Modern technique to study Cauchy-type problem of fractional variable order differential equations

Abstract

In this work, we have explored the theoretical and numerical aspects of fractional calculus, with a particular focus on fractional derivatives and their approximation using finite difference methods. The study began with foundational concepts, such as the Riemann-Liouville fractional integral and derivative, the Gamma and Beta functions,and the notion of phase space. These elements provided the essential mathematical tools required to model and analyze systems governed by fractional differential equations. The second part of the study dealt with the existence and uniqueness of solutions. By examining fixed point theorems and different types of stability, especially Ulam’s stability, we were able to build a rigorous theoretical framework ensuring the validity and reliability of the fractional models being used.