Abstract
In this communication, using a generalized conformable differential operator, a simulation of the well-known Newton’s law of cooling is made. In particular, we use the conformable t1−α, e(1−α)t and non-conformable t−α kernels. The analytical solution for each kernel is given in terms of the conformable order derivative 0<α≤1. Then, the method for inverse problem solving, using Bayesian estimation with real temperature data to calculate the parameters of interest, is applied. It is shown that these conformable approaches have an advantage with respect to ordinary derivatives.
Highlights
The fractional calculus idea was first suggested by Leibniz and L’Hopital in a letter three centuries ago; it is an area of classical mathematics which deals with derivatives and integrals of arbitrary orders [1,2]
The conformable derivative definition was first given by Khalil et al in [5,6], with 0 < α < 1, this operator shows a similarity to the integer order derivative and overcomes many of the shortcomings of the classical fractional derivatives [1,2]
We model the Newton’s law of cooling using a conformable and nonconformable kernels in a generalized conformable differential operator
Summary
The fractional calculus idea was first suggested by Leibniz and L’Hopital in a letter three centuries ago; it is an area of classical mathematics which deals with derivatives and integrals of arbitrary orders [1,2]. Many fractional derivative definitions have been introduced [3,4]. In recent years, the notion of conformable derivative was introduced in terms of an incremental quotient, which opened a new direction in this area: the conformable calculus. A new non conformable derivative definition has been introduced in [7]. These definitions are valid and work in the case 0 < α < 1, a general definition was needed for conformable derivatives of any order, integer or not, generalizing the well known conformable derivatives to higher orders [8]. In [20], it was performed an experimental setup to verify the effectiveness of the conformable derivative
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