A relativistic position--dependent mass system of bosonic field in cosmic string space--time background

In this work, we investigate the relativistic quantum motions of spin–zero scalar bosons via the Duffin–Kemmer–Petiau (DKP) equation with a position–dependent mass (PDM) system in the background of the topological defect space–time produced by a cosmic string. We determine the radial wave equation and obtain the exact analytical solutions of the wave equation for the linear and Cornell–type potential through the Bi–Confluent Heun differential equation. In fact, we have obtained the ground state energy for both potentials.