Progress in Fractional Differentiation & Applications

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Newton cooling-law equation in terms of a fractional non-local time Caputo derivative of order 0 < a ≤ 1 is solved analytically by the conventional Laplace transform. Smooth solutions in terms of Mittag-Leffler function show two different behaviors when compared to the exponential decay solution from the classical integer-order model: 1) fast heat dissipation at short times, this is characterized by transient solutions showing faster cooling as a tends to 0; 2) slow heat dissipation at medium-large times, solutions in this regime exhibit slower cooling as a approaches 0. Moreover, for a < 1 and as time tends to infinity, the temperature decays algebraically with time rather than exponentially. On the other hand, we used the fractional complex transform method to derive the local fractional Newton’s law of cooling differential equation of order a. This model defined on Cantor sets, is analytically solved via the Laplace transform. Our staircase shaped solutions are compared with those from the model with Caputo derivative; similarities and differences between these two approaches are pointed out. Hopefully, this generalization of Newton’s law of cooling will allow both gaining a better insight into heat convection processes through fractal media and developing a wide variety of new applications.

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