Fifth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 10, 2003, pp. 23-31.

An adaptive numerical method for the wave equation with a nonlinear boundary condition

Azmy S. Ackleh, Keng Deng, & Joel Derouen

Abstract:
We develop an efficient numerical method for studying the existence and non-existence of global solutions to the initial-boundary value problem
$$\displaylines{ u_{tt}=u_{xx}\quad 0 less than x less than \infty,\; tgreater than 0,\cr -u_{x}(0,t)=h(u(0,t)) \quad t greater than 0,\cr u(x,0)=f(x),\quad u_{t}(x,0)=g(x) \quad 0 less than x less than \infty. }$$
The results by this numerical method corroborate the theory presented in [1]. Furthermore, they suggest that blow-up can occur for more general nonlinearities $h(u)$ with weaker conditions on the initial data $f$ and $g$.

Published February 28, 2003.
Subject classifications: 35B40, 35L05, 35L20, 65M25, 65N50.
Key words: Time-adaptive numerical method, blow-up time, blow-up rate.

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Azmy S. Ackleh
Department of Mathematics
University of Louisiana at Lafayette
Lafayette, Louisiana 70504, USA.
e-mail: ackleh@louisiana.edu
Keng Deng
Department of Mathematics
University of Louisiana at Lafayette
Lafayette, Louisiana 70504, USA.
e-mail: deng@louisiana.edu
  Joel Derouen
Department of Mathematics
University of Louisiana at Lafayette
Lafayette, Louisiana 70504, USA.
e-mail: jbd8438@louisiana.edu

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