scholarly journals Fractional stochastic heat equation with piecewise constant coefficients

2020 ◽  
Vol 21 (01) ◽  
pp. 2150002
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Mounir Zili ◽  
Eya Zougar
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Divergence Form ◽  
Stochastic Heat Equation ◽  
Piecewise Constant ◽  
Constant Diffusion Coefficient ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Diffusion ◽  
Constant Coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

2021 ◽  
Vol 24 (02) ◽  
pp. 2150010
Author(s):  
Jingqi Han ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Fractional Brownian Motion ◽  
Stochastic Heat Equation ◽  
Index Formula ◽  
Hurst Index ◽  
Skorohod Integral ◽  
Fractional Brownian Motions ◽  
Infinite Dimensional ◽  
Lp Theory

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

scholarly journals Time-Dependent Simulation of Cosmic-Ray Shocks, Including Alfvén Transport

1994 ◽  
Vol 142 ◽  
pp. 969-973
Author(s):  
T. W. Jones
Keyword(s):  
Diffusion Coefficient ◽  
Cosmic Rays ◽  
Wave Energy ◽  
Cosmic Ray ◽  
Alfven Waves ◽  
Alfven Wave ◽  
Alfvén Waves ◽  
Constant Diffusion Coefficient ◽  
Constant Diffusion ◽  
Transport Effects

AbstractTime evolution of plane, cosmic-ray modified shocks has been simulated numerically for the case with parallel magnetic fields. Computations were done in a “three-fluid” dynamical model incorporating cosmic-ray and Alfvén-wave energy transport equations. Nonlinear feedback from the cosmic rays and Alfvén waves is included in the equation of motion for the underlying plasma, as is the finite propagation speed and energy dissipation of the Alfvén waves. Exploratory results confirm earlier, steady state analyses that found these Alfvén transport effects to be potentially important when the upstream Alfvén speed and gas sound speeds are comparable. As noted earlier, Alfvén transport effects tend to reduce the transfer of energy through a shock from gas to energetic particles. These studies show as well that the timescale for modification of the shock is altered in nonlinear ways. It is clear, however, that the consequences of Alfvén transport are strongly model dependent and that both advection of cosmic rays by the waves and dissipation of wave energy in the plasma will be important to model correctly when quantitative results are needed. Comparison is made between simulations based on a constant diffusion coefficient and more realistic diffusion models allowing the diffusion coefficient to vary in response to changes in Alfvén wave intensity. No really substantive differences were found between them.Subject headings: cosmic rays — MHD — shock waves

scholarly journals On Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIM

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Ásdís Helgadóttir ◽  
Arthur Guittet ◽  
Frédéric Gibou
Keyword(s):  
Diffusion Coefficient ◽  
Poisson Equation ◽  
Real Life ◽  
Ghost Fluid Method ◽  
Constant Diffusion Coefficient ◽  
The Poisson Equation ◽  
Constant Diffusion ◽  
Physical Phenomena ◽  
Irregular Interface ◽  
Better Than

We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the L∞-norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand.

DIRICHLET PROBLEMS IN A HALF-SPACE OF A HILBERT SPACE

2002 ◽  
Vol 05 (02) ◽  
pp. 257-291 ◽  
Author(s):  
ENRICO PRIOLA
Keyword(s):  
Hilbert Space ◽  
Half Space ◽  
Regularity Result ◽  
Interpolation Theory ◽  
Dirichlet Problems ◽  
Optimal Regularity ◽  
Hölder Continuous ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Coefficients

We study a homogeneous infinite dimensional Dirichlet problem in a half-space of a Hilbert space involving a second-order elliptic operator with Hölder continuous coefficients. Thanks to a new explicit formula for the solution in the constant coefficients case, we prove an optimal regularity result of Schauder type. The proof uses nonstandard techniques from semigroups and interpolation theory and involves extensive computations on Gaussian integrals.

The computer simulation of the boost diffusion oxidation process for the deep-case hardening of titanium alloys

2004 ◽  
Vol 120 ◽  
pp. 259-268
Author(s):  
J. Luo ◽  
Z. Zhang ◽  
H. Dong ◽  
T. Bell
Keyword(s):  
Diffusion Coefficient ◽  
Titanium Alloys ◽  
Concentration Dependence ◽  
Oxygen Diffusion ◽  
Oxidation Process ◽  
Case Hardening ◽  
One Dimensional ◽  
Constant Diffusion Coefficient ◽  
Constant Diffusion ◽  
Simulation Results

A one dimensional finite difference diffusion model for simulating the Boost Diffusion Oxidation (BDO) process of titanium alloys is developed and implemented as a window-based program. The program can simulate the BDO process for both constant diffusion coefficient and concentration dependent diffusion coefficient. It is found that to accurately simulate the BDO process, the concentration dependence of oxygen diffusion has to be taken into account. If the concentration dependence is taken as the Shamblen and Redden’s equation, the simulation results agree well with the experimental results.

Sign in / Sign up

Export Citation Format

Share Document