A combination of time-scale calculus and a cross-validation technique used in fitting and evaluating fractional models

In this work, we demonstrate how fractional calculus and time-scale calculus can be combined beautifully to solve and fit a modeling problem. In addition, a cross-validation technique is used to evaluate the fitted model. The specific application that we consider is the one-compartment model. The one-compartment model is a first-order differential equation that describes drug concentration over time. It turns out that approximating the solution by using a fractional model allows us to get more accurate results for model fitting. To quantitatively verify this insight, we compare between a first-order model and an α -order fractional model using real data for drug concentration. Then the mean squared error and a cross-validation method are used to determine the model that provides the best fit and predictions for unseen data.