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.

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 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.

Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander S. Leonov ◽  
Alexander N. Sharov ◽  
Anatoly G. Yagola
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Differential Operators ◽  
The Finite Element Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Element Method

Abstract A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young’s modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.

scholarly journals On Complex Singularity Analysis for Some Linear Partial Differential Equations in

2013 ◽  
Vol 2013 ◽  
pp. 1-30
Author(s):  
A. Lastra ◽  
S. Malek ◽  
C. Stenger
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equation ◽  
Singularity Analysis ◽  
Singular Set ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Majorant Series ◽  
Formal Power

We investigate the existence of local holomorphic solutionsYof linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solutionXof some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutionsYare shown to develop singularities along with estimates of exponential type depending on the growth's rate ofXnear the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders ofXin one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.

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