Fluctuations near homogeneous states of chemical reactions with diffusion

1987 ◽  
Vol 19 (2) ◽  
pp. 352-370 ◽  
Author(s):  
Peter Kotelenez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Limit Theorem ◽  
Chemical Reactions ◽  
Stochastic Partial Differential Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Homogeneous State ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process

Conditions are given under which a space-time jump Markov process describing the stochastic model of non-linear chemical reactions with diffusion converges to the homogeneous state solution of the corresponding reaction-diffusion equation. The deviation is measured by a central limit theorem. This limit is a distribution-valued Ornstein–Uhlenbeck process and can be represented as the mild solution of a certain stochastic partial differential equation.

Fluctuations near homogeneous states of chemical reactions with diffusion

1987 ◽  
Vol 19 (02) ◽  
pp. 352-370
Author(s):  
Peter Kotelenez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Limit Theorem ◽  
Chemical Reactions ◽  
Stochastic Partial Differential Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Homogeneous State ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process

Conditions are given under which a space-time jump Markov process describing the stochastic model of non-linear chemical reactions with diffusion converges to the homogeneous state solution of the corresponding reaction-diffusion equation. The deviation is measured by a central limit theorem. This limit is a distribution-valued Ornstein–Uhlenbeck process and can be represented as the mild solution of a certain stochastic partial differential equation.

scholarly journals A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Wenyuan Liao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Time Integration ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Rosenbrock Method ◽  
Ode System ◽  
Nonlinear Parabolic

The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.

scholarly journals An Elliptic PDE with Convex Solutions

2018 ◽  
Vol 61 (1) ◽  
pp. 201-214
Author(s):  
Jon Warren
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Partial Differential Equation ◽  
Elliptic Pde ◽  
Uhlenbeck Process ◽  
Partial Differential ◽  
Ornstein Uhlenbeck Process

AbstractUsing a mixture of classical and probabilistic techniques, we investigate the convexity of solutions to the elliptic partial differential equation associated with a certain generalized Ornstein–Uhlenbeck process.

Stochastic partial differential equation with reflection driven by fractional noises

Stochastics ◽  
2019 ◽  
Vol 92 (1) ◽  
pp. 46-66
Author(s):  
Suxin Wang ◽  
Yiming Jiang ◽  
Yongjin Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stochastic Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Noises
Sign in / Sign up

Export Citation Format

Share Document