Quintic B-spline collocation method for numerical solution of the Generalized Benjamin-Bona-MahonyBurgers equation

144 TRNG I HC TH  H NI QUINTIC B-SPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTION OF THE GENERALIZED BENJAMIN-BONA-MAHONY- BURGERS EQUATION Nguyen Van Tuan1(1), Nguyen Duc Thuyet 2 1Hanoi Metropolitan University 2Vinh Phuc Vocational College Abstract: In this paper, numerical solutions of the Generalized Benjamin-Bona-Mahony- Burgers (GBBMB) equation are obtained by collocation of quintic B-splines-based method. 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 GBBMB equation. Keywords: GBBMB equation; quintic B-spline; collocation method; finite difference. 1. INTRODUCTION In this paper we consider the solution of the GBBMB equation: (1) with the initial condition: (2) and the boundary condition: (3) where are constants, is an integer. GBBMB equations play a dominant role in many branches of science and engineering. In the past several years, many different methods have been used to solution of the GBBMB equation and some their cases, see [1, 3, 5]. The paper is used quintic B-spline collocation method for equation (1). (1) Nhận bài ngày 15.7.2016; gửi phản biện và duyệt đăng ngày 15.9.2016 Liên hệ tác giả: Nguyễn Văn Tuấn; Email: [email protected] TP CH KHOA HC − S 8/2016 145 2. QUINTIC B – SPLINE COLLOCATION METHOD The interval is partitioned in to a mesh of uniform length by the knots such that: Our numerical study for GBBMB equation using the collocation method with quintic B-spline is to find an approximate solution to exact solution in the form: (4) are the quintic B-spline basis functions at knots, given by [4]. The value of and its derivatives may be tabulated as in Table 1. Table 1. and at the node points x 0 1 26 66 26 1 0 0 0 0 0 0 Using the finite difference method, from the equation (1), we have:

(5)
146 TRNG I HC TH  H NI The nolinear term in Eq. (5) can be approximated by using the following formulas which obtainted by applying the Taylor expansion So Eq. (5) can be rewritten as (6) Using the value given in Table 1, Eq. (6) can be calculated at the knots so that at Eq. (6) reduces to (7) Where: At Eq. (7) becames (8) Where: TP CH KHOA HC − S 8/2016 147 The system (8) consists of equations in the knowns To get a solution to this system, we need four additional constraints. These constraints are obtained from the boundary conditions (3) and can be used to eliminate from the system (8). Then, we get the matrix system equation: (9) where the matrix are penta-diagonal matrices and is the dimensional colum vector. The algorithm is then used to solve the system (8). We apply first the intial condition:










(10) then we need that the approximately solution is satisfied folowing conditions: (11) Eliminating and from the system (11), we get: where is the penta-diagonal matrix given by: and 54 60 6 0 0 0 ... 0 101 135 105 1 0 0 ... 0 4 2 4 1 26 66 26 1 0 ... 0 ... ... ... A ... ... ... 0 ... 0 1 26 66 26 1 105 135 101 0 ... 0 0 1 4 2 4 0 ... 0 0 0 6 60 54               =                 148 TRNG I HC TH  H NI 3. STABILITY ANALYSIS To apply the Von-Neumann stability for the system (7), we must first linearize this system. We have: (12) where is the mode number and is the element size. Being applicable to only linear schemes the nonlinear term is linearized by taking as a locallyconstant value The linearized form of proposed scheme is given as (13) Where: TP CH KHOA HC − S 8/2016 149 Substitretion of into Eq. (13) leads to: (14) Simplifying Eq. (14), we get Where: It is clear that Therefore, the linearized numerical scheme for the GBBMB equation is unconditionally stable. 4. NUMERICAL EXAMPLE We now obtain the numerical solution of the GBBMB equation for a problem. To show the efficiency of the present method for our problem in comparison with the exact solution, we report and using formula where is numerical solution and denotes exact solution. Example. Consider the GBBMB equation with . The exact of Eq. (1) is given in [7] where and represent the amplitude and veloeity of a single solitary wave initially centered at We choose the following parameters . 150 TRNG I HC TH  H NI Fig.1. The Physical Behaviour of Numerical Solutions of Example at Different Time Levels 0 ≤ t ≤ 5 Table 2. Erros at different time levels Errors t =1 t =2 t =3 t =4 t =5 0.0054252098 0.0109452151 0.0165516810 0.0223096252 0.0337610187 0.008207923949 0.01642287375 0.02463674556 0.0328621363 0.06382919069 5. CONCLUSIONS A numerical method based on collocation of quintic B-spline had been described in the previous section for solving GBBMB equation. A finite difference scheme had been used for discretizing time derivatives and quintic B-spline for interpolating the solution at is capable time level. From the test problem, the obtained resulft show that the present method is capable for solving GBBMB equation. REFERENCES 1. G. Arora, R. C. Mittal, B. K. Singh (2014), "Numerical solution of BBM-Burger equation with quartic B-spline collocation method", J. of Engineering Sci. and Technology, special Issue on ICMTEA 2013 conference, pp.104-116. 2. D. J. Evans and K. R. Raslan (2005), "Solitar waves for the generalized equal width (GEW) equation", International J. Computer Mathematics, Vol. 82, No. 4, April., pp.445-455. 3. C. Hai – tao, P. Xin – tian, Z. Lu – ming and W. Yi – ju (2012), "Numerical analysis of a linear – implicit average scheme for generalized Benjamin-Bona-Mahony-Burgers equation", J. of Applied Mathematics, Vol. 2012, Artich ID 308410. 4. P. M. Prenter (2008), "Spline and variational methods", Dover Publications, New York. TP CH KHOA HC − S 8/2016 151 5. K. Shin – ichi and M. Ming and O. Seiro (2000), Convergence to diffusion waves of the solutions for Benjamin-Bona-Mahony-Burgers equations, Applicable Analysis, Vol. 75 (3 – 40), pp.317-340. 6. Siraj-Ul-Islam, Fazal-I-Had and Ikram A. Tirmizi (2010),"Collocation method using quartic B-spline for numerical solution of the modified equal width wave equation", J. Appl. Math. Informatics, Vol. 28, No.3-4, pp.611-624. 7. G. Turabi, B. G. K. Seydi (2011), "Septic B – spline collocation Method for the numerical solution of the modified equal width wave equation", Applied Mathematics, 2, pp.739-749. 8. M. Zarebnia and R. Parvaz (2017), Numerical study of the Benjamin-Bona-Mahony-Burgers equation, Bol. Soc. Paran. Mat. Vol. 35 1, pp.127-138. PHƯƠNG PHÁP COLLOCATION VỚI CƠ SỞ B-SPLINE BẬC 5 GIẢI PHƯƠNG TRÌNH GENERALIZED BENJAMIN-BONA- MAHONY-BURGERS Tóm tắt: Trong bài báo này chúng ta sử dụng phương pháp collocation với cơ sở B – spline bậc 5 giải xấp xỉ phương trình generalized Benjamin – Bona – mahony – Burgers. Sử dụng phương pháp Von – Neumann hệ phương trình sai phân ổn định vô điều kiện. Kết quả số chứng tỏ phương pháp đưa ra hữu hiệu để giải phương trình trên. Từ khoá: Phương trình GBBMB, spline bậc 5, phương pháp collocation, phương pháp sai phân hữu hạn.