Proportional diet model on time scales based on energy balance

In this study, a diet model based on energy balance is reconstructed using time-scale calculus and proportional derivative theory. The solutions to the model are obtained, and the solutions are examined on different time scales, such as Z, R, and 2^N₀. This approach allows for the evaluation of energy intake expenditure dynamics under a common framework. By selecting appropriate parameters, numerical examples are presented to illustrate the effects of time-scale selection and proportional order on body weight development. The obtained results indicate that the use of time scales and proportional derivatives provides a flexible and interpretable framework for evaluating dietary strategies.