Hilbert Space Approach to Partial Differential Equations

1969 ◽  
pp. 401-421
Author(s):  
ZALMAN RUBINSTEIN
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Space Approach

scholarly journals On some stochastic parabolic differential equations in a Hilbert space

2005 ◽  
Vol 2005 (2) ◽  
pp. 167-173 ◽  
Author(s):  
Khairia El-Said El-Nadi
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Partial Differential Operator ◽  
Parabolic Differential Equations ◽  
Partial Differential ◽  
Mixed Problems ◽  
Uniqueness Of The Solution ◽  
Weiner Process

We consider some stochastic difference partial differential equations of the form du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order, M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.

scholarly journals Correct solvability and representation of the solutions of Volterra integro-differential equations with exponential-fractional kernels

2019 ◽  
Vol 488 (5) ◽  
pp. 476-480
Author(s):  
V. V. Vlasov ◽  
N. A. Rautian
Keyword(s):  
Hilbert Space ◽  
Spectral Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Unbounded Operator ◽  
Strong Solutions ◽  
Abstract Form ◽  
Partial Differential ◽  
Operator Functions ◽  
Well Posed

For abstract integro-differential equations with unbounded operator coefficients in a Hilbert space, we study the well-posed solvability of initial problems and carry out spectral analysis of the operator functions that are symbols of these equations. This allows us to represent the strong solutions of these equations as series in exponentials corresponding to points of the spectrum of operator functions. The equations under study are the abstract form of linear integro-partial differential equations arising in viscoelasticity and several other important applications.

scholarly journals Two-scale homogenization of abstract linear time-dependent PDEs

2020 ◽  
pp. 1-41
Author(s):  
Stefan Neukamm ◽  
Mario Varga ◽  
Marcus Waurick
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Linear Time ◽  
Random Coefficients ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Equations Of Mathematical Physics

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

scholarly journals On behavior of solutions of non-linear differential equations in Hilbert space II

1988 ◽  
Vol 11 (2) ◽  
pp. 297-313
Author(s):  
Vladimir Schuchman
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Linear Differential

This paper deals with the behavior of solutions of non-linear ordinary differntial equations in a Hilbert space with applications to non-linear partial differential equations.

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