  In this article, we seek exact charged spherically symmetric black holes
(BHs) with considering $f(\mathcal{R})$ gravitational theory. These BHs are
characterized by convolution and error functions. Those two functions depend on
a constant of integration which is responsible to make such a solution deviate
from the Einstein general relativity (GR). The error function which constitutes
the charge potential of the Maxwell field depends on the constant of
integration and when this constant is vanishing we can not reproduce the
Reissner-Nordstr\"om BH in the lower order of $f(\mathcal{R})$. This means that
we can not reproduce Reissner-Nordstr\"om BH in lower-order-curvature theory,
i.e., in GR limit $f(\mathcal{R})=\mathcal{R}$, we can not get the well known
charged BH. We study the physical properties of these BHs and show that it is
asymptotically approached as a flat spacetime or approach AdS/dS spacetime.
Also, we calculate the invariants of the BHS and show that the singularities
are milder than those of BH's of GR. Additionally, we derive the stability
condition through the use of geodesic deviation. Moreover, we study the
thermodynamics of our BH and investigate the impact of the
higher-order-curvature theory. Finally, we show that all the BHs are stable and
have radial speed equal to one through the use of odd-type mode.
