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 consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Grünwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the likelihood function, we propose an explicit numerical scheme based on the truncated series of the derivative definition. By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations. To evaluate the methodology, a simulation was performed using synthetic data, where the bias and mean square error are calculated, the results give evidence of the effectiveness for the method and the suitable performance of the proposed model. Moreover, an example with real data is presented as evidence of the relevance of using a fractional model.

We use a conformable fractional derivative $G_T^{\alpha }$ through two kernels $T(t,\alpha )=e^{(\alpha -1)t}$ and $T(t,\alpha )=t^{1-\alpha }$ in order to model the alcohol concentration in blood; we also work with the conformable Gaussian differential equation (CGDE) of this model, to evaluate how the curve associated with such a system adjusts to the data corresponding to the blood alcohol concentration. As a practical application, using the symmetry of the solution associated with the CGDE, we show the advantage of our conformable approaches with respect to the usual ordinary derivative.