The numerical analysis of stochastic time-fractional equations exhibits a significantly low-order convergence rate since the limited regularity of model caused by the nonlocal operator and the presence of noise. In this work, we consider stochastic time-fractional equations driven by integrated white noise, where CDt alpha psi(x, t), 0 G alpha G 2 and It gamma W-center dot (x, t), 0 G gamma G 1. We first establish the regularity of the mild solution. Then superlinear convergence rate (E||psi(& centerdot;,tn)-psi n||2)21=O(tau alpha+gamma-alpha d4-2 1-epsilon) with sufficiently small epsilon term in the exponent is established based on the modified twostep backward difference formula methods. Here d represents the spatial dimension, psi n denotes the approximate solution at the n-th time step, and E is the expectation operator. Numerical experiments are performed to verify the theoretical results. To the best of our knowledge, this is the first topic on the superlinear convergence analysis for the stochastic time-fractional equations with integrated white noise.
This paper investigates two spectral volume (SV) methods applied to 2D linear hyperbolic conservation laws on rectangular meshes. These methods utilize upwind fluxes and define control volumes using Gauss-Legendre (LSV) and right-Radau (RRSV) points within mesh elements. Within the framework of Petrov-Galerkin method, a unified proof is established to show that the proposed LSV and RRSV schemes are energy stable and have optimal error estimates in the L2 norm. Additionally, we demonstrate superconvergence properties of the SV method at specific points and analyze the error in cell averages under appropriate initial and boundary discretizations. As a result, we show that the RRSV method coincides with the standard upwind discontinuous Galerkin method for hyperbolic problems with constant coefficients. Numerical experiments are conducted to validate all theoretical findings.
The saturation assumption plays a central role in much of the analysis of a posteriori error estimates and refinement algorithms for adaptive finite element methods. In this work we provide an analysis of this assumption in the simple setting of interpolation. We have proved elsewhere [Bank and Yserentant, Numer. Math., 131:1 (2015)] that interpolation error is both reliable and efficient as an a posteriori error estimate. Thus behavior of interpolation error is indicative of the behavior of the error in the exact finite element solution of a PDE as well as any practical a posteriori error estimate that is also reliable and efficient.
In this paper, a numerical method for solving nonlinear stochastic delay differential equations is proposed: two-step Milstein method. The mean square consistent and mean square convergence of the numerical method are studied. Through the relevant derivation, the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained, and it is proved that the mean-square convergence order of the numerical method is 1. Finally, the theoretical results are verified by numerical experiments.
In this research article,we present convenient analytical-approximate solutions for fluid flow models known as multi-dimensional Navier-Stokes equations containing time-fractional order by using a relatively new analytical method called modified generalized Mittag-Leffler function method.The Caputo fractional derivative is used to describe fractional mathematical formalism.The approximate solutions for five problems are implemented to demonstrate the validity and accuracy of the proposed method.It is also demonstrated that the solutions obtained from our method when α=1 coincide with the exact solutions,this is displayed by using some 2D and 3D plots for each problem.Moreover,the comparison between our outcomes with given exact solutions and results obtained by other methods in the literature besides absolute error is provided in some tables.Additionally,we offer some plots when α has different values to present the effect of fractional order on the solution of each suggested problem.The numerical simulation presented in this work indicates that the proposed method is efficient,reliable,accurate and easy which has less computational ability to give analytical-approximate solution form.So,this method can be extended to implement on different related problems arising in various areas of innovation and research.
