Analytical Solution of the Generalized Fractional Newton’s Cooling Law by Generalized Fractional Laplace Transform Method

Abstract

This study presents an analytical framework for a generalized fractional formulation of Newton’s law of cooling based on a newly defined generalized fractional (GF) Laplace transform. The proposed formulation extends existing fractional models by incorporating both a fractional order and a non-negative continuous kernel function, thereby providing increased modelling flexibility and a more general representation of memory-dependent thermal processes. A rigorous definition of the GF Laplace transform is established, together with its fundamental properties, including linearity, existence conditions, and operational rules for generalized fractional derivatives and integrals. Within this framework, the classical Newton’s cooling equation is reformulated in terms of the GF derivative, and an explicit analytical solution is obtained using the proposed transform method. The resulting solution generalizes the classical exponential decay law and reduces to the standard model under appropriate parameter selection. In addition, the effects of the fractional order and kernel function on cooling dynamics were examined through several representative cases. The analysis showed that both parameters played a significant role in determining the rate of temperature decay. In particular, smaller fractional orders lead to stronger memory effects and slower convergence, whereas the choice of the kernel function determines the effective time scale of the system. These results demonstrate the capability of the generalized fractional framework to describe a wider range of thermal behaviors than classical and existing fractional models.