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Two Fractal Versions of Newton’s Law of Cooling |
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PP: 133-143 |
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Author(s) |
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Francisco A. God´ınez,
Margarita Navarrete,
Oscar A. Ch´avez,
Alexis Merlin,
Jos´e R. Vald´es,
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Abstract |
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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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