scholarly journals Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise

2018 ◽  
Vol 88 (318) ◽  
pp. 1715-1741 ◽  
Author(s):  
Max Gunzburger ◽  
Buyang Li ◽  
Jilu Wang
Keyword(s):  
White Noise ◽  
Convergence Rates ◽  
Time Discretization ◽  
Space Time ◽  
Fractional Pdes ◽  
Stochastic Time

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

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 380
Author(s):  
Wensheng Wang
Keyword(s):  
Differential Equations ◽  
Heat Equation ◽  
White Noise ◽  
Harmonic Analysis ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Space Time ◽  
The Other ◽  
Moduli Of Continuity ◽  
Fractional Pdes

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.

scholarly journals Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1917
Author(s):  
Junmei Wang ◽  
James Hoult ◽  
Yubin Yan
Keyword(s):  
White Noise ◽  
Linear Growth ◽  
Convergence Rates ◽  
Finite Difference Methods ◽  
Growth Conditions ◽  
Space Time ◽  
Spatial Discretization ◽  
Initial Value ◽  
Spatial Convergence ◽  
Lipschitz Conditions

Spatial discretization of the stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative space-time white noise is considered. The nonlinear terms f and σ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the fractionally integrated multiplicative space-time white noise are discretized by using the finite difference methods. Based on the approximations of the Green functions expressed by the Mittag–Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under some suitable smoothness assumptions of the initial value.

scholarly journals Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1251
Author(s):  
Wensheng Wang
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Fourth Order ◽  
Gaussian Processes ◽  
Three Dimensional ◽  
Space Time ◽  
Symmetry Analysis ◽  
Stochastic Heat Equation ◽  
Moduli Of Continuity

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. 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. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

2021 ◽  
Author(s):  
Annie Millet ◽  
Svetlana Roudenko ◽  
Kai Yang
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Random Variable ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Deterministic Setting

Abstract We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.

scholarly journals Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

2019 ◽  
Vol 27 (3) ◽  
pp. 155-182 ◽  
Author(s):  
Igor Voulis ◽  
Arnold Reusken
Keyword(s):  
Discontinuous Galerkin ◽  
Error Bounds ◽  
Convergence Rates ◽  
Time Discretization ◽  
Linear Constraints ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Parabolic Problems ◽  
Discretization Methods ◽  
Optimal Convergence

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

scholarly journals Auditory Space-Time Receptive Field Dynamics Revealed by Spherical White-Noise Analysis

2001 ◽  
Vol 21 (12) ◽  
pp. 4408-4415 ◽  
Author(s):  
Rick L. Jenison ◽  
Jan W. H. Schnupp ◽  
Richard A. Reale ◽  
John F. Brugge
Keyword(s):  
Receptive Field ◽  
White Noise ◽  
Noise Analysis ◽  
Space Time ◽  
White Noise Analysis ◽  
Auditory Space
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