  This article primarily aims at introducing a novel operational calculus of
Mikusi\'nski's type for the Dzherbashian-Nersesian operator. Using this
calculus, we are able to derive exact solutions for the forward and backward
source problems (BSPs) of a differential equation that features
Dzherbashian-Nersesian operator in time and intertwined with nonlocal boundary
conditions. The initial condition is expressed in terms of Riemann-Liouville
integral (RLI). Solution is presented using Mittag-Leffler type functions
(MLTFs). The outcomes related to the existence and uniqueness subject to
certain conditions of regularity on the input data are established.
