  In this work we focus on substantial fractional integral and differential
operators which play an important role in modeling anomalous diffusion. We
introduce a new generalized substantial fractional integral. Generalizations of
fractional substantial derivatives are also introduced both in
Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental
properties of these operators. Finally, we consider a class of generalized
substantial fractional differential equations and discuss the existence,
uniqueness and continuous dependence of solutions on initial data.
