Turabi Geyikli

Petrov-Galerkin finite element method for solving the MRLW equation

AbstractIn this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and L2, L∞ error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigated.

Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation

A numerical solution to a generalized Korteweg-de Vries (KdV) equation is obtained using the Galerkin method with quadratic B-spline finite element method (FEM) over which the nonlinear term is locally linearized and using the Adomians Decomposition Method (ADM). Test problems concerning the motion and interaction of soliton solutions are used to compare the FEM with the ADM. The present methods extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.

Modelling solitary waves of a fifth-order non-linear wave equation

A numerical solution of a fifth-order non-linear dispersive wave equation is set up using collocation of seventh-order B-spline interpolation functions over finite elements. A linear stability analysis shows that this numerical scheme, based on a Crank–Nicolson approximation in time, is unconditionally stable. The method is used to model the behaviour of solitary waves.