scholarly journals Existence and uniqueness of global solutions to the stochastic heat equation with superlinear drift on an unbounded spatial domain

2021 ◽  
Author(s):  
Michael Salins
Keyword(s):  
Heat Equation ◽  
Existence And Uniqueness ◽  
Global Solutions ◽  
Spatial Domain ◽  
Stochastic Heat Equation ◽  
The Stochastic Heat Equation ◽  
Dynamic Weighting ◽  
Osgood Condition

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition [Formula: see text] along with additional restrictions. For example, consider the forcing [Formula: see text]. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.

scholarly journals Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

2010 ◽  
Vol 34 (3) ◽  
pp. 243-260
Author(s):  
Nathalie Eisenbaum ◽  
Mohammud Foondun ◽  
Davar Khoshnevisan
Keyword(s):  
Heat Equation ◽  
Stochastic Heat Equation ◽  
Isomorphism Theorem ◽  
The Stochastic Heat Equation

scholarly journals Initial measures for the stochastic heat equation

2014 ◽  
Vol 50 (1) ◽  
pp. 136-153 ◽  
Author(s):  
Daniel Conus ◽  
Mathew Joseph ◽  
Davar Khoshnevisan ◽  
Shang-Yuan Shiu
Keyword(s):  
Heat Equation ◽  
Stochastic Heat Equation ◽  
The Stochastic Heat Equation

scholarly journals Solution of the Stochastic Heat Equation with Nonlinear Losses Using Wiener-Hermite Expansion

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohamed A. El-Beltagy ◽  
Noha A. Al-Mulla
Keyword(s):  
Heat Equation ◽  
Perturbation Technique ◽  
Stochastic Heat Equation ◽  
Nonlinear Pdes ◽  
Deterministic System ◽  
Hermite Expansion ◽  
Expansion Technique ◽  
Stochastic Solution ◽  
The Stochastic Heat Equation ◽  
Non Gaussian

In the current work, the Wiener-Hermite expansion (WHE) is used to solve the stochastic heat equation with nonlinear losses. WHE is used to deduce the equivalent deterministic system up to third order accuracy. The solution of the equivalent deterministic system is obtained using different techniques numerically and analytically. The finite-volume method (FVM) with Pickard iteration is used to solve the equivalent system iteratively. The WHE with perturbation technique (WHEP) is applied to deduce more simple and decoupled equivalent deterministic system that can be solved numerically without iterations. The system resulting from WHEP technique is solved also analytically using the eigenfunction expansion technique. The Monte-Carlo simulations (MCS) are performed to get the statistical properties of the stochastic solution and to verify other solution techniques. The results show that higher-order solutions are essential especially in case of nonlinearities where non-Gaussian effects cannot be neglected. The comparisons show the efficiency of the numerical WHE and WHEP techniques in solving stochastic nonlinear PDEs compared with the analytical solution and MCS.

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