Fractional Stochastic Differential Equations Research Articles

Averaging principle for Hifer–Katugampola fractional stochastic differential equations

In this paper, we mainly study the averaging principle for a class of Hifer–Katugampola fractional stochastic differential equations driven by standard Brownian motion. Firstly, we establish the existence and uniqueness of mild solution for the considered system using Banach contraction principle. Then, under suitable assumptions, we demonstrate that the solution to the original differential equations converges to that of the averaged differential equations in the sense of mean square and probability as the time scale goes to zero. Finally, an illustrative example is provided to verify our theoretical results.

Advanced stability analysis of a fractional delay differential system with stochastic phenomena using spectral collocation method.

In recent years, there has been a growing interest in incorporating fractional calculus into stochastic delay systems due to its ability to model complex phenomena with uncertainties and memory effects. The fractional stochastic delay differential equations are conventional in modeling such complex dynamical systems around various applied fields. The present study addresses a novel spectral approach to demonstrate the stability behavior and numerical solution of the systems characterized by stochasticity along with fractional derivatives and time delay. By bridging the gap between fractional calculus, stochastic processes, and spectral analysis, this work contributes to the field of fractional dynamics and enriches the toolbox of analytical tools available for investigating the stability of systems with delays and uncertainties. To illustrate the practical implications and validate the theoretical findings of our approach, some numerical simulations are presented.

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Fractional Stochastic Partial Differential Equations: Numerical Advances and Practical Applications—A State of the Art Review

This article aims to provide a comprehensive review of the latest advancements in numerical methods and practical implementations in the field of fractional stochastic partial differential equations (FSPDEs). This type of equation integrates fractional calculus, stochastic processes, and differential equations to model complex dynamical systems characterized by memory and randomness. It introduces the foundational concepts and definitions essential for understanding FSPDEs, followed by a comprehensive review of the diverse numerical methods and analytical techniques developed to tackle these equations. Then, this article highlights the significant expansion in numerical methods, such as spectral and finite element methods, aimed at solving FSPDEs, underscoring their potential for innovative applications across various disciplines.

Non-confluence of fractional stochastic differential equations driven by Lévy process

Non-confluence of fractional stochastic differential equations driven by Lévy process

Statistical methods for the computation and parameter estimation of a fractional SIRC model with Salmonella infection

This study analyzes the fractional order SIRC epidemic model under stochastic fractional differential equations in the Caputo sense. This article describes Salmonella infection in animal herds. For the deterministic system, we explain solution positivity and boundness. We also prove that the fractional stochastic solution exists and is unique. Other criteria considered include non-negativity of solutions, local and global stability analyses, Hyers-Ulam stability analysis, and sensitivity analysis for the deterministic system. Furthermore, according to the truncated Ito-Taylor expansion, we apply a numerical method to solve the stochastic fractional SIRC epidemic model, namely the Milstein method, to solve the stochastic fractional SIRC epidemic model. A comparison of the approximation solution and the corresponding deterministic model for different sample paths shows the efficiency of the numerical method. In addition, graphs and error tables provide insight into numerical experiments' results. The stochastic nature of the model allows random fluctuations in Salmonella infection spread. By incorporating uncertainty into the model, we gain a more realistic understanding of the epidemic dynamics and can better evaluate the effectiveness of control measures. Additionally, the numerical method used to solve the stochastic fractional SIRC epidemic model provides valuable insights into the variability of the results. This enhances our ability to make informed decisions about managing and preventing bacterial infections in animal herds.

Strong convergence analysis of spectral fractional diffusion equation driven by Gaussian noise with Hurst parameter less than [formula omitted]

We propose and analyze an efficient time discretization for the spectral fractional stochastic partial differential equation with Hurst parameter less than 12. By using variable substitution, the original equation is transformed into a system of equations, which includes a partial differential equation and a stochastic integral equation. Then the time discretization is composed of two parts. We discretize the partial differential equation in time by using a semi-implicit Euler scheme. Integration by parts is used to obtain the approximation of stochastic integral. We show the optimal error estimates of the time discretization in the strong sense. Finally, we provide some numerical examples to verify these theoretical results.

Approximate controllability for Hilfer fractional stochastic differential systems of order 1

In this paper, we analyse the approximate controllability of Hilfer fractional stochastic differential systems of order 1 < μ < 2 in Hilbert spaces. The primary findings are carried out by using fractional calculus, stochastic analysis theory, cosine family, and Schauder's fixed point theorem. In particular, we draw a new set of sufficient conditions for the approximate controllability of Hilfer fractional stochastic differential equations of order 1 < μ < 2 under the assumption that the corresponding linear system is approximately controllable. In the end, an example is provided to illustrate the derived theory.

Annealed fractional Lévy–Itō diffusion models for protein generation

Protein generation has numerous applications in designing therapeutic antibodies and creating new drugs. Still, it is a demanding task due to the inherent complexities of protein structures and the limitations of current generative models. Proteins possess intricate geometry, and sampling their conformational space is challenging due to its high dimensionality. This paper introduces novel Markovian and non-Markovian generative diffusion models based on fractional stochastic differential equations and the Lévy distribution, allowing for a more effective exploration of the conformational space. The approach is applied to a dataset of 40,000 proteins and evaluated in terms of Fréchet distance, fidelity, and diversity, outperforming the state-of-the-art by 25.4%, 35.8%, and 11.8%, respectively.

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements.

Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system’s solutions, we investigate stochasticity’s influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

Qualitative Analysis of Fractional Stochastic Differential Equations with Variable Order Fractional Derivative

Qualitative Analysis of Fractional Stochastic Differential Equations with Variable Order Fractional Derivative

The Delayed Effect of Multiplicative Noise on the Blow-Up for a Class of Fractional Stochastic Differential Equations

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.

Khasminskii Approach for ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo Fractional Stochastic Pantograph Problem

In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-sense accompanied by Brownian movement. Under certain assumptions, we are able to approximate solutions for FSPEs by solutions to averaged stochastic systems in the sense of mean square. Analysis of system solutions before and after the average allows extending the classical Khasminskii approach to random fractional differential equations in the sense of ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo. For clarity, we present at the end an applied example to facilitate the clarification of the theoretical results obtained

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise

Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.

A numerical approach based on Pell polynomial for solving stochastic fractional differential equations

A numerical approach based on Pell polynomial for solving stochastic fractional differential equations

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

In this paper, we first prove the existence, uniqueness and continuity dependence on the initial value of multi-term Caputo tempered fractional stochastic differential equations with initial value. Then, an Euler-Maruyama (EM) method is presented for solving the considered equation. The strong convergence of the presented EM method is strictly investigated with the order to be min{αm−αm−1,αm−0.5} with 0 < α 1 < α 2 < ⋯ < α m−1 < α m < 1, α m > 0.5, and m is a given positive integer. Finally, three numerical examples are provided to support our theoretical findings.

Euler–Maruyama methods for Caputo tempered fractional stochastic differential equations

In this paper, we introduce the initial value problem of Caputo tempered fractional stochastic differential equations and then study the well-posedness of its solution. Further, a Euler–Maruyama (EM) method is derived for solving the considered problem. The strong convergence order of the derived EM method is proved to be α − 1 2 with 1 2 < α < 1 . Additionally, a fast EM method is also developed which is based on the sum-of-exponentials approximation. Finally, numerical experiments are given to support the theoretical findings of the above two methods and verify computational efficiency of the fast EM method. The fast EM method can greatly improve the computational performance of the original EM method.