Conformable Fractional Newton’s Law of Cooling for Extended Time Periods

This article presents an improved formulation of Newton’s law of cooling using the conformable fractional derivative to model long-term thermal behavior more accurately. A key feature of our approach is the use of the fractional time variable tγ, which introduces a simple scaling symmetry: the structure of the model remains unchanged even when time is proportionally stretched or compressed. This symmetry-based property provides additional flexibility compared to the classical formulation and enables the derivation of analytical solutions under both constant and non-constant ambient temperature. In particular, we incorporate sinusoidal models for ambient temperature to capture realistic environmental fluctuations over extended periods. Experimental measurements confirm that the conformable model achieves significantly better accuracy than traditional integer-order models. These results highlight the relevance of symmetry and fractional calculus in describing physical processes and demonstrate the potential of conformable methods for improving long-term thermal predictions.