A variational formulation of the Nagumo reaction–diffusion equation and the Nagumo telegraph equation

We consider the Nagumo reaction–diffusion equation ut=uxx+f(u)ut=uxx+f(u) and the Nagumo telegraph equation τutt+(1−τf′(u))ut=uxx+f(u)τutt+(1−τf′(u))ut=uxx+f(u), where f(u)=u(a−u)(1−u)f(u)=u(a−u)(1−u) as boundary value problems over the finite spatial interval x∈[0,L]x∈[0,L] and finite time t∈[0,t∗]t∈[0,t∗]. We are able to obtain approximate solutions for the boundary value problems via a variational technique. Also, we obtain numerical solutions for some special cases, in order to demonstrate the validity of the variational technique. Also, these numerical solutions are used to discuss the qualitative characteristics of the solutions for the specific initial data considered.