An Averaging Principle for Caputo Fractional Stochastic Differential Equations with Compensated Poisson Random Measure

This article deals with an averaging principle for Caputo fractional stochastic differential equations with compensated Poisson random measure.The main contribution of this article is impose some new averaging conditions to deal with the averaging principle for Caputo fractional stochastic differential equations.Under these conditions,the solution to a Caputo fractional stochastic differential system can be approximated by that of a corresponding averaging equation in the sense ofmean square.

ERROR ESTIMATES OF FINITE ELEMENT METHODS FOR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.

The Averaging Principle for Stochastic Fractional Partial Differential Equations with Fractional Noises

The purpose of this paper is to establish an averaging principle for stochastic fractional partial differential equation of order a.>1 driven by a fractional noise.We prove the existence and uniqueness of the global mild solution for the considered equation by the fixed point principle.The solutions for SPDEs with fractional noises can be approximated by the solution for the averaged stochastic systems in the sense of p-moment under some suitable assumptions.

Bi-Orthogonal fPINN:A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs

Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal interactions.However,inherent uncertainties arise naturally in real applications due to random forcing or unknown material properties.Mathematical models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and nonlocal.Here,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent SFPDEs.Specifically,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak formulation.Since automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network output.The weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue crossings.Moreover,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward problems.We demonstrate the effectiveness of the BO-fPINN method for different benchmark problems.Specifically,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method fails.We then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial conditions.We study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN method.The results demonstrate the flexibility and efficiency of the proposed method,especially for inverse problems.We also present a simple example of transfer learning(for the fractional order)that can help in accelerating the training of BO-fPINN for SFPDEs.Taken together,the simulation results show that the BO-fPINN method can be employed to effectively solve time-dependent SFPDEs and may provide a reliable computational strategy for real applications exhibiting anomalous transport.

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

A class of fractional stochastic neutral functional differential equation is analyzed in this paper.With the utilization of the fractional calculations,semigroup theory,fixed point technique and stochastic analysis theory,a sufficient condition of the existence for p-mean almost periodic solution is obtained,which are supported by two examples.

ALMOST PERIODIC SOLUTIONS TO STOCHASTIC FRACTIONAL PARTIAL EQUATIONS IN FRACTIONAL POWER SPACE

In this paper, we prove the existence of pth moment almost periodic mild solutions in fractional power space to a class of stochastic fractional partial equations using the techniques of contraction fixed point principle, Schauder fixed point theorem and semigroup theorem.

Controllability of nonlinear stochastic fractional systems with distributed delays in control

In this paper we study the controllability of linear and nonlinear stochastic fractional systems with bounded operator having distributed delay in control.The necessary and sufficient conditions for controllability of the linear system is obtained.Also,the nonlinear system is shown controllable under the assumption that the corresponding linear system is controllable and using the Banach contraction principle.