, and impulsive effects that manifest over finite intervals rather than occurring instantaneously. This combination of features offers a more precise framework for capturing critical aspects of many real-world processes. Recent findings demonstrate the existence, uniqueness, and Ulam–Hyers stability of standard fractional stochastic systems. In this study, we extend these results to include systems characterized by FSDEs that incorporate time delays and non-instantaneous impulses. We prove the existence and uniqueness of the solution for this system using Krasnoselskii’s and Banach’s fixed-point theorems. Additionally, we present findings related to Ulam–Hyers stability. To illustrate the practical application of our results, we develop a population model that incorporates memory effects, randomness, and non-instantaneous impulses. This model is solved numerically via the Euler–Maruyama method, and graphical simulations effectively depict the dynamic behavior of the system. Full article

This paper investigates the existence of square-mean S-asymptotically $(\omega ,c)$-periodic solutions for a class of neutral impulsive stochastic differential equations driven by fractional Brownian motion, addressing the challenge of modeling long-range dependencies, delayed feedback, and abrupt changes in systems like biological networks or mechanical oscillators. By employing semigroup theory to derive mild solution representations and the Banach contraction principle, we establish sufficient conditions–such as Lipschitz continuity of nonlinear terms and growth bounds on the resolvent operator—that guarantee the uniqueness and existence of such solutions in the space ${\mathcal{S}AP}_{\omega ,c}([0,\infty ),L^2(\Omega ,\mathbb{H}))$. The important results demonstrate that under these assumptions, the mild solution exhibits square-mean S-asymptotic $(\omega ,c)$-periodicity, enabling robust asymptotic analysis beyond classical periodicity. We illustrate these findings with examples, such as a neutral stochastic heat equation with impulses, revealing stability thresholds and decay rates and highlighting the framework’s utility in predicting long-term dynamics. These outcomes advance stochastic analysis by unifying neutral, impulsive, and fractional noise effects, with potential applications in control theory and engineering. Full article

In this paper, we develop a systematic review of the existing literature on the mathematical modeling of several aspects of pollinators. We selected the MathSciNet and Wos databases and performed a search for the words “pollinator” and “mathematical model”. This search yielded a total of 236 records. After a detailed screening process, we retained 107 publications deemed most relevant to the topic of mathematical modeling in pollinator systems. We conducted a bibliometric analysis and categorized the studies based on the mathematical approaches used as the central tool in the mathematical modeling and analysis. The mathematical theories used to obtain the mathematical models were ordinary differential equations, partial differential equations, graph theory, difference equations, delay differential equations, stochastic equations, numerical methods, and other types of theories, like fractional order differential equations. Meanwhile, the topics were positive bounded solutions, equilibrium and stability analysis, bifurcation analysis, optimal control, and numerical analysis. We summarized the research findings and identified some challenges that could inform the direction of future research, highlighting areas that will aid in the development of future research. Full article

Campylobacteriosis has been described as an ever-changing disease and health issue that is rather dangerous for different population groups all over the globe. The World Health Organization (WHO) reports that 33 million years of healthy living are lost annually, and nearly one in ten persons have foodborne illnesses, including Campylobacteriosis. This explains why there is a need to develop new policies and strategies in the management of diseases at the intergovernmental level. Within this framework, an advanced stochastic fractional delayed model for Campylobacteriosis includes new stochastic, memory, and time delay factors. This model adopts a numerical computational technique called the Grunwald–Letnikov-based Nonstandard Finite Difference (GL-NSFD) scheme, which yields an exponential fitted solution that is non-negative and uniformly bounded, which are essential characteristics when working with compartmental models in epidemic research. Two equilibrium states are identified: the first is an infectious Campylobacteriosis-free state, and the second is a Campylobacteriosis-present state. When stability analysis with the help of the basic reproduction number $R_0$ is performed, the stability of both equilibrium points depends on the $R_0$ value. This is in concordance with the actual epidemiological data and the research conducted by the WHO in recent years, with a focus on the tendency to increase the rate of infections and the necessity to intervene in time. The model goes further to analyze how a delay in response affects the band of Campylobacteriosis spread, and also agrees that a delay in response is a significant factor. The first simulations of the current state of the system suggest that certain conditions can be achieved, and the eradication of the disease is possible if specific precautions are taken. The outcomes also indicate that enhancing the levels of compliance with the WHO-endorsed SOPs by a significant margin can lower infection rates significantly, which can serve as a roadmap to respond to this public health threat. Unlike most analytical papers, this research contributes actual findings and provides useful recommendations for disease management approaches and policies. Full article

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in ${\mathcal{L}}^{\mathrm{p}}$ space. Our findings generalize existing results for fractional derivatives and space, as we formulate them in the Caputo–Katugampola fractional derivative and ${\mathcal{L}}^{\mathrm{p}}$ space. To support our theoretical results, we present an illustrative example. Full article

Cassava is the sixth most important food crop worldwide and the third most important source of calories in the tropics. More than 800 million people depend on this plant’s tubers and sometimes leaves. To protect cassava crops and the livelihoods depending on them, we developed a stochastic fractional delayed model based on stochastic fractional delay differential equations (SFDDEs) to analyze the dynamics of cassava mosaic disease, focusing on two equilibrium states, the state of being absent from cassava mosaic disease and the state of being present with cassava mosaic disease. The basic reproduction number and sensitivity of parameters were estimated to characterize the level beyond which cassava mosaic disease prevails or declines in the plants. We analyzed the stability locally and globally to determine the environment that would ensure extinction and its persistence. To support the theoretical analysis, as well as the reliable results of the model, the present study used a nonstandard finite difference (NSFD) method. This numerical method not only improves the model’s accuracy but also guarantees that cassava mosaic probabilities are positive and bounded, which is essential for the accurate modeling of the cassava mosaic processes. The NSFD method was applied in all the scenarios, and it was determined that it yields adequate performance in modeling cassava mosaic disease. The ideas of the model are crucial for exploring key variables, which affect the scale of cassava mosaic and the moments of intervention. The present work is useful for discerning the mechanism of cassava mosaic disease as it presents a solid mathematical model capable of determining the stage of cassava mosaic disease. Full article

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations. Full article

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing $\epsilon$ in front of the drift term and $\sqrt{\epsilon }$ in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in ${\mathrm{L}}^{\mathrm{p}}$ space, generalizing the case of $\mathrm{p}=2$. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives. Full article

This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties of the proposed method are rigorously analyzed. Afterward, the practical utility of the proposed numerical technique is validated through a series of test examples. These examples illustrate the method’s capability to produce accurate and efficient solutions. Full article

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in ${\mathcal{L}}^{\mathrm{p}}$ spaces with $\mathrm{p}\ge 2$ under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end. Full article

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study’s originality is the use of fresh data simulation techniques and different solution methodologies. Full article

We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity. Full article

This article describes a new form of an anticipated backward stochastic differential equation (BSDE) with a time-delayed generator driven by fractional Brownian motion, further known as fractional BSDE, with a Hurst parameter $H\in (1/2,1)$. This study expands upon the findings of the anticipated BSDE by considering the scenario when the driver is fractional Brownian motion rather instead of standard Brownian motion. Additionally, the generator incorporates not only the present and future but also the past. We will demonstrate the existence and uniqueness of the solutions to these equations by employing the fixed point theorem. Furthermore, an equivalent comparison theorem is derived. Full article

The method of studying the stability in the probability for nonlinear systems of stochastic difference equations is demonstrated on two systems with exponential and fractional nonlinearities. The proposed method can be applied to nonlinear systems of higher dimensions and with other types of nonlinearity, both for difference equations and for differential equations with delay. Full article

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results. Full article