In this paper, we present an exact solution of the Klein–Gordon equation in the framework of the fractional-dimensional space, in which the momentum and position operators satisfying the R-deformed Heisenberg algebras. Accordingly, three essential problems have been solved such as: the free Klein–Gordon equation, the Klein–Gordon equation with mixed scalar and vector linear potentials and with mixed scalar and vector inversely linear potentials of Coulomb-type. For all these considered cases, the expressions of the eigenfunctions are determined and expressed in terms of the special functions: the Bessel functions of the first kind for the free case, the biconfluent Heun functions for the second case and the confluent hypergeometric functions for the end case, and the corresponding eigenvalues are exactly obtained.

In this work, we investigate the relativistic quantum motions of spinzero scalar bosons via the Duffin-Kemmer-Petiau (DKP) equation with a positiondependent mass (PDM) system in the background of the topological defect spacetime 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 Cornelltype potential through the Bi-Confluent Heun differential equation. In fact, we have obtained the ground state energy for both potentials.

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