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.