On an almost free damping vibration equation using N-fractional calculus

In this paper, an almost free damping vibration equation is discussed by means of N-fractional calculus. Let ϕ∈℘°={ϕ,0≠|ϕν|<∞,ν∈R}. We focus on the following type of equation:ϕm+ε+ϕ1+(ε/m)·a+ϕ·b=0,where m is an integer and ab≠0, a2<4b, a>0, ϕ=ϕ(t), |ε|<1, ε,t∈R. In the case of m=2 and |ε|⪡1, we call this equation as an almost free damping vibration equation. So the solutions are investigated to be given using N-fractional calculus in the case of m=2. Furthermore, we illustrate the shapes of the solution according to the vibration of ε.