Combined Liouville–Caputo Fractional Differential Equation

Abstract

This paper studies a fractional differential equation combined with a Liouville–Caputo fractional differential operator, namely, LCDηβ,γQ(t)=λϑ(t,Q(t)),t∈[c,d],β,γ∈(0,1],η∈[0,1], where Q(c)=qc is a bounded and non-negative initial value. The function ϑ:[c,d]×R→R is Lipschitz continuous in the second variable, λ>0 is a constant and the operator LCDηβ,γ is a convex combination of the left and the right Liouville–Caputo fractional derivatives. We study the well-posedness using the fixed-point theorem, estimate the growth bounds of the solution and examine the asymptotic behaviours of the solutions. Our findings are illustrated with some analytical and numerical examples. Furthermore, we investigate the effect of noise on the growth behaviour of the solution to the combined Liouville–Caputo fractional differential equation.

Keywords

• well-posedness
• growth estimate
• asymptotic behaviours
• combined Liouville–Caputo fractional derivative
• numerical simulations
• stochastic models