Analytical investigation of vesicle dynamics via the modified Riemann-Liouville fractional derivative: Mittag-Leffler function solution and comparative analysis with Caputo's derivative

The solution of fractional differential equations is a significant focus of current research, given their prevalence in various fields of application. This paper introduces an innovative exploration of vesicle dynamics using Jumarie's modified Riemann-Liouville fractional derivative within a five-dimensional fractional rigid sphere model. The study reveals an exact solution through the Mittag-Leffler function, providing a deep understanding of intricate vesicle dynamics, including alternative motions, such as tank-treading with over-damped and under-damped vesicle oscillations, respectively, TT-OD and TT-UD. A comparative analysis with Caputo's derivative emphasizes the effectiveness of these fractional derivatives, contributing not only to theoretical insights but also practical implications in applied mathematics and biophysical systems. The findings advance our understanding of complex vesicle behaviors, particularly in mimicking real cell-like behaviors, and pave the way for further research and applications in the field.