First-order linear proportional dynamic equations

There are many applications of proportional fractional calculus that is a valuable tool for describing and understanding the dynamics of complex systems, providing greater flexibility and diversity compared to standard fractional operators in many areas. In the first part of this study, the place and importance of proportional derivative in the literature is stated. In the second part, some important concepts and definitions on the time scale are given. In the third section, the definition of proportional derivative on time scale and some of its basic properties are given. In the fourth chapter, first order linear proportional dynamic equations and solution methods are given. In the fifth chapter, the proportional Laplace transform and its properties on the time scale are given. In the last section, solutions of the Noyes-Whitney dynamic equation are given using the proportional Laplace transform.