NONLINEAR IMPLICIT FRACTIONAL DYNAMIC EQUATION ON TIME SCALE

Abstract

In this chapter we studied the existence and uniqueness of solution of a fractional dynamic equation with initial condition involving Caputo nabla (∇) - derivative on time scale. The proof of existence relies on Schauder’s fixed point theorem and the Arzela`-Ascoli theorem, where the uniqueness of the solution is given by the Banach contraction theorem on time scale. Beyond theoretical implications, our findings have tangible relevance in modeling various real-world phenomena. This includes biomechanical systems, where fractional dynamic equations aid in representing complex physiological processes. Additionally, applications in economics enable a more accurate portrayal of fractional-order systems, enhancing predictions in financial markets. In the realm of environmental science, these equations find utility in simulating pollutant dispersion, offering improved environmental management strategies. Overall, our work not only contributes to mathematical understanding but also broadens the scope of practical applications in diverse fields.