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In this paper, we have studied some theoretical properties concerning the Caputo-Fabrizio fractional operators. Also we obtained expressions for the fractional integral and derivative of some elementary functions. As auxiliary results, general integration formulas for functions with radicals of any order are presented.

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This contribution describes the second stage of the creation of a language training system programmed in Python with the aim of application to speech therapy in spanish-speaking countries, starting the study in Cuba. The first stage of this research was carried out in Matlab by analyzing the dynamics of change of the centroids of the codebooks, extracted from words pronounced by a locutor. As second stage, the Variational Coefficient formula is used in order to estimate the percentage of effectiveness with which the announcer performs voice training. A modified approach to programming the variational coefficient is taken into account as a measure of dispersion of a group of vectors. The modification is given by taking the mean of the group of vectors as the vector that represents the phonetic boundaries of the word to be trained. Besides, a novel approach for word recognition is used, based on the K-Nearest Training Matrix (KNTM) algorithm that lays its foundations in the analysis of matrix similarity taken the Frobenius norm as a measure to distinguish similar or non-similar characteristics of a matrix with respect to a database of matrices. To reduce the computational cost of the program and speed up its proper functioning, the training matrices of the database are saved in files with a .tex extension, in this way after training process, the program should only read them and not recalculate them, which significantly reduces the running time of the algorithm.

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