  Accelerating cosmological models are constructed in a modified gravity theory
dubbed as $f(R,T)$ gravity at the backdrop of an anisotropic Bianchi type-III
universe. $f(R,T)$ is a function of the Ricci scalar $R$ and the trace $T$ of
the energy-momentum tensor and it replaces the Ricci scalar in the
Einstein-Hilbert action of General Relativity. The models are constructed for
two different ways of modification of the Einstein-Hilbert action. Exact
solutions of the field equations are obtained by a novel method of integration.
We have explored the behaviour of the cosmic transit from an decelerated phase
of expansion to an accelerated phase to get the dynamical features of the
universe. Within the formalism of the present work, it is found that, the
modification of the Einstein-Hilbert action does not affect the scale factor.
However the dynamics of the effective dark energy equation of state is
significantly affected.
