On the probabilistic approach to the solution of generalized fractional differential equations of Caputo and Riemann-Liouville type

This dissertation focuses on the study of generalized fractional differential equations involving a general class of non-local operators which are referred to as the generalized fractional derivatives of Caputo and Riemann-Liouville (RL) type. These operators were introduced recently as a probabilistic extension of the classical fractional Caputo and Riemann-Liouville derivatives of order β e (0,1) (when acting on regular enough functions). The main contribution of this work lies in displaying the use of stochastic analysis as a valuable approach for the study of fractional differential equations and their generalizations. The stochastic representations presented here also lead to many interesting potential applications, e.g., by providing new numerical approaches to approximate solutions to equations for which an explicit solution is not available.