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

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.