A new method for solving the mGRLW equation using a base of quintic b spline

42   TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI A NEW METHOD FOR SOLVING THE mGRLW EQUATION USING A BASE OF QUINTIC B - SPLINE Nguyen Van Tuan, Nguyen Thi Tuyet 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 quintic B – splines Applying the von – Neumann stability analysis, the proposed method is shown to be unconditionallystable The numerical algorithm is applied to some test problems consisting of a single solitary wave The numerical result shows that the present method is a successful numerical technique for solving the mRGLW equations Keywords: mGRLW equation, quintic B-spline, collocation method, finite differences Email: nvtuan@daihocthudo.edu.vn  Received 01 December 2017  Accepted for publication 25 December 2017  INTRODUCTION In this work, we consider the solution of the mGRLW equation  u + αu + εu u − μu − βu = 0,          (1)  x ∈ [a, b], t ∈ [0, T], with the initial condition  u(x, 0) = f(x), x ∈ [a, b],            (2)  u(a, t) = 0, u(b, t) = u (a, t) = u (a, t) =     u (a, t) = u (b, t) = 0,         (3)  and the boundary condition  where α, ε, μ, β, p are constants, μ > 0, > 0,  is an integer.  The equation (1) is called the modified generalized regularized long wave (mGRLW)  equation if μ = 0, the generalized regularized long wave (GRLW) equation if  μ = 0, the  regularized long wave (RLW) equation or Benjamin – Bona – Mohony (BBM) equation if  β = 1, p = 1,etc.      TẠP CHÍ KHOA HỌC  SỐ 20/2017 43 Equation (1) describes the mathematical model of wave formation and propagation in  fluid dynamics, turbulence, acoustics, plasma dynamics, ect.  So in recent years, researchers  solve the GRLW and mGRLW equation by both analytic and numerical methods.  In this present work, we have applied the quintic B – spline collocation method to the  mGRLW  equations.  This  work  is  built  as  follow:  in  Section  2,  numerical  scheme  is  presented. The stability analysis of the method is established in Section 3. The numerical  results are discussed in Section 4. In the last Section, Section 5, conclusion is presented.  QUINTIC B – SPLINE COLLOCATION METHOD The interval [ , ] is partitioned in to a mesh of uniform length h = x − x  by the  knots x , i = 0, N such that:  a=x