QUINTIC BQUINTIC B----SPLINE COLLOCATION METHOD FOR NUMERICAL SPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTION A MODI SOLUTION A MODIFIED GRLW EQUATIONS FIED GRLW EQUATIONS FIED GRLW EQ
Trang 1QUINTIC B
QUINTIC B SPLINE COLLOCATION METHOD FOR NUMERICAL SPLINE COLLOCATION METHOD FOR NUMERICAL
SOLUTION A MODI SOLUTION A MODIFIED GRLW EQUATIONS FIED GRLW EQUATIONS FIED GRLW EQUATIONS
Nguyen Van Tuan
Hanoi Metropolitan University
Abstract
Abstract: In this paper, numerical solution of a modified generalized regularized long wave (mGRLW) equation are obtained by a method based on collocation of quintic 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 and mRGLW equations that they have real exact solutions
Keywords
Keywords: mGRLW equation; quintic B-spline; collocation method; finite difference
Email: nvtuan@daihocthudo.edu.vn
Received 12 July 2017
Accepted for publication 10 September 2017
1 INTRODUCTION
In this paper we consider the solution of the mGRLW equation:
x ∈ a, b, t ∈ 0, T,
with the initial condition:
and the boundary condition:
¬ ua, t 0, ub, t 0u£a, t u£a, t 0
where α, ε, μ, β, p are constants, μ f 0, β f 0, p is an integer
The mGRLW (1) is called the generalized regularized long wave (GRLW) equation if
μ 0, the generalized equal width (GEW) equation if α 0, μ 0, the regularized long wave (RLW) equation or Benjamin – Bona – Mohony (BBM) equation if β 1,
p 1, etc
Trang 2Equation (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 The GRLW equation is solved by finite difference method [6], Petrov – Galerkin method [8], distributed approximating functional method [7], IMLS – Ritz method [3], methods use the
B – spline as the basis functions [2], exact solution methods [9] The mGRLW is solved by reproducing kernel method [4], time – linearrization method [5], exact solution method [1]
In this present paper, we have applied the pentic B – spline collocation method to the GRLW and 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
2 QUINTIC B – SPLINE COLLOCATION METHOD
knots x¯, i 0, NTTTTT such that:
a x~; x: ; ⋯ ; x²:; x² b
Our numerical study for mGRLW equation using the collocation method with quintic B-spline is to find an approximate solution Ux, t to exact solution ux, t in the form: Ux, t ∑²9 δ¯tB¯x,
B¯x are the quintic B-spline basis functions at knots, given by [4]
B¯x h1¶
·
¸
¸
¸
¸
¹
¸
¸
¸
¸
º x / x¯»¶, x¯»Q x Q x¯
x / x¯»¶/ 6x / x¯¶, x¯Q x Q x¯:
x / x¯»¶/ 6x / x¯¶# 15x / x¯:¶, x¯:Q x Q x¯
x / x¯»¶/ 6x / x¯¶# 15x / x¯:¶/ /20x / x¯¶, x¯Q x Q x¯9:
x / x¯»¶/ 6x / x¯¶# 15x / x¯:¶/ 20x / x¯¶# #15x / x¯9:¶, x¯9:Q x Q x¯9
x / x¯»¶/ 6x / x¯¶# 15x / x¯:¶/ 20x / x¯¶# #15x / x¯9:¶/ 6x / x¯9¶, x¯9Q x Q x¯9»
0, x ; x¯»∪ x f x¯9» The value of B¯x and its derivatives may be tabulated as in Table 1
U¯ δ¯# 26δ¯:# 66δ¯# 26δ¯9:# δ¯9
U′¯5h /δ¯/ 10δ¯:# 10δ¯9:# δ¯9 U′′¯ 20hδ¯# 2δ¯:/ 6δ¯# 2δ¯9:# δ¯9
Trang 3Table 1 ¿, ¿′, and ¿′′ at the node points
h
40
Using the finite difference method, from the equation (1), we have:
u / βu££Ç9:/ u / βu££Ç
¥u£Ç9:# u¥u£Ç
Ç9:# u£Ç
2 / μu££Ç9:2# u££Ç 0
(5) The nolinear term u¥u£Ç9: in Eq (5) can be approximated by using the following formulas which obtainted by applying the Taylor expansion:
u¥u£Ç9: u¥Çu£Ç9:# puÇ¥:u£ÇuÇ9:/ puÇ¥u£Ç
So Eq (5) can be rewritten as
u / βu££Ç9:#Δt
2 /u££
Ç9:# εu¥Çu£Ç9:# pεuÇÉ:u£ÇuÇ9:# αu£Ç9:
u / βu££Ç#Δt
2 μu££
Ç # p / 1εu¥Çu£Ç/ αu£Ç
(6) Using the value given in Table 1, Eq (6) can be calculated at the knots x¯, i 0, NTTTTT so that at - x~, Eq (6) reduces to:
a¯:δ¯Ç9:# a¯δ¯:Ç9:# a¯»δ¯Ç9:# a¯Êδ¯9:Ç9:# a¯¶δ¯9Ç9: b¯:δ¯Ç # b¯δ¯:Ç # b¯»δ¯Ç #b¯Êδ¯9:Ç # b¯¶δ¯9Ç , (7) Where:
a¯: 2h/ 5hαΔt / 5hεΔtL¥¯:# 5hpεΔtL¯:¥: L¯/ 20μΔt / 40β;
a¯ 52h/ 50hαΔt / 50hεΔtL¥¯:# 130hpεΔtL¯:¥: L¯/ 40μΔt / 80β;
a¯» 132h# 330hpεΔt # 330hβΔtL¯:¥: L¯# 240β;
a¯Ê 52h# 50hαΔt # 50hεΔtL¥¯:# 130hpεΔtL¯:¥: L¯/ 40μΔt / 80β;
a¯¶ 2h# 5hαΔt # 5hεΔtL¥¯:# 5hpεΔtL¯:¥: L¯/ 20μΔt / 40β;
b¯: 2h# 5hαΔt / 5hp / 1εΔtL¥¯:# 20μΔt / 40β;
b¯ 52h# 50hαΔt / 50hp / 1εΔtL¥¯:# 40μΔt / 80β;
b¯» 266h/ 60μΔt # 120β;
Trang 4b¯Ê 52h/ 50hαΔt # 50hp / 1εΔtL¥¯:# 40μΔt / 80β;
b¯¶ 2h/ 5hαΔt # 5hp / 1εΔtL¥¯:# 20μΔt / 40β;
L¯: δ¯# 26δ¯:# 66δ¯# 26δ¯9:# δ9
L¯ /δ¯/ 10δ¯:# 10δ¯9:# δ¯9
The system (7) consists of N # 1 equations in the N # 5 knowns
δ, δ:, … , δ²9:, δ²9Ï
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 (7) Then, we get the matrix system equation
where the matrix AδÇ, BδÇ are penta-diagonal N # 1 Ò N # 1 matrices and r is the N # 1 dimensional colum vector The algorithm is then used to solve the system (8)
We apply first the intial condition:
Ux, 0 ∑²9 δ¯~B¯x,
¯ (9)
then we need that the approximately solution is satisfied folowing conditions:
·
¸
¹
¸
º U Ux¯, 0 fx¯
£x~, 0 U£a, 0 0
U£x², 0 U£b, 0 0
U££x~, 0 U££a, 0 0
U££x², 0 U££b, 0 0
i 0,1, … , N
Eliminating δ~ , δ:~ , δ²9:~ and δ²9~ from the system (11), we get:
Aδ~ r, where A is the penta-diagonal matrix given by:
and δ~ δ~, δ~, … , δ~Ï, r fx , fx , … , fx Ï
101 135 105
A
105 135 101
Trang 53 STABILITY ANALYSIS
To apply the Von-Neumann stability for the system (6), we must first linearize this system
We have:
δÓÇ ξÇexpiγjh , i √/1, (11) where γ is the mode number and h is the element size
Being applicable to only linear schemes the nonlinear term U¥U£ is linearized by
taking U as a locally constant value k The linearized form of proposed scheme is given as:
p:δ¯Ç9:# pδ¯:Ç9:# p»δ¯Ç9:# pÊδ¯9:Ç9:# p¶δ¯9Ç9: p′:δ¯Ç # p′δ¯:Ç #
p′»δ¯Ç# # p′Êδ¯9:Ç # p′¶δ¯9Ç (12) Where:
p: 1 / M / N:/ P
p 26 / 10M / 2N:/ 2P
p» 66 # 6N:# 6P
pÊ 26 # 10M / 2N:/ 2P
p¶ 1 # M / N:/ P
p′: 1 # M # N:/ P
p′ 26 # 10M # 2N:/ 2P
p′» 66 / 6N:# 6P
p′Ê 26 / 10M # 2N:/ 2P
p′¶ 1 / M # N:/ P,
M 5α # εkh ¥∆t,
N: 10μ∆th , P 10βh
Substitretion of δÓÇ expiγjhξÇ, into Eq (12) leads to:
ξp:exp/2ihγ # pexp/iγh # p»# pÊexpiγh # p¶exp2iγh p′:exp/2iγh # #p′exp/iγh # p′»# p′Êexpiγh # p′¶exp2iγh (13) Simplifying Eq (13), we get:
I A / iBC # iB
Trang 6Where:
A 21 # N:/ Pcos 2ϕ # 413 # N:/ Pcosϕ # 66 / 6N:# 6P;
B 2Msin 2ϕ # 10;
C 21 / N:/ Pcos 2ϕ # 413 / N:/ Pcosϕ # 66 # 6N:# 6P;
α, γ f 0, ϕ γh
It is clear that CC A
Therefore, the linearized numerical scheme for the mGRLW 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 Lá and L using formula:
Lá max¯|Ux¯, t / ux¯, t|,
L âh |Ux¯, t / ux¯, t|
¯
ã
:
,
where U is numerical solution and u denotes exact solution
Three invariants of motion which correspond to the conservation of mass, momentum, and energy are given as
I: å udx,æ
ç I å uæ # βu£dx,
I» å èuæ Ê/2βp # 1ε u£é dx
ç
Eq (1) is given in [7]
ux, t ëì # 1ì # 22ê sechí ì
2îïx / x~/ ctð
ñ
We choose the following parameters
a 0; b 80; x~ 30; T 20; p 3; 6; 8; 10; c 0.03; 0.01; h 0.1; 0.2 The obtained results are tabulated in Table 2
Trang 7Table 2 Errors for single solitary wave with t = 20, x ∈ [0,80]
c
| Ã
Ò Äò ó
| á
Ò Äò ó
exact of Eq (1) is given:
É
,
where ρ ùúûü¥9Ê: bαβp# 5p # 4 # p # 1Ac, k úûý¥9: /αβp # 4 # A,
ω û¥9Ê¥ý , A îβp # 4αβp # 4 / 8μ
We choose the following parameters:
a 0, b 80, x~ 30, t ∈ 0, 20, p 8, h 0.1; 0.2, ∆t 0.01; 0.05 The obtained results are tabulated in Table 3 and Table 4
Table 3 Errors and invariants for single solitary wave with x ∈ [0,80],
Ä 104.32 104.32 104.32 104.32 104.32 104.45 104.45 104.45 104.45 104.45
à 104.32 104.32 104.32 104.32 104.32 104.45 104.45 104.45 104.45 104.45
 230.42 230.42 230.42 230.42 230.42 230.71 230.71 230.71 230.71 230.71
à 6.3Ò
10 0.48Ò
10 ¶ 0.44Ò
10 ¶ 0.44Ò
10 ¶ 0.46Ò
10 ¶ 6.3Ò
10 0.45Ò
10 ¶ 0.44Ò
10 ¶ 0.44Ò
10 ¶ 0.46Ò
10 ¶
á 10 6.5Ò
10
6.9Ò
10
6.8Ò
10
6.5Ò
10 10 9.4 Ò
10
9 4 Ò
10
9 4 Ò
10
9 5 Ò
10
Trang 8Table 4 Errors and invariants for single solitary wave with x ∈ [0,80],
Ä 104.32 104.32 104.32 104.32 104.32 104.45 104.45 104.45 104.45 104.45
à 104.32 104.32 104.32 104.32 104.32 104.45 104.45 104.45 104.45 104.45
 230.42 230.42 230.42 230.42 230.42 230.71 230.71 230.71 230.71 230.71
à 6.3Ò
10
3.57 Ò
10
2.92 Ò
10
3.58 Ò
10
3.56 Ò
10
6.3 Ò
10
2.29 Ò
10
2.33 Ò
10
2.22
Ò 10 2.21
Ò 10
á 10 6.5Ò
10 ú
6.5Ò
10 ú
8.4 Ò
10 ú
6.4Ò
10 ú 10 3.6 Ò
10 ú
4.9 Ò
10 ú
3.7 Ò
10 ú
4.2 Ò
10 ú
5 CONCLUSIONS
A numerical method based on collocation of quintic B-spline had been described in the previous section for solving mGRLW 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 problems, the obtained resulft show that the present method is capable for solving mGRLW equation
REFERENCES
1 Baojian Hong, Dianchen Lu (2008), “New exact solutions for the generalized BBM and
Burgers-BBM equations”, World Journal of Modelling and Simulation, Vol 4, No 4,
pp.243–249
2 S Battal Gazi Karakoça, Halil Zeybek (2016), “Solitary-wave solutions of the GRLW
equation using septic B-spline collocation method”, Applied Mathematics and computation,
289, pp.159–171
3 Dong-MeiHuang, L.W.Zhang (2014), “Element-Free Approximation of Generalized
Regularized Long Wave Equation”, Mathematical Problems in Engineering, Vol 2014
4 M.J Du, Y.L Wang, C.L Temuer, X Liu (2016), “Numerical Comparison of two Reproducing Kernel Methods for solving Nonlinear Generalized Regularized Long Wave
Equation”, Universal Journal of Engineering Mechanics, 4, pp.19-25
5 C M Garcia – Lopez, J I Ramos (2012), Effect of convection on a modified GRLW equation,
Applied Mathematics and computation, 219, pp.4118–4132
6 D A Hammad, M S EI – Azad (2015), “A 2N order compact finite difference method for
solving the generalized regularized long wave (GRLW) equation”, Applied Mathematics and
computation, 253, pp.248–261
Trang 97 E Pindza, E Mare (2014), “Solving the generalized regularized long wave equation using a
distributed approximating functional method”, International J Computational Mathematics,
Vol 2014
8 Thoudam Roshan (2012), “A Petrov – Galerkin method for solving the generalized regularized
long wave (GRLW) equation”, Computers and Mathematics with applications, 63, pp.943-956
9 Wang Ju-Feng, Bai Fu-Nong and Cheng Yu-Min (2011), “A meshless method for the
nonlinear generalized regularized long wave equation”, Chin Phys B Vol 20, No 3,
p.030206
PHƯƠNG PHÁP COLLOCATION VỚI CƠ SỞ B-SPLINE BẬC 5 GIẢI PHƯƠNG TRÌNH GENERALIZED
BENJAMIN-BONA-MAHONY-BURGERS
Tóm t
Tóm tắt ắt ắt: Trong bài báo này chúng ta sử dụng phương pháp collocation với cơ sở B – spline bậc 5 giải xấp xỉ phương trình mGRLW Sử dụng phương pháp Von – Neumann hệ phương trình sai phân ổn ñịnh vô ñiều kiện Kết quả số chứng tỏ phương pháp ñưa ra hữu hiệu ñể giải phương trình trên
T
Từ khóa ừ khóa ừ khóa: Phương trình mGRLW, spline bậc 5, phương pháp collocation, phương pháp sai phân hữu hạn