In this study, we define a new fractional derivative using the normalization \(t e^{\frac{\rho -1}{\rho }t}\) by means of the proportionality parameter \(\rho \) , which allows us to proportionally adjust the concept of the memory effect, one of the most powerful aspects of the fractional derivative. Then, we consider the Malthusian and Verhulst equations using this derivative and solve them using the power series method. We show that the series solutions are convergent and numerically estimate the radii of convergence of these series. We analyze the behavior of the solution series via rich numerical calculations.
