scholarly journals Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xichao Sun ◽  
Junfeng Liu
Keyword(s):  
Weak Convergence ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Random Perturbation ◽  
Continuous Functions ◽  
Dirichlet Boundary Conditions ◽  
Fractional Noise ◽  
Fractional Heat Equation ◽  
Fractional Equations

We consider a class of stochastic fractional equations driven by fractional noise ont,x∈0,T×0,1  ∂u/∂t=Dδαu+ft,x,u+∂2BHt,x/∂t ∂x, with Dirichlet boundary conditions. We formally replace the random perturbation by a family of sequences based on Kac-Stroock processes in the plane, which approximate the fractional noise in some sense. Under some conditions, we show that the real-valued mild solution of the stochastic fractional heat equation perturbed by this family of noises converges in law, in the space𝒞0,T×0,1of continuous functions, to the solution of the stochastic fractional heat equation driven by fractional noise.

WEAK CONVERGENCE FOR QUASILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE WITH HURST PARAMETER H ∈ (½, 1)

2013 ◽  
Vol 13 (03) ◽  
pp. 1250024 ◽  
Author(s):  
TARIK EL MELLALI ◽  
YOUSSEF OUKNINE
Keyword(s):  
Heat Equation ◽  
Dirichlet Boundary ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Stochastic Heat Equation ◽  
Dirichlet Boundary Conditions ◽  
One Dimension ◽  
Hurst Parameter ◽  
Fractional Noise

In this paper, we consider a quasi-linear stochastic heat equation in one dimension on [0, 1], with Dirichlet boundary conditions driven by an additive fractional white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ ℕ which can approximate the fractional noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space [Formula: see text] of continuous functions, to the solution of the fractional noise driven SPDE.

scholarly journals Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

2020 ◽  
Vol 22 ◽  
pp. 1-9
Author(s):  
Kahsay Godifey Wubneh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Constant Coefficient ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Nonhomogeneous Heat ◽  
Interesting Solution

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

scholarly journals Backward stochastic differential equations with non-Markovian singular terminal values

2019 ◽  
Vol 19 (02) ◽  
pp. 1950006 ◽  
Author(s):  
Ali Devin Sezer ◽  
Thomas Kruse ◽  
Alexandre Popier
Keyword(s):  
Boundary Condition ◽  
Brownian Motion ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Constant Function ◽  
Dirichlet Boundary Conditions ◽  
Sample Paths ◽  
Terminal Values

We solve a class of BSDE with a power function [Formula: see text], [Formula: see text], driving its drift and with the terminal boundary condition [Formula: see text] (for which [Formula: see text] is assumed) or [Formula: see text], where [Formula: see text] is the ball in the path space [Formula: see text] of the underlying Brownian motion centered at the constant function [Formula: see text] and radius [Formula: see text]. The solution involves the derivation and solution of a related heat equation in which [Formula: see text] serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain, the BSDE has continuous sample paths with the prescribed terminal value.

scholarly journals On the exact controllability of a nonlinear stochastic heat equation

2006 ◽  
Vol 2006 ◽  
pp. 1-12
Author(s):  
Bui An Ton
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Exact Controllability ◽  
Stochastic Heat Equation ◽  
Dirichlet Boundary Conditions ◽  
Target Values

The exact controllability of a nonlinear stochastic heat equation with null Dirichlet boundary conditions, nonzero initial and target values, and an interior control is established.

The semilinear heat equation with a Heaviside source term

1992 ◽  
Vol 3 (4) ◽  
pp. 367-379 ◽  
Author(s):  
Roberto Gianni ◽  
Josephus Hulshof
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Initial Value Problem ◽  
Source Term ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Bounded Interval ◽  
Semilinear Heat Equation

We consider the initial value problem for the equation ut = uxx + H(u), where H is the Heaviside graph, on a bounded interval with Dirichlet boundary conditions, and discuss existence, regularity and uniqueness of solutions and interfaces.

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