Parametric nonlinear PDEs with multiple solutions: A PGD approach

Marianne  Beringhier; Adrien  Leygue; Francisco  Chinesta

doi:10.3934/dcdss.2016002

Article Contents

2016, Volume 9, Issue 2: 383-392. Doi: 10.3934/dcdss.2016002

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Parametric nonlinear PDEs with multiple solutions: A PGD approach

1.
P' Institute, UPR CNRS - ISAE-ENSMA, 1 avenue Clément Ader, BP 40109, F-86961 Futuroscope Chasseneuil Cedex
2.
GeM Institute, UMR CNRS - Ecole Centrale de Nantes, 1 rue de la Noë, BP 30179, F-44321 Nantes cedex 3

Received: February 28, 2015

Revised: September 30, 2015

Published: March 2016

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Abstract

This paper presents some insights into the determination, using the Proper Generalized Decomposition, of multiple solutions of nonlinear parametric partial differential equations. Although the Proper Generalized Decomposition (PGD) is well suited for computing the solution of, possibly nonlinear, parametric problems that vary smoothly with a physical parameter, no work has been achieved for the case of problems that exhibit multiple solutions for some values of a parameter. For two representative cases, we show how an appropriate parametrization, combined to a nonlinear solution procedure can be devised to describe and compute the multiple solutions of a PDE.

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Mathematics Subject Classification: Primary: 65P30, 68U20; Secondary: 74S30.

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