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We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.

This paper focuses on a numerical study about the stochastic Navier–Stokes equations. Unlike previous studies, this paper focuses on studying these equations from the perspective of vortices. The vorticity–stream function method was proposed to deal with incompressible fluid flow. And a Crank–Nicolson Fourier pseudo-spectral method was put forward to solve the formulation of stream function equation. In addition, we have conducted some numerical experiments to observe the effects of random forcing on vortices in the fluid flow by utilizing stochastic solution.

Elastic compression stockings (ECSs) are essential for the prevention and treatment of venous disorders of the lower limbs. Finite element modeling (FEM) is an effective method for numerically analyzing ECS pressure performance for guiding ECS material design and pressure dose selection in treatment. However, existing FEM studies have primarily used the two-dimensional (2D) mechanical properties (i.e., properties along the wale and course directions) of ECS fabrics and ignored their three-dimensional (3D) mechanical properties (i.e., those along the thickness direction), causing deviations in pressure predictions. To address this limitation, the present study developed a new approach for determining the 3D mechanical properties of ECS fabrics through orthotropic theoretical analysis, analytical model development, FEM, and experimental testing and validation. The results revealed that the deviation ratios between the experimental and simulated pressure values of ECS fabrics was 19.3% obtained using the 2D material mechanical properties that was reduced to 10.3% obtained using the 3D material mechanical properties. Equivalently, the FEM simulation precision increased by 46.6%. These results indicate that the proposed approach can improve finite element analysis efficiency for ECS pressure prediction, thus facilitating the functional design of elastic compression materials for improving compression therapeutic efficacy.

Abstract In this paper, we investigate a class of Hilfer fractional stochastic differential equations with nonlocal conditions. We first study the existence of mild solutions of these equations by means of stochastic analysis theory, fractional calculations, and operator semigroup theory. Further, the existence of optimal pairs for the corresponding Lagrange control systems is investigated. Finally, an example is presented to illustrate our obtained results.

Abstract In this paper, we consider a class of nonlocal fractional stochastic differential equations driven by fractional Brownian motion with Hurst index H > 1 2 $H>\frac{1}{2}$ . Sufficient conditions for the existence and uniqueness of mild solutions are obtained. Finally, an example is presented to illustrate our obtained results.

In recent years, differential inclusions with fractional order have been extensively studied by many authors. However, up to now fractional evolution inclusions with Riemann- Liouville derivative have not been considered in the literature. In order to fill this gap, in this paper we establish sufficient conditions for the existence of solutions of fractional evolution inclusions involving the Riemann-Liouville fractional derivative. The cases of convex-valued and nonconvex-valued right-hand sides are considered and we present a version of Filippov's theorem for fractional semilinear differential inclusions with Riemann-Liouville derivative. [ABSTRACT FROM AUTHOR]