Galerkin reduced-order modeling scheme for time-dependent randomly parametrized linear partial differential equations

2012 ◽  
Vol 92 (4) ◽  
pp. 370-398 ◽  
Author(s):  
Christophe Audouze ◽  
Prasanth B. Nair
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Reduced Order Modeling ◽  
Time Dependent ◽  
Partial Differential ◽  
Reduced Order ◽  
Linear Partial Differential Equations

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

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.

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