An efficient computational technique for higher order KdV equation arising in shallow water waves

Abstract

This paper focuses on the computational study of seventh-order KdV equations via the quintic B-spline (QBS) finite element method (FEM). The KdV equations contain higher-order derivative terms. We transform the higher order into lower order and obtain a coupled system to ease the solution procedure. Forward difference combined with the theta-weighted scheme is used to discretize the temporal part. In addition, the quintic B-spline approximates the functions and their derivatives. The Von Neumann method is used to check stability of the scheme. The performance of the proposed method is evaluated using four test problems. The numerical results are presented in the tables to evaluate the accuracy and proficiency of the scheme at various collocation points. Furthermore, the approximate and exact solutions are compared graphically which match each other, showing validation of the proposed method.