This study aims to extend the classical model using fractional calculus, a Caputo derivative of order is used in place of the integer-order time derivative. The first example explores Newton’s Law of Cooling by adding a fractional-order time derivative that accounts for memory effects and the strange cooling patterns observed in real systems. The second model uses fractional dynamics to elucidate the nonlinear cooling of a lumped system characterized by a variable heat transfer coefficient, depicted through a power-law relationship with temperature.
In this paper, the authors examine two examples of physically interconnected heat transfer problems within the framework of fractional calculus using two effective numerical methods, the predictor-corrector method and the Genocchi wavelet collocation method.
The Caputo derivative introduces hereditary features that are more consistence with experimental data, and numerical methods ensure stability and accuracy. Graphical illustrations show how fractional-order affects cooling dynamics.
The results emphasize the potential of fractional differential equations not only in modeling complex heat transfer phenomena but also for providing a robust numerical framework applicable to engineering, biomedical and environmental applications. Moreover, the model shows more freedom in fitting real-world cooling curves and is thus applicable to thermal systems where non-exponential decay occurs. Future studies could consider the inclusion of space-fractional dynamics or mechanisms of thermal feedback to enhance the physical relevance of such models.
