In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

Accurate turbidity classification is essential for maintaining water quality in various contexts, from drinking water to industrial processes. Traditional turbidimeters face challenges, including interference from colored substances, particle shape and size variations, and the need for regular calibration and maintenance. This paper implements a convolutional neural network (CNN) to classify water samples based on their turbidity levels. The dataset consisted of images captured under controlled laboratory conditions, with turbidity levels measured using a 2100P Portable Turbidimeter. The CNN achieved a classification accuracy of 97.00% in laboratory settings. When tested on real-world water body samples, the model maintained an accuracy of 85.00%. The results demonstrate that deep learning can effectively classify turbidity levels, offering a promising solution to overcome the limitations of traditional methods. The study highlights the potential of CNNs for accurate and efficient turbidity measurement, balancing accuracy with practical applicability in field conditions.

In this paper, some estimators are proposed for nonlinear dynamical systems with the general conformable derivative. In order to analyze the stability of these estimators, some Lyapunov-like theorems are presented, taking into account finite-time stability. Thus, to prove these theorems, a stability function is defined based on the general conformable operator, which implies exponential stability. The performance of the estimators is assessed by means of numerical simulations. Furthermore, a comparison is made between the results obtained with the integer, fractional, and general conformable derivatives.