Numerical solution of the Good Boussinesq equation using a modified hybrid quintic B-spline-based differential quadrature method

This paper proposes a modified hybrid quintic B-spline differential quadrature method for solving the “Good” Boussinesq (GBQ) equation. The novelty of the proposed work lies in the constructing a new hybrid quintic B-spline basis function, by combining and modifying the quintic B-spline and quintic trigonometric B-spline basis functions within a unified differential quadrature framework. The convergence and stability of the proposed scheme are theoretically analyzed, and the corresponding spatial and temporal convergence rates are computed to validate the theoretical findings. Numerical experiments are conducted for problems with and without exact analytical solutions, and the numerical errors and CPU times are compared with those of existing methods to assess the accuracy, conservation properties, and computational efficiency of the proposed approach. The method is further applied to simulate single and double solitary wave solutions as well as the interaction of three solitary waves governed by the GBQ equation, accompanied by a systematic investigation of their dynamical behaviors. The results demonstrate that the proposed method accurately captures key physical phenomena, including solitary-wave propagation, elastic collisions, blow-up behavior, interactions with stationary solitons, and overtaking collisions, with reasonable agreement between the numerical and theoretical results.