University of Kashan Soft Computing Journal 2322-3707 13 1 2024 08 22 Numerical simulation of water long waves modeled by nonlinear Boussinesq partial differential equation using a spectral approximation Numerical simulation of water long waves modeled by nonlinear Boussinesq partial differential equation using a spectral approximation 172 181 114264 10.22052/scj.2024.253459.1178 FA Hoorieh Fakhari Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran. Akbar Mohebbi Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran. Journal Article 2023 08 22 In this paper, the Galerkin method is proposed for the solution of the nonlinear Boussinesq partial differential equation describing water waves. The main idea is to use generalized Jacobi polynomials (GJPs) as basis functions to deal with spatial derivatives such that boundary conditions are satisfied. To avoid solving nonlinear equations, the Leap-frog and Crank-Nicolson method is proposed for time discretization of the equation. The error estimate of the proposed method is investigated and numerical results show the high accuracy and low CPU time of proposed method and confirmed the theoretical ones. Also, the obtained results show that the method is suitable for nonlinear and even-order partial differential equations.  In this paper, the Galerkin method is proposed for the solution of the nonlinear Boussinesq partial differential equation describing water waves. The main idea is to use generalized Jacobi polynomials (GJPs) as basis functions to deal with spatial derivatives such that boundary conditions are satisfied. To avoid solving nonlinear equations, the Leap-frog and Crank-Nicolson method is proposed for time discretization of the equation. The error estimate of the proposed method is investigated and numerical results show the high accuracy and low CPU time of proposed method and confirmed the theoretical ones. Also, the obtained results show that the method is suitable for nonlinear and even-order partial differential equations.  https://scj.kashanu.ac.ir/article_114264_22fb24fd8bf3f4c9c912f9d47e35e25f.pdf