This talk presents a numerical approach for solving the Korteweg-de Vries (KdV) equation on a time-dependent domain with moving boundaries. The KdV equation is fundamentally given by $u_t + \alpha u u_x + \beta u_{xxx} = 0$. While problems on fixed domains are well-studied, moving boundaries introduce significant analytical and computational challenges.To address this, we map the non-cylindrical moving boundary domain into a fixed reference domain using a coordinate transformation. We then apply a Galerkin finite-element formulation based on cubic B-spline basis functions for spatial discretization, and a Crank-Nicolson scheme for temporal integration. Numerical simulations, including benchmarking against known solutions, are provided to illustrate wave dynamics and the interaction of solutions with the boundaries. The results demonstrate that the proposed method attains high accuracy and efficiency for tracking nonlinear waves in non-cylindrical domains.