Septic B-spline collocation method spline collocation method spline collocation method for numerical solution of the mGRLW equation

In this paper, numerical solution of a modified generalized regularized long wave (mGRLW) equation are obtained by a new method based on collocation of septic B – splines. Applying the von – Neumann stability analysis, the proposed method is shown to be unconditionally stable.

TẠP CHÍ KHOA HỌC − SỐ 31/2019 51 SEPTIC B – SPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTION OF THE mGRLW EQUATION Nguyen Van Tuan Hanoi Metropolitan University Abstract: In this paper, numerical solution of a modified generalized regularized long wave (mGRLW) equation are obtained by a new method based on collocation of septic B – splines Applying the von – Neumann stability analysis, the proposed method is shown to be unconditionally stable The numerical result shows that the present method is a successful numerical technique for solving the GRLW equations and the mGRLW equations Keyword: mGRLW equation; septic B-spline; collocation method; finite difference Email: nvtuan@hnmu.edu.vn Received 27 March 2019 Accepted for publication 25 May 2019 INTRODUCTION In this work, we consider the solution of the mGRLW equation uÞ + uò + uỏ uò uòò uòòị = 0, x ∈ :a, b;, t ∈ :0, T;, with the initial condition u x, = f x , x ∈ :a, b;, (1) (2) and the boundary condition uß a, t = 0, uò b, t = ỗ ußß a, t = ußß b, t = ußßß a, t = ußßß b, t = 0, (3) where ε, μ, β, p are constants, μ > 0, è > 0, \ is a positive integer The equation (1) is called the modified generalized regularized long wave (mGRLW) equation If μ = 0, the equation (1) is called the generalized regularized long wave (GRLW) Equation (1) describes the mathematical model of wave formation and propagation in fluid dynamics, turbulence, acoustics, plasma dynamics, ect So in recent years, TRƯỜNG ĐẠI HỌC THỦ ĐÔ H 52 NỘI researchers solve the GRLW and mGRLW equation by both analytic and numerical methods In this present work, we have applied the septic B – spline collocation method to the mGRLW equations and GRLW equations This work is built as follow: in Section 2, numerical scheme is presented The stability analysis of the method is established in Section The numerical results are discussed in Section In the last Section, Section 5, conclusion is presented SEPTIC B – SPLINE COLLOCATION METHOD The interval :‚, é; is partitioned in to a mesh of uniform length h = x ´ − x by the 0, N such that knots x , i = ëëëëë a=x