In this research, the Smoothed Particle Hydrodynamics (SPH) method has been used to model the dynamic behavior of solids under large deformation. In this regard, a set of combined equations for linear momentum, deformation gradient tensor, volume mapping, and area mapping (the co-factor of the deformation gradient tensor) is derived in form of first-order conservation laws. Subsequently, the corrected SPH method has been employed for spatial discretization within the solution domain. To reduce computational costs, explicit temporal discretization is considered using the third-order Runge-Kutta method. In the SPH meshless method, neighboring point information is utilized for computations. The primary objective of employing these equations in Updated Lagrangian description, where neighboring points may change at each time step, is enabling the simulation of severe deformation, which poses challenges for mesh-based approaches. Hence, the modeling capability of the proposed numerical method is examined by simulating the behavior of materials undergoing severe plastic deformation in the forging process. The simulation of the necking example demonstrates the model's capability to predict plastic behavior accompanied by large deformations. Additionally, simulation in the forging process accompanied by very large and complex deformations showed that the proposed meshless model can simulate such deformations without the need for ALE formulation or remeshing.