Hossein Hosseinzadeh

This thesis presents a comprehensive numerical framework for solving the nonlinear Rubenchik-Zakharov system of partial differential equations using a novel meshless approach. The study begins with a concise review of the historical development and foundational principles of finite difference methods. Subsequently, a detailed exposition of partial differential equations is provided, leading to the derivation of numerical schemes based on Taylor series expansions. Through a rigorous comparison of local and global truncation errors, a high-order, stable discretization method is selected for the PDE system. The core of this research involves the numerical solution of the coupled nonlinear Rubenchik-Zakharov system using a hybridized Radial Basis Function-Finite Difference method. The proposed scheme leverages polyharmonic splines as radial basis functions for spatial discretization, combined with a fourth-order Runge-Kutta method for temporal integration. A principal advantage of this methodology is its local approximation of differential operators, which generates sparse system matrices and significantly reduces computational overhead compared to global Radial Basis Function methods. The accuracy and robustness of the proposed solver are rigorously validated through comparisons with analytical solutions and a series of numerical examples. The results demonstrate that the method achieves high precision with minimal error, establishing its efficacy for tackling complex problems across various scientific domains. This work constitutes a significant advancement in the development of accurate and stable numerical methods within the fields of applied mathematics and numerical analysis.