Vol. 7, No. 6, 2014

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Numerical results on existence and stability of steady state solutions for the reaction-diffusion and Klein–Gordon equations

Miles Aron, Peter Bowers, Nicole Byer, Robert Decker, Aslihan Demirkaya and Jun Hwan Ryu

Vol. 7 (2014), No. 6, 723–742
Abstract

In this paper, we study numerically the existence and stability of the steady state solutions of the reaction-diffusion equation, ${u}_{t}-a{u}_{xx}-u+{u}^{3}=0$, and the Klein–Gordon equation, ${u}_{tt}+c{u}_{t}-a{u}_{xx}-u+{u}^{3}=0$, with the boundary conditions: $u\left(-1\right)=u\left(1\right)=0$. We show that as $a$ varies, the number of steady state solutions and their stability change.

Keywords
reaction-diffusion, Klein–Gordon equation, stability, steady state solutions
Mathematical Subject Classification 2010
Primary: 35B30, 35B32, 35B35, 35K57, 35L71