Galerkin-Averaging Method in Infinite-Dimensional Spaces for Weakly Nonlinear Problems

2001 ◽  
pp. 269-279
Author(s):  
Michal Fečkan
Keyword(s):  
Averaging Method ◽  
Nonlinear Problems ◽  
Weakly Nonlinear ◽  
Infinite Dimensional

Model of induction brazing of nonmagnetic metals using model order reduction approach

2018 ◽  
Vol 37 (4) ◽  
pp. 1515-1524 ◽  
Author(s):  
Pavel Karban ◽  
David Pánek ◽  
Ivo Doležel
Keyword(s):  
Heat Treatment ◽  
Model Order Reduction ◽  
Computation Time ◽  
Order Reduction ◽  
Nonlinear Problems ◽  
Model Order ◽  
Content Type ◽  
Weakly Nonlinear ◽  
Induction Brazing ◽  
Multiphysics Problems

Purpose A novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing. Design/methodology/approach The complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM. Findings The proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability. Research limitations/implications The quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated. Practical implications The presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials. Originality/value Proposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.

NONLINEAR PROBLEMS IN INFINITE INTERACTING PARTICLE SYSTEMS

2008 ◽  
Vol 11 (02) ◽  
pp. 179-211 ◽  
Author(s):  
ROBERT OLKIEWICZ ◽  
LIHU XU ◽  
BOGUSŁAW ZEGARLIŃSKI
Keyword(s):  
Configuration Space ◽  
Interacting Particle Systems ◽  
Nonlinear Problems ◽  
Particle Systems ◽  
Markov Semigroups ◽  
Infinite Dimensional ◽  
Interacting Particle

We introduce and study a class of nonlinear jump type Markov semigroups for systems with infinite dimensional configuration space.

A note on nonlinear problems depending on infinite— dimensional parameters

1982 ◽  
Vol 6 (11) ◽  
pp. 1185-1191 ◽  
Author(s):  
Allan L. Edelson ◽  
Maria Patrizia Pera
Keyword(s):  
Nonlinear Problems ◽  
Infinite Dimensional ◽  
Dimensional Parameters

scholarly journals A center manifold application: existence of periodic travelling waves for the 2D abcd-Boussinesq system

2017 ◽  
Vol 11 (12) ◽  
pp. 641-667
Author(s):  
Jose R. Quintero ◽  
Alex M. Montes
Keyword(s):  
Center Manifold ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Type System ◽  
Boussinesq System ◽  
Wave Regime ◽  
Center Manifold Reduction ◽  
Small Initial Data ◽  
Weakly Nonlinear ◽  
Infinite Dimensional

In this paper we study the existence of periodic travelling waves for the 2D abcd Boussinesq type system related with the three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation form an infinite-dimensional family, by characterizing them using a center manifold reduction of infinite dimension and codimension due to the fact that at the linear level we are dealing with an ill-posed mixed-type initial-value problem. As happens for the Benney-Luke model and the KP II model for wave speed large enough and large surface tension, we show that a unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). As a consequence of this fact, we show that the spatial evolution of bottom velocity is governed by a dispersive, nonlocal, nonlinear wave equation.

scholarly journals Asymptotical estimation of oscillatory integral

2009 ◽  
Vol 50 ◽  
Author(s):  
Aleksandras Krylovas
Keyword(s):  
Averaging Method ◽  
Wave Interaction ◽  
Oscillatory Integral ◽  
Parameters Estimation ◽  
Differential Systems ◽  
Weakly Nonlinear ◽  
Interaction Modelling ◽  
Nonlinear Differential Systems

The oscillatory integral is important for averaging of weakly nonlinear differential systems. Uniformly valid for parameters estimation of the integral can be use for substantiation of averaging method for wave interaction modelling.

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