STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

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Reduction of Asymptotic Approximate Expansion of Navier–Stokes Equation and Solution of Inviscid Burgers Equation by Similarity Transformation

Scientific Programming ◽

10.1155/2021/9054328 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Farhad Ali ◽

Wali Khan Mashwani ◽

Hamayat Ullah ◽

Ahmed Hussein Msmali ◽

Ikramullah Ikramullah ◽

...

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Similarity Transformation ◽

Nonlinear Differential Equations ◽

The Other ◽

Navier Stokes ◽

Partial Differential ◽

Navier Stokes Equation ◽

Analytic Methods

Symmetry methods for differential equations are a powerful tool for the solutions of differential equations. It linearizes nonlinear differential equations, reduces the order of differential equations, reduces the number of independent variables in partial differential equations, and solves almost all those differential equations for which the other analytic methods fail to solve them. Similarity transformation is a particular case of symmetries, but it is easy and often used to deal with differential equations. The similarity transformation can do all the aforementioned works. In this research, we use the similarity transformation to solve different nonlinear differential equations. Particularly, we will apply this transformation to the nonlinear Navier–Stokes partial differential equations to reduce them to ordinary differential equations. Ordinary differential equations are easy to deal with than partial differential equations. Some nonlinear physical examples of ODEs and PDEs are given to show that the similarity transformation solves those problems where the other analytic methods fail to work.

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DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025706002287 ◽

2006 ◽

Vol 09 (01) ◽

pp. 155-168 ◽

Cited By ~ 6

Author(s):

FULVIA CONFORTOLA

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Hilbert Spaces ◽

Stochastic Partial Differential Equations ◽

Existence And Uniqueness ◽

Backward Stochastic Differential Equations ◽

Uniqueness Result ◽

Infinite Dimensions ◽

Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

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