In this paper, a meshless method for analyzing the elastodynamic behavior of two-dimensional problems is introduced. The nonsingular method of fundamental solutions (NMFS) is combined with the method of particular solutions. To overcome singularities arising from point-shaped sources, a technique known as the boundary distributed source (BDS) method is used, which involves distributing sources across disks centered on the boundary. The NMFS is regarded as a mesh-free boundary method due to source point independence to the neighboring source details. The dual reciprocity method (DRM) with a suitable approximation is used to obtain the particular solution corresponding to the nonhomogeneous inertial term; moreover, added benefits of eliminating domain integration and readiness for computer algorithms anchor desirability. Cautious investigation of solutions to various numerical examples using DRM reinforces reliability and accuracy. Finally, the study comprehensively examines factors such as the number of collocation points, internal points, the radius of the circular disk, and the time-step size to assess their impact on achieving convergence.