Nowadays, the expanding application of differential equations in describing physical phenomenons associated with their limited analytical solutions leads to an increase in the importance of various numerical methods. In this thesis, Time-Weighted Residual Method (TWRM) solution is developed to investigate the effect of anisotropy, dispersion, convection, and damping of a medium on wave propagation, also to evaluate the solution of advection-diffusion problem. TWRM, a meshless method, is based on using Exponential Basis Functions (EBFs) which satisfy the governing equation. The main idea of this method is using displacement, velocity and acceleration pre-integration equations along with the equilibrium equation. In this method, the initial conditions are satisfied precisely and the equilibrium equation is satisfied using the time-weighted residual method. Boundary conditions are also satisfied by utilizing point collocation method and a set of boundary points at the end of each time step. The main advantage of this method is saving the information of each time step through the coefficients of EBFs, so the time solution advances without requiring any domain discretization, using just a feasible recursive relation for updating coefficients of EBFs. The content of this dissertation is arranged in three main parts. In the first part, considering two approaches based on the shape of wave propagation, TWRM is formulated to reach the solution of wave propagation problem in an anisotropic medium. In the next season, the governing equation of dispersive and convective wave propagation along with the PDE of advection-diffusion problem are defined, respectively. Afterward, the efficiency of TWRM in the solution of above-mentioned equations is evaluated in terms of accuracy and stability of results. In the last part of this study, the capability of TWRM is indicated by the analysis of the axial wave propagation in the medium including the effect of damping. To this end, the axial wave propagation in a pile is considered as a physical example to simulate this effect. At the end of each season, the considerable accuracy and stability of TWRM results are verified through a series of numerical examples. Key Words Time-Weighted Residual Method, Exponential Basis Functions, Anisotropic wave propagation, Dispersive wave propagation, Convective wave propagation, Advection-diffusion problem.