On Newton’s law of cooling with time delay and Ψ-Caputo fractional derivatives

Abstract In this paper, we study fractional delayed Newton’s law of cooling involving Ψ-Caputo fractional derivatives of order α ∈ (0, 1). The existence and uniqueness of solutions for our proposed model are derived from Banach fixed point theorem, Henry–Gronwall inequalities and some basic tools of Ψ-Caputo fractional calculus. In addition, a novel finite time stability criterion and some estimate results of solutions with time delay are established by using heat transfer model. We give also some specific examples with graphs and numerical experiment to illustrate the obtained results. More importantly, the comparison of model predictions versus experimental data, classical model and non-delayed model show the effectiveness of our proposed model with a reasonable precision.