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 $t^{1-\alpha }$, $e^{(1-\alpha )t}$ and non-conformable $t^{-\alpha }$ kernels. The analytical solution for each kernel is given in terms of the conformable order derivative $0<\alpha \le 1$. 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.

In this paper, we present a general definition of a generalized integral operator which contains as particular cases, many of the well-known, fractional and integer order integrals.

In this paper, we present a number of Chebyshev type inequalities involving generalized integral operators, essentially motivated by the earlier works and their applications in diverse research subjects.

In this work we obtain integral inequalities of the Hermite-Hadamard type, using generalized derivatives of the Caputo type. Throughout the work, we see that several results reported in the literature are particular cases of those presented here.