Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a mems capacitor

A. E. Lindsay, Joceline C Lega

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

Original languageEnglish (US)
Pages (from-to)935-958
Number of pages24
JournalSIAM Journal on Applied Mathematics
Volume72
Issue number3
DOIs
StatePublished - 2012

Fingerprint

Singular Nonlinearity
Parabolic Partial Differential Equations
Quenching
Capacitor
Partial differential equations
Fourth Order
Capacitors
Singularity
Modeling
Strip
Partial differential equation
MEMS
Numerical methods
Finite-time Singularities
Derivatives
Adaptive Method
Diverge
Micro-electro-mechanical Systems
Nonlinear Partial Differential Equations
Unit Disk

Keywords

  • Biharmonic equations
  • Self-similar solutions
  • Singularity formation
  • Touchdown

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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abstract = "Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.",
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