Fractional Stochastic Integro-differential Equations Research Articles

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

Stochastic fractional integrodifferential equations with continuous conditions

In this paper, we use a smoothness method to obtain an existence of a strong solution for a general class of stochastic fractional integrodifferential equations. We established also the uniqueness of solution. We consider the case of continuous coefficients, which satisfy some so-called ρ-condition.

EULER MARUYAMA APPROXIMATIONS FOR A GENERAL CLASS OF STOCHASTIC FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

In this paper, we study a class of stochastic fractional integro differential equations under the non-Lipschitz conditions. Thanks to Euler Maruy- ama's numerical scheme, we prove existence and uniqueness of a solution. We obtain also the strong convergence.

On collocation-Galerkin method and fractional B-spline functions for a class of stochastic fractional integro-differential equations

In recent years, as detailed in several monographs, derivations of the fractional differential equations and fractional integral equations are based on random functional or stochastic equations, with the output that physical interpretation of the resulting fractional derivatives and fractional integrals has been elusive. In many different sciences and problems such as biological systems, environmental quality and natural resources engineering, and so on, stochastic equations and in some cases random functional have appeared. Despite the widespread use of stochastic fractional integro-differential equations (SFIDE), the analytical solution of this equation is not easy and in some cases it is impossible. Therefore, the existence of an efficient and appropriate numerical method can solve this problem.In this study and based on fractional derivative in the Caputo sense, we investigate a class of SFIDE due to Brownian motion by using fractional B-spline basis functions (FB-spline) and with the help of the collocation-Galerkin method as well as the trapezoidal integral law. In other words, a continuous operator problem (here was named as SFIDE) is transformed into a discrete problem by limited sets of basis functions with common assumptions and approximation methods. In the follow-up a system of linear equations is generated, which makes the analysis of the method be efficient. As an important advantage of this combined method is its flexible and easy implementation. Another advantage of the method is its ability to be implemented for different types of linear, non-linear and system of SFIDE, which are discussed in the body of manuscript. An accurate upper bound is obtained and some theorems are established on the stability and convergence analysis. The computational cost is estimated from the sum of the number arithmetic operations and a lower bound for approximation of this equation is formulated. Finally, by examining several examples, the computational performance of the proposed method effectively verifies the applicability and validity of the suggested scheme.

Optimal control of conformable fractional neutral stochastic integrodifferential systems with infinite delay

The main concern of the manuscript deals with the optimal control problem of conformable fractional neutral stochastic integrodifferential systems with infinite delay. This study is motivated by SAR and RAR systems advantage of providing control over many factors such as power, frequency, phase, polarization, incidence angle, spatial resolution and swath width, all of which are important when designing and operating a radar system. Initially, we investigate the existence of mild solutions for the conformable fractional stochastic integrodifferential equations with infinite delay using stochastic analysis techniques and Banach fixed point theorem. In the later part we establish the existence of mild solutions of the conformable fractional neutral stochastic integrodifferential system with infinite time delay. Furthermore, the existence of optimal control of the corresponding Lagrange optimal control problem is investigated. An example is provided to illustrate the applications of the obtained results. We explain the limitation we have with the existence software, and developed a numerical scheme to justify the theory.

A new effective coherent numerical technique based on shifted Vieta–Fibonacci polynomials for solving stochastic fractional integro-differential equation

A new effective coherent numerical technique based on shifted Vieta–Fibonacci polynomials for solving stochastic fractional integro-differential equation

Weighted fractional stochastic integro-differential equation with infinite delay

In this paper, we consider a weighted fractional stochastic integro-differential equation with infinite delay and nonzero initial values involving a Riemann–Liouville fractional derivative of order 1/2<alpha <1. The existence of a mild solution is investigated using fractional calculus, stochastic analysis, and the fixed point theorem. An example is also provided to illustrate the obtained result.

A Meshless Method for the Numerical Solution of Fractional Stochastic Integro-Differential Equations Based on the Moving Least Square Approach

A Meshless Method for the Numerical Solution of Fractional Stochastic Integro-Differential Equations Based on the Moving Least Square Approach

Fractional Stochastic Integro-Differential Equations with Nonintantaneous Impulses: Existence, Approximate Controllability and Stochastic Iterative Learning Control

In this paper, existence/uniqueness of solutions and approximate controllability concept for Caputo type stochastic fractional integro-differential equations (SFIDE) in a Hilbert space with a noninstantaneous impulsive effect are studied. In addition, we study different types of stochastic iterative learning control for SFIDEs with noninstantaneous impulses in Hilbert spaces. Finally, examples are given to support the obtained results.

Exponential stability of second-order fractional stochastic integro-differential equations

In this paper studies the exponential stability result is derived for the second-order fractional stochastic integro-differential equations (FSIDEs) driven by sub-fractional Brownian motion (sub-fBm). By constructing a successive approximation method, we present pth moment exponential stability result of second-order FSIDEs using stochastic analysis techniques and fractional calculus (FC). At last, an example is demonstrated to illustrate the obtained theoretical result.

Well-posedness and numerical simulations employing Legendre-shifted spectral approach for Caputo–Fabrizio fractional stochastic integrodifferential equations

This paper investigates the well-posedness of a class of FSIDEs utilizing the fractional Caputo–Fabrizio derivative. Herein, the well-posedness proofs are constructed by considering some applicable conditions and combining theories of Banach space, AAT, and FPST. Approximating the solutions of such equations is still challenging for many mathematicians today due to their randomness and the hardness of finding the exact one. For the numerical aim, we introduce some useful properties of the Legendre-shifted polynomials and employ them as a basis of the collocation spectral method. The idea of this scheme is to convert such stochastic equations into algebraic systems subject to [Formula: see text]-measurable independent parameters. The stochastic term is driven by one-dimensional standard Brownian motion which is the most familiar type and for simulating its trajectories we discuss an easy method. We rigorously analyze the convergence of the proposed technique and other error behavior-bound results. Finally, various tangible numerical applications are performed to verify the present scheme’s accuracy and great feasibility and support theoretical results. The acquired results reveal that the methodology used is effective and appropriate to deal with various issues in light of the fractional Caputo–Fabrizio derivative.

A modified Euler–Maruyama method for Riemann–Liouville stochastic fractional integro-differential equations

In this paper, we propose a modified Euler–Maruyama (EM) method for Riemann–Liouville stochastic fractional integro-differential equations with weakly singular kernels, and then analyse the strong convergence of the proposed EM method. Specifically, we transform the considered stochastic fractional integro-differential equation into its equivalent form of stochastic Volterra integral equations and derive the corresponding modified EM method. Then, the strong convergence order of the proposed method is established, where α is the order of Riemann–Liouville fractional derivative with . Finally, numerical experiments are demonstrated to support our theoretical results.

Lipschitz continuity in the Hurst index of the solutions of fractional stochastic volterra integro-differential equations

The problem of investigating the continuity in the Hurst index arises naturally in statistical inferences related to fractional Brownian motion. In this paper, based on the techniques of the Malliavin calculus, we introduce a method to deal with this problem. We first provide an explicit bound on the difference between two non-smooth functionals of Malliavin differentiable random variables. Then, we apply the obtained bound to show Lipchitz continuity of fractional stochastic Volterra integro-differential equations and its additive functionals.

A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations

&lt;abstract&gt;&lt;p&gt;A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.&lt;/p&gt;&lt;/abstract&gt;

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

&lt;p style='text-indent:20px;'&gt;This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \alpha = 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Finally, several numerical tests are reported for verification of the theoretical findings.&lt;/p&gt;

Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump

Abstract In this paper, we investigate the sufficient conditions for null controllability of noninstantaneous impulsive Hilfer fractional stochastic integrodifferential system with the Rosenblatt process and Poisson jump. The required results are obtained based on fractional calculus, stochastic analysis, and Sadovskii’s fixed point theorem. Finally, an example is given to illustrate the obtained results.

Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations

This work is an extended version of the ICCS 2020 conference paper [1]. The paper aims to present an efficient numerical method to quantify the uncertainty in the solution of stochastic fractional integro-differential equations. The numerical method presented here is a wavelet collocation method based on Legendre polynomials, and their deterministic and stochastic operational matrix of integration. The operational matrices are used to convert the stochastic fractional integro-differential equation to a linear system of algebraic equations. The accuracy and efficiency of the proposed method are validated through numerical experiments. Moreover, the results are compared with the numerical methods based on the Gaussian radial basis function (GA RBF) and thin plate splines radial basis function (TBS RBF) to show the superiority of the proposed method. Finally, concerning the real-world application, a stock market model has been simulated and the results are demonstrated.

Construction of operational matrices based on linear cardinal B-spline functions for solving fractional stochastic integro-differential equation

The main purpose of this paper is to develop a new method based on operational matrices of the linear cardinal B-spline (LCB-S) functions to numerically solve of the fractional stochastic integro-differential (FSI-D) equations. To reach this aim, LCB-S functions are introduced and their properties are considered, briefly. Then, the operational matrices based on LCB-S functions are constructed, for the first time, including the fractional Riemann-Liouville integral operational matrix, the stochastic integral operational matrix, and the integer integral operational matrix. The main characteristic of the new scheme is to convert the FSI-D equation into a linear system of algebraic equations which can be easily solved by applying a suitable method. Also, the convergence analysis and error estimate of the proposed method are studied and an upper bound of error is obtained. Numerical experiments are provided to show the potential and efficiency of the new method. Finally, some numerical results, for various values of perturbation in the parameters of the main problem are presented which can indicate the stability of the suggested method.

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.