Existence, uniqueness, and finite-time stability of solutions for Ψ-Caputo fractional differential equations with time delay

Document Type : Research Paper


1 LMACS Laboratory, Sultan Moulay Slimane University, Beni Mellal, Morocco.

2 Superior School of Technology, Sultan Moulay Slimane University, Khenifra, Morocco.


In this paper, we study the existence, uniqueness, and finite-time stability results for fractional delayed Newton cooling law equation involving Ψ-Caputo fractional derivatives of order α ∈ (0, 1). By using Banach fixed point theorem, Henry Gronwall type retarded integral inequalities, and some techniques of Ψ-Caputo fractional calculus, we establish the existence and uniqueness of solutions for our proposed model. Based on the heat transfer model, a new criterion for finite time stability and some estimated results of solutions with time delay are derived. In addition, we give some specific examples with graphs and numerical experiments to illustrate the obtained results. More importantly, the comparison of model predictions versus experimental data, classical model, and non-delayed model shows the effectiveness of our proposed model with a reasonable precision.


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