Open Access

Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines

  • Seydi Battal Gazi Karakoc1,
  • Nuri Murat Yagmurlu2Email author and
  • Yusuf Ucar2
Boundary Value Problems20132013:27

DOI: 10.1186/1687-2770-2013-27

Received: 19 November 2012

Accepted: 29 January 2013

Published: 14 February 2013

Abstract

In this work, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by the method based on collocation of quintic B-splines over the finite elements. A linear stability analysis shows that the numerical scheme based on Von Neumann approximation theory is unconditionally stable. Test problems including the solitary wave motion, the interaction of two and three solitary waves and the Maxwellian initial condition are solved to validate the proposed method by calculating error norms L 2 and L that are found to be marginally accurate and efficient. The three invariants of the motion have been calculated to determine the conservation properties of the scheme. The obtained results are compared with other earlier results.

MSC: 97N40, 65N30, 65D07, 76B25, 74S05.

Keywords

MRLW equation collocation finite element method B-spline solitary waves

1 Introduction

The modified regularized long wave (MRLW) equation, based upon the regularized long wave (RLW) equation,
U t + U x + δ U U x μ U x x t = 0 ,
(1)
which was proposed at first by Peregrine [1] to describe the development of an undular bore, has the form
U t + U x + 6 U 2 U x μ U x x t = 0 ,
(2)
where δ and μ are positive parameters and the subscripts x and t denote the differentiation. The RLW equation is one of the best known partial differential equations because it describes a large number of important physical phenomena with weak nonlinearity and dispersion waves such as magneto hydrodynamic and ion-acoustic waves in plasma, phonon packets in non-linear crystals, the transverse waves in shallow water, rotating flow down a tube and pressure waves in liquid-gas bubble mixtures. Bona and Pryant [2] have studied the existence and uniqueness of the equation. Benjamin et al. [3] have proposed the RLW equation as a numerically superior modification of the Korteweg de-Vries (KdV) equation. This superiority arises because, unlike the KdV equation, the dispersion relation associated with the linearized RLW equation yields the frequency that is bounded for large wave numbers [4]. But they have found an analytical solution of the RLW equation under the restricted initial and boundary conditions. So, various numerical techniques have been introduced to solve the equation. These include the finite difference [57], finite element [822], Fourier pseudo-spectral [23] methods and the meshfree method [24]. One of the special properties of the equation is that the solutions may exhibit solitons whose magnitudes, shapes and velocities are not changed after the collision. The RLW equation is a special case of the generalized long wave (GRLW) equation having the form
U t + U x + δ U p U x μ U x x t = 0 ,
(3)

where p is a positive integer. Zhang [25] has used the finite difference method to solve the GRLW equation for a Cauchy problem. The quasilinearization method based on finite differences was used by Ramos [26] for solving the GRLW equation. Kaya et al. [27] have also studied the GRLW equation with the Adomian decomposition method. Roshan [28] has solved the GRLW equation numerically by the Petrov-Galerkin method using a linear hat function as the trial function and a quintic B-spline function as the test function. Gardner et al. [29] have developed a collocation solution to the MRLW equation using quintic B-splines finite elements. Khalifa et al. [30, 31] have obtained the numerical solutions of the MRLW equation using the finite difference method and the cubic B-spline collocation finite element method. Solutions based on the collocation method with quadratic B-spline finite elements and the central finite difference method for time have been investigated by Raslan [32]. Raslan and Hassan [33] have solved the MRLW equation by the collocation finite element method using quadratic, cubic, quartic and quintic B-splines to obtain the numerical solutions of a single solitary wave. Fazal-i-Haq et al. [34] have designed a numerical scheme based on the quartic B-spline collocation method for the numerical solution of the MRLW equation. Ali [35] has formulated a classical radial basis functions (RBFs) collocation method for solving the MRLW equation. In this paper, we have obtained a type of the quintic B-spline collocation procedure in which a nonlinear term in the equation is linearized by using the form introduced by Rubin and Graves [36] to solve the MRLW equation. The proposed method is shown to represent accurately the migration of a single solitary wave. Then the interaction of two and three solitary waves and the Maxwellian initial condition are studied. The linear stability analysis based on the Von Neumann method is also investigated.

2 Quintic B-spline finite element solution

Let us consider MRLW equation (2) with the following initial,
U ( x , 0 ) = f ( x ) , a x b ,
(4)
and boundary conditions:
U ( a , t ) = 0 , U ( b , t ) = 0 , U x ( a , t ) = 0 , U x ( b , t ) = 0 , U x x ( a , t ) = 0 , U x x ( b , t ) = 0 , t > 0 .
(5)
For the numerical calculation, the solution domain of the problem is restricted over an interval a x b . The interval is partitioned into uniformly-sized finite elements of length h by the knots x m such that a = x 0 < x 1 < < x N = b . The set of quintic B-spline functions { ϕ 2 ( x ) , ϕ 1 ( x ) , , ϕ N + 1 ( x ) , ϕ N + 2 ( x ) } forms a basis over the problem domain [ a , b ] . We seek the numerical solution U N ( x , t ) to the exact solution U ( x , t ) in the form of
U N ( x , t ) = j = 2 N + 2 ϕ j ( x ) δ j ( t ) ,
(6)

where δ j ( t ) are time dependent parameters to be determined from the boundary and collocation conditions.

Quintic B-splines ϕ m ( x ) ( m = 2 ( 1 ) N + 2 ), at the knots x m are defined over the interval [ a , b ] by [37].
ϕ m ( x ) = 1 h 5 { ( x x m 3 ) 5 , [ x m 3 , x m 2 ] , ( x x m 3 ) 5 6 ( x x m 2 ) 5 , [ x m 2 , x m 1 ] , ( x x m 3 ) 5 6 ( x x m 2 ) 5 + 15 ( x x m 1 ) 5 , [ x m 1 , x m ] , ( x x m 3 ) 5 6 ( x x m 2 ) 5 + 15 ( x x m 1 ) 5 20 ( x x m ) 5 , [ x m , x m + 1 ] , ( x x m 3 ) 5 6 ( x x m 2 ) 5 + 15 ( x x m 1 ) 5 20 ( x x m ) 5 + 15 ( x x m + 1 ) 5 , [ x m + 1 , x m + 2 ] , ( x x m 3 ) 5 6 ( x x m 2 ) 5 + 15 ( x x m 1 ) 5 20 ( x x m ) 5 + 15 ( x x m + 1 ) 5 6 ( x x m + 2 ) 5 , [ x m + 2 , x m + 3 ] , 0 , otherwise .
(7)
Each quintic B-spline covers six elements so that each element [ x m , x m + 1 ] is covered by six B-splines. Substituting trial function (7) into Eq. (6), the nodal values of U, U , U at the knots x m are obtained in terms of the element parameters δ m by
U N ( x m , t ) = U m = δ m 2 + 26 δ m 1 + 66 δ m + 26 δ m + 1 + δ m + 2 , U m = 5 h ( δ m 2 10 δ m 1 + 10 δ m + 1 + δ m + 2 ) , U m = 20 h 2 ( δ m 2 + 2 δ m 1 6 δ m + 2 δ m + 1 + δ m + 2 ) ,
(8)

where the symbols ′ and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_IEq20_HTML.gif represent first and second differentiation with respect to x, respectively. The splines ϕ m ( x ) and their four principle derivatives vanish outside the interval [ x m 3 , x m + 3 ] .

Using a first-order forward difference formula for the time derivative of the U and Crank-Nicolson approximation for the space derivatives U x and U xx in Eq. (2) leads to
U n + 1 U n Δ t + U x n + 1 + U x n 2 + 6 ( U 2 U x ) n + 1 + ( U 2 U x ) n 2 μ U x x n + 1 U x x n Δ t = 0 .
(9)
Now, if we apply a linearization technique similar to the one first introduced by Rubin and Graves [36] to Eq. (9),
( U 2 U x ) n + 1 = U n + 1 U n U x n + U n U n + 1 U x n + U n U n U x n + 1 2 U n U n U x n ,
we obtain
U n + 1 + Δ t 2 U x n + 1 + 3 Δ t ( U n + 1 U n U x n + U n U n + 1 U x n + U n U n U x n + 1 ) μ U x x n + 1 = U n Δ t 2 U x n 3 Δ t ( U 2 U x ) n μ U x x n + 6 Δ t ( U n U n U x n ) .
If we substitute the nodal values of U, U x and U xx given by (8) into (10), we obtain the following iterative system:
δ m 2 n + 1 ( 1 α 1 + 2 α 2 α 3 α 4 ) + δ m 1 n + 1 ( 26 10 α 1 + 52 α 2 10 α 3 2 α 4 ) + δ m n + 1 ( 66 + 132 α 2 + 6 α 4 ) + δ m + 1 n + 1 ( 26 + 10 α 1 + 52 α 2 + 10 α 3 2 α 4 ) + δ m + 2 n + 1 ( 1 + α 1 + 2 α 2 + α 3 α 4 ) = δ m 2 n ( 1 + α 1 + α 2 α 4 ) + δ m 1 n ( 26 + 10 α 1 + 26 α 2 2 α 4 ) + δ m n ( 66 + 66 α 2 + 6 α 4 ) + δ m + 1 n ( 26 10 α 1 + 26 α 2 2 α 4 ) + δ m + 2 n ( 1 α 1 + α 2 α 4 ) , m = 0 ( 1 ) N ,
(10)
where
α 1 = 5 Δ t 2 h , α 2 = 15 Δ t h ( δ m 2 n + 26 δ m 1 n + 66 δ m n + 26 δ m + 1 n + δ m + 2 n ) ( δ m 2 n 10 δ m 1 n + 10 δ m + 1 n + δ m + 2 n ) , α 3 = 15 Δ t h ( δ m 2 n + 26 δ m 1 n + 66 δ m n + 26 δ m + 1 n + δ m + 2 n ) 2 , α 4 = 20 μ h 2 .
This newly obtained iterative system (10) consists of N + 1 linear equations including N + 5 unknown parameters ( δ 2 , δ 1 , , δ N + 1 , δ N + 2 ) T . To obtain a unique solution to this system, we need four additional constraints. These are obtained from the boundary conditions U ( a , t ) = U ( b , t ) = 0 and U x ( a , t ) = U x ( b , t ) = 0 and can be used to eliminate δ 2 , δ 1 and δ N + 1 , δ N + 2 from system (10), which then becomes a matrix equation for the N + 1 unknowns d = ( δ 0 , δ 1 , , δ N ) T of the form
Ad n + 1 = Bd n .
(11)
The matrices A and B are pentagonal ( N + 1 ) × ( N + 1 ) matrices given as
A = [ a 11 a 12 a 13 a 21 a 22 a 23 a 24 a m , m 2 a m , m 1 a m , m a m , m + 1 a m , m + 2 a n , n 2 a n , n 1 a n , n a n , n + 1 a n + 1 , n 1 a n + 1 , n a n + 1 , n + 1 ] , m = 3 ( 1 ) n 1 , B = [ b 11 b 12 b 13 b 21 b 22 b 23 b 24 b m , m 2 b m , m 1 b m , m b m , m + 1 b m , m + 2 b n , n 2 b n , n 1 b n , n b n , n + 1 b n + 1 , n 1 b n + 1 , n b n + 1 , n + 1 ] , m = 3 ( 1 ) n 1 ,
where
a 11 = 27 α 4 , a 12 = 30 α 4 , a 13 = 3 α 4 , b 11 = 27 α 4 , b 12 = 30 α 4 , b 13 = 3 α 4 , a 21 = 175 8 47 8 α 1 + 175 4 α 2 47 8 α 3 + 17 8 α 4 , a 22 = 255 4 + 9 4 α 1 + 255 2 α 2 + 9 4 α 3 + 33 4 α 4 , a 23 = 207 8 + 81 8 α 1 + 207 4 α 2 + 81 8 α 3 15 8 α 4 , a 24 = 1 + α 1 + 2 α 2 + α 3 α 4 , b 21 = 175 8 + 47 8 α 1 + 175 8 α 2 + 17 8 α 4 , b 22 = 255 4 9 4 α 1 + 255 4 α 2 + 33 4 α 4 , b 23 = 207 8 81 8 α 1 + 207 8 α 2 15 8 α 4 , b 24 = 1 α 1 + α 2 α 4 , a m , m 2 = 1 α 1 + 2 α 2 α 3 α 4 , a m , m 1 = 26 10 α 1 + 52 α 2 10 α 3 2 α 4 , a m , m = 66 + 132 α 2 + 6 α 4 , a m , m + 1 = 26 + 10 α 1 + 52 α 2 + 10 α 3 2 α 4 , a m , m + 2 = 1 + α 1 + 2 α 2 + α 3 α 4 , b m , m 2 = 1 + α 1 + α 2 α 4 , b m , m 1 = 26 + 10 α 1 + 26 α 2 2 α 4 , b m , m = 66 + 66 α 2 + 6 α 4 , b m , m + 1 = 26 10 α 1 + 26 α 2 2 α 4 , b m , m + 2 = 1 α 1 + α 2 α 4 , m = 3 ( 1 ) n 1 , a n 1 , n 3 = 1 α 1 + 2 α 2 α 3 α 4 , a n 1 , n 2 = 207 8 81 8 α 1 + 207 4 α 2 81 8 α 3 15 8 α 4 , a n 1 , n 1 = 255 4 9 4 α 1 + 255 2 α 2 9 4 α 3 + 33 4 α 4 , a n 1 , n = 175 8 + 47 8 α 1 + 175 4 α 2 + 47 8 α 3 + 17 8 α 4 , b n 1 , n 3 = 1 + α 1 + α 2 α 4 , b n 1 , n 2 = 207 8 + 81 8 α 1 + 207 8 α 2 15 8 α 4 , b n 1 , n 1 = 255 4 + 9 4 α 1 + 255 4 α 2 + 33 4 α 4 , b n 1 , n = 175 8 47 8 α 1 + 175 8 α 2 + 17 8 α 4 , a n , n 2 = 3 α 4 , a n , n 1 = 30 α 4 , a n , n = 27 α 4 , b n , n 2 = 3 α 4 , b n , n 1 = 30 α 4 , b n , n = 27 α 4 .
To proceed with iterative formula (11), we need the initial vector d 0 which is determined from the initial and boundary conditions. For this purpose, approximation (6) must be rewritten for the initial condition as
U N ( x , 0 ) = m = 2 N + 2 δ m ( 0 ) ϕ m ( x ) ,
(12)
where the δ m ’s are unknown element parameters. Now, if we require the initial numerical approximation U N ( x , 0 ) to satisfy the following boundary conditions to eliminate δ 1 and  δ N + 1 :
U N ( x , 0 ) = U ( x m , 0 ) , m = 0 , 1 , , N , ( U N ) x ( a , 0 ) = 0 , ( U N ) x ( b , 0 ) = 0 , ( U N ) x x ( a , 0 ) = 0 , ( U N ) x x ( b , 0 ) = 0 ,
(13)
we obtain the following matrix form for the initial vector d 0 :
Wd 0 = b ,
(14)
where
W = [ 54 60 6 25.25 67.50 26.25 1 1 26 66 26 1 1 26 66 26 1 1 26 66 26 1 1 26.25 67.50 25.25 6 60 54 ] , d 0 = ( δ 0 , δ 1 , δ 2 , , δ N 2 , δ N 1 , δ N ) T
and
b = ( U ( x 0 , 0 ) , U ( x 1 , 0 ) , U ( x 2 , 0 ) , , U ( x N 2 , 0 ) , U ( x N 1 , 0 ) , U ( x N , 0 ) ) T .

2.1 A linear stability analysis

The stability analysis is based on the Von Neumann theory in which the growth factor of a typical Fourier mode is defined as
δ j n = ζ ˆ n e i j k h ,
(15)
where k is a mode number and h is the element size. The non-linear term U 2 U x of the MRLW equation cannot be handled by the Fourier mode method. Thus, this term is linearized by making the quantity U 2 in the nonlinear term a local constant such as Z m . Then substituting Eq. (15) into system (10) gives
ζ ˆ n + 1 = g ζ ˆ n ,
(16)

where g is the growth factor.

Now, we identify the collocation points with the knots and use Eq. (8) to evaluate U m and its necessary space derivatives and substitute into Eq. (2) to obtain the following equation:
δ ˙ m 2 + 26 δ ˙ m 1 + 66 δ ˙ m + 26 δ ˙ m + 1 + 26 δ ˙ m + 2 + 5 h ( 1 + 6 Z m ) ( δ m 2 10 δ m 1 + 10 δ m + 1 + δ m + 2 ) 20 μ h 2 ( δ ˙ m 2 + 2 δ ˙ m 1 6 δ ˙ m + 2 δ ˙ m + 1 + 26 δ ˙ m + 2 ) = 0 .
(17)
Here denotes derivative with respect to time. If time parameters δ i ’s and their time derivatives δ ˙ i ’s in Eq. (17) are discretized by the Crank-Nicolson formula and usual forward finite difference approximation, respectively:
δ i = δ n + δ n + 1 2 , δ ˙ i = δ n + 1 δ n Δ t ,
(18)
we obtain a recurrence relationship between two time levels n and n + 1 relating two unknown parameters δ i n + 1 , δ i n for i = m 2 , m 1 , , m + 1 , m + 2
γ 1 δ m 2 n + 1 + γ 2 δ m 1 n + 1 + γ 3 δ m n + 1 + γ 4 δ m + 1 n + 1 + γ 5 δ m + 2 n + 1 = γ 5 δ m 2 n + 1 + γ 4 δ m 1 n + γ 3 δ m n + γ 2 δ m + 1 n + γ 1 δ m + 2 n ,
(19)
where
γ 1 = ( 1 E M ) , γ 2 = ( 26 10 E 2 M ) , γ 3 = ( 66 + 6 M ) , γ 4 = ( 26 + 10 E 2 M ) , γ 5 = ( 1 + E M ) , m = 0 , 1 , , N , E = ( 1 + 6 Z m ) 5 2 h Δ t , M = 20 h 2 μ .
(20)
Substituting the Fourier mode (15) into (19) gives the growth factor g of the form
g = a i b a + i b ,
(21)
where
a = 33 + 3 M + ( 26 2 M ) cos [ h k ] + ( 1 M ) cos [ 2 h k ] , b = 10 E sin [ h k ] + E sin [ 2 h k ] .
(22)

The modulus of | g | is 1, therefore the linearized scheme is unconditionally stable.

3 Results and discussion

In this section, we consider the following four test problems: the motion of a single solitary wave, the interaction of two and three solitary waves and the Maxwellian initial condition. Accuracy and efficiency of the method are measured by the error norms L 2
L 2 = U exact U N 2 h J = 1 N | U j exact ( U N ) j | 2 ,
and L
L = U exact U N max j | U j exact ( U N ) j | , j = 1 , 2 , , N 1 .
The MRLW equation satisfies only three conservation laws given by [38]
I 1 = a b U d x h j = 1 N U j n , I 2 = a b [ U 2 + μ ( U x ) 2 ] d x h j = 1 N [ ( U j n ) 2 + μ ( U x ) j n ] , I 3 = a b ( U 4 μ U x 2 ) d x h j = 1 N [ ( U j n ) 4 μ ( U x ) j n ] ,

which correspond to conversation of mass, momentum and energy, respectively. In the simulation of a solitary wave motion, the invariants I 1 , I 2 and I 3 are monitored to check the conversation of the numerical algorithm.

3.1 The motion of a single solitary wave

For this problem, MRLW Eq. (2) is considered with the boundary condition U 0 as x ± and the initial condition
U ( x , 0 ) = c sech ( p ( x x 0 ) ) .
Note that the analytical solution of this problem can be written as
U ( x , t ) = c sech ( p ( x ( c + 1 ) t x 0 ) ) ,
where p = c μ ( c + 1 ) , x 0 and c are arbitrary constants. The constants of motion, for a solitary wave of amplitude c and width depending on p may be evaluated analytically as [29]
I 1 = U ( x , 0 ) d x = π c p , I 2 = ( U 2 ( x , 0 ) + μ U x 2 ( x , 0 ) ) d x = 2 c p + 2 μ p c 3 , I 3 = ( U 4 ( x , 0 ) μ U x 2 ( x , 0 ) ) d x = 4 c 2 3 p 2 μ p c 3 .
(23)
For our computational work, we have chosen two sets of parameters. Firstly, we have used the parameters c = 1 , μ = 1 , h = 0.2 , x 0 = 40 , k = 0.025 over the interval [ 0 , 100 ] to coincide with those of earlier papers [2830, 35]. So, the solitary wave has amplitude 1.0 and the computations are done up to time t = 10 to obtain the invariants and error norms L 2 and L at various times. Error norms L 2 , L and three invariants of the MRLW equation are listed in Table 1. It is seen that the error norms are found to be small enough and the computed values of invariants are in good agreement with their analytical values I 1 = 4.4428829 , I 2 = 3.2998316 , I 3 = 1.4142135 . Percentage values of the relative error of the conserved quantities I 1 , I 2 and I 3 are calculated with respect to the conserved quantities at t = 0 . Percentage values of relative changes of I 1 , I 2 and I 3 are found to be 0.001 × 10 3 % , 3.389 × 10 3 % , 7.909 × 10 3 % , respectively. Thus, the invariants remain almost constant during the computer run. Table 2 displays a comparison of the values of the invariants and error norms obtained by the present method with those obtained by other methods [2830, 35]. It can be seen from Table 2 that the error norms obtained by the present method are smaller than other methods [2830, 35]. Figure 1 shows the motion of a solitary wave with c = 1 , h = 0.2 , k = 0.025 at different time levels. It is observed that the soliton moves to the right at a constant speed and almost unchanged amplitude with increasing time, as expected. At t = 0 the amplitude is 1.0 which is located at x = 40 , while it is 0.999946 which is located at x = 60 . At times t = 0 and t = 10 , the absolute difference in amplitude is 5 × 10 5 so there is a little change between the amplitudes.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_Fig1_HTML.jpg
Figure 1

Single solitary wave with c = 1 , h = 0.2 , Δ t = 0.025 , 0 x 100 , t = 0 , 2 , 4 , 6 , 8 and 10.

Table 1

Invariants and error norms for a single solitary wave with c = 1 , h = 0.2 , k = 0.025 , 0 x 100

t

I 1

I 2

I 3

L 2 × 10 3

L × 10 3

0

4.4428660

3.2998226

1.4142046

0.00000000

0.00000000

1

4.4428660

3.2998068

1.4142204

0.28867055

0.17189210

2

4.4428660

3.2997776

1.4142496

0.56818930

0.32423631

3

4.4428660

3.2997536

1.4142736

0.83577886

0.46169463

4

4.4428660

3.2997375

1.4142897

1.09458149

0.59234819

5

4.4428660

3.2997272

1.4143000

1.34807894

0.72040575

6

4.4428660

3.2997206

1.4143066

1.59852566

0.84732049

7

4.4428660

3.2997164

1.4143108

1.84722430

0.97368288

8

4.4428660

3.2997137

1.4143135

2.09491698

1.09976163

9

4.4428661

3.2997119

1.4143153

2.34203425

1.22568849

10

4.4428661

3.2997108

1.4143165

2.58891199

1.35164457

Table 2

Errors and invariants for a single solitary wave with c = 1 , h = 0.2 , k = 0.025 , 0 x 100 , at t = 10

Method

I 1

I 2

I 3

L 2 × 10 3

L × 10 3

Analytical

4.4428829

3.2998316

1.4142135

0

0

Present

4.4428661

3.2997108

1.4143165

2.58891

1.35164

Pet-Gal.[28]

4.44288

3.29981

1.41416

3.00533

1.68749

Cubic B-splines coll-CN[29]

4.442

3.299

1.413

16.39

9.24

Cubic B-splines coll+PA-CN[29]

4.440

3.296

1.411

20.3

11.2

Cubic B-splines coll[30]

4.44288

3.29983

1.41420

9.30196

5.43718

MQ[35]

4.4428829

3.29978

1.414163

3.914

2.019

IMQ[35]

4.4428611

3.29978

1.414163

3.914

2.019

IQ[35]

4.4428794

3.29978

1.414163

3.914

2.019

GA[35]

4.4428829

3.29978

1.414163

3.914

2.019

TPS[35]

4.4428821

3.29972

1.414104

4.428

2.306

For the second set, the parameters μ = 1 , c = 0.3 , h = 0.1 , k = 0.01 and x 0 = 40 with range [ 0 , 100 ] are chosen to compare the results obtained by the present method with those obtained given in Refs. [28, 30, 32, 34, 35]. So, the solitary wave has amplitude 0.547723 and the computations are done up to time t = 20 to obtain the invariants and error norms L 2 and L at various times. Error norms L 2 and L and conserved quantities are tabulated in Table 3 together with the results obtained with Refs. [28, 30, 32, 34, 35]. As it is seen from the table, the error norms obtained by the present method are smaller than those given in Refs. [30, 32] and almost the same as those in Refs. [28, 34, 35]. The agreement between numerical and analytic solutions is perfect which is given by Eq. 23. Percentage values of relative changes of I 1 , I 2 and I 3 are found to be 0.001 × 10 3 % , 0.246 × 10 3 % , 2.152 × 10 3 % , respectively. Moreover, from Table 3 , the changing of the invariants I 1 , I 2 and I 3 during the computer run is less than 1 × 10 7 , 3.3 × 10 6 , 3.3 × 10 6 , respectively. The profiles of the solitary wave at different time levels have been shown in Figure 2. The distributions of the errors at time t = 10 and t = 20 are shown graphically for solitary wave amplitudes 1 and 0.3 in Figure 3. It is seen that the maximum errors are about at the tip of the solitary waves and between 6 × 10 3 and 6 × 10 3 , 2 × 10 4 and 2 × 10 4 , respectively.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_Fig2_HTML.jpg
Figure 2

Single solitary wave with c = 0.3 , h = 0.1 , Δ t = 0.01 , 0 x 100 at times t = 0 , 5 , 10 , 15 and 20.

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_Fig3_HTML.jpg
Figure 3

Error with a) c = 1 , h = 0.2 , Δ t = 0.025 , t = 10 , 0 x 100 , b) c = 0.3 , h = 0.1 , Δ t = 0.01 , t = 20 , 0 x 100 .

Table 3

Invariants and error norms for a single solitary wave with c = 0.3 , h = 0.1 , k = 0.01 , 0 x 100

t

I 1

I 2

I 3

L 2 × 10 4

L × 10 4

0

3.5820205

1.3450941

0.1537283

0.0000000

0.0000000

2

3.5820205

1.3450944

0.1537280

0.0082694

0.0034843

4

3.5820205

1.3450950

0.1537274

0.0162937

0.0070162

6

3.5820206

1.3450955

0.1537268

0.0242346

0.0105732

8

3.5820206

1.3450960

0.1537264

0.0322064

0.0141521

10

3.5820206

1.3450964

0.1537260

0.0402374

0.0177376

12

3.5820206

1.3450966

0.1537257

0.0483276

0.0213278

14

3.5820206

1.3450969

0.1537255

0.0564695

0.0249138

16

3.5820206

1.3450971

0.1537253

0.0646548

0.0285146

18

3.5820206

1.3450972

0.1537251

0.0728758

0.0321067

20

3.5820204

1.3450974

0.1537250

0.8112594

0.3569076

20[30]

3.58197

1.34508

0.153723

6.06885

2.96650

20[34]

3.581967

1.345076

0.153723

0.508927

0.222284

20[35]MQ

3.5819665

1.3450764

0.153723

0.51498

0.22551

20[35]IMQ

3.5819664

1.3450764

0.153723

0.51498

0.22551

20[35]IQ

3.5819654

1.3450764

0.153723

0.51498

0.22551

20[35]GA

3.5819665

1.3450764

0.153723

0.51498

0.22551

20[35]TPS

3.5819663

1.3450759

0.153723

0.51498

0.26605

3.2 Interaction of two solitary waves

Here the interaction of two solitary waves is studied by using the initial condition given by the linear sum of two well-separated solitary waves having various amplitudes
U ( x , 0 ) = j = 1 2 A j sech ( p j ( x x j ) ) ,
(24)
where A j = c j , p j = c j μ ( c j + 1 ) , j = 1 , 2 , c j and x j are arbitrary constants. The analytical values of the invariants are found by [29]
I 1 = j = 1 2 π c j p j , I 2 = j = 1 2 ( 2 c j p j + 2 μ p j c j 3 ) , I 3 = j = 1 2 ( 4 c j 2 3 p j 2 μ p j c j 3 ) .
(25)
For the numerical simulation, the parameters μ = 1 , h = 0.2 , k = 0.025 , c 1 = 4 , c 2 = 1 , x 1 = 25 , x 2 = 55 are used over the range 0 x 250 to coincide with those used by Refs. [28, 30, 34, 35]. The experiment is run from t = 0 to t = 20 and the values of invariant quantities I 1 , I 2 and I 3 are recorded in Table 4. The analytical values of the invariants for this case are I 1 = 11.467698 , I 2 = 14.629243 , I 3 = 22.880466 . A comparison of the values of the invariants obtained by the present method with those obtained in Refs. [28, 30, 34, 35] are listed in Table 4. It is seen that the obtained values of the invariants remain almost constant during the computer run. The development of the interaction of two solitary waves is shown in Figure 4. It can be seen from the figure that at t = 0 the wave with larger amplitude is to the left of the second wave with smaller amplitude. Since the taller wave moves faster than the shorter one, it catches up and collides with the shorter one at t = 8 and then moves away from the shorter one as time increases. At t = 20 , the amplitude of larger waves is 2.001090 at the point x = 127.4 whereas the amplitude of the smaller one is 0.996399 at the point x = 92 . It is found that the absolute difference in amplitude is 3.60 × 10 3 for the smaller wave and 1.09 × 10 3 for the larger wave for this algorithm.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_Fig4_HTML.jpg
Figure 4

Interaction of two solitary waves with t = 0 , 4 , 8 , 10 , 14 , 20 .

Table 4

Comparison of invariants for the interaction of two solitary waves with results from [34] with h = 0.2 , k = 0.025 in the region 0 x 250

Present method

[34]

t

I 1

I 2

I 3

I 1

I 2

I 3

0

11.4676542

14.6292080

22.8803584

11.467698

14.629277

22.880432

2

11.4678169

14.6282301

22.8813363

11.467698

14.624259

22.860365

4

11.4679819

14.6282293

22.8813371

11.467698

14.619226

22.840279

6

11.4681349

14.6181053

22.8914611

11.467699

14.614169

22.820069

8

11.4675390

14.1393389

23.3702275

11.467700

14.606821

22.787857

10

11.4674118

14.0502062

23.4593602

11.467700

14.603687

22.771773

12

11.4685494

14.6816556

22.8279107

11.467699

14.603056

22.775766

14

11.4687073

14.6648742

22.8446922

11.467699

14.598059

22.756029

16

11.4688627

14.6459207

22.8636457

11.467700

14.593048

22.736127

18

11.4690242

14.6370095

22.8725569

11.467700

14.588061

22.716289

20

11.4691886

14.6331334

22.8764330

11.467701

14.583089

22.696510

20[28]

11.4677

14.6299

22.8806

   

20[30]

11.4677

14.6292

22.8809

   

20[35]MQ

11.467698

14.583052

22.696539

   

20[35]IMQ

11.467679

14.583052

22.696539

   

20[35]IQ

11.467690

14.583052

22.696539

   

20[35]GA

11.467698

14.583052

22.696539

   

20[35]TPS

11.467742

14.582424

22.694269

   

3.3 Interaction of three solitary waves

For this problem, the behavior of interaction of three solitary waves having different amplitudes and traveling in the same direction is studied. So, we consider Eq. (2) with the initial condition given by the linear sum of three well-separated solitary waves of different amplitudes
U ( x , 0 ) = j = 1 3 A j sech ( p j ( x x j ) ) ,
(26)
where A j = c j , p j = c j μ ( c j + 1 ) , j = 1 , 2 , 3 , c j and x j are arbitrary constants. The analytical values of the conservation laws are found from Eq. (23) as follows:
I 1 = j = 1 3 π c j p j , I 2 = j = 1 3 ( 2 c j p j + 2 μ p j c j 3 ) , I 3 = j = 1 3 ( 4 c j 2 3 p j 2 μ p j c j 3 ) .
(27)
For the purpose of comparison, parameters μ = 1 , h = 0.2 , k = 0.025 , c 1 = 4 , c 2 = 1 , c 3 = 0.25 , x 1 = 15 , x 2 = 45 , x 3 = 60 are used over the region 0 x 250 . During the simulation, time is taken up to t = 45 . The analytical values of the invariants for this case are I 1 = 14.9801 , I 2 = 15.8218 , I 3 = 22.9923 . A comparison of the values of the invariants obtained by the present method with those obtained in Refs. [30, 34, 35] are shown in Table 5. It is observed from the table that the obtained values of the invariants remain almost constant during the computer run which are all in good agreement with their analytical values given by Eq. (27). The absolute difference between the values of the conservative constants obtained by the present method at times t = 0 and t = 45 are Δ I 1 = 2.6 × 10 1 , Δ I 2 = 3.4 × 10 1 , Δ I 3 = 3.4 × 10 3 , respectively. Figure 5 shows the interaction of these solitary waves at different times. As it is seen from the Figure 5, the interaction started at about time t = 10 , overlapping processes occurred between time t = 15 and t = 40 and waves started to resume their original shapes after the time t = 40 .
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_Fig5_HTML.jpg
Figure 5

Interaction of three solitary waves with t = 0 , 5 , 8 , 15 , 20 , 40 .

Table 5

Comparison of invariants for the interaction of three solitary waves with results from [34] with h = 0.2 , k = 0.025 in the region 0 x 250

Present method

[34]

t

I 1

I 2

I 3

I 1

I 2

I 3

0

14.9800762

15.8374849

23.0081806

14.980099

15.837528

23.008136

5

14.9381371

15.7382326

23.1074329

14.980105

15.824928

22.957891

10

14.9071292

14.1781087

24.6675567

14.980109

15.807025

22.877972

15

14.8836886

15.3648852

23.4807802

14.980106

15.807032

22.885947

20

14.8503851

15.5659364

23.2797291

14.980106

15.795022

22.837454

25

14.8194163

15.6235556

23.2221098

14.980107

15.782840

22.788852

30

14.7905616

15.5976717

23.2479938

14.980107

15.770634

22.740419

35

14.7636015

15.5610664

23.2845991

14.980108

15.758480

22.692279

40

14.7383184

15.5256320

23.3200335

14.980108

15.746389

22.644448

45

14.7145273

15.4927592

23.3529062

14.968030

15.734374

22.596591

45[30]

13.7043

15.6563

22.9303

   

45[35]MQ

14.96814

15.73434

22.596625

   

45[35]IMQ

14.96808

15.73434

22.596625

   

45[35]IQ

14.96813

15.73434

22.596625

   

45[35]GA

14.96810

15.73433

22.596626

   

45[35]TPS

14.96824

15.73376

22.594494

   

3.4 The Maxwellian initial condition

Finally, we have studied the development of the Maxwellian initial condition
U ( x , 0 ) = exp ( ( x 40 ) 2 )
(28)
into a train of solitary waves. As it is known, with the Maxwellian condition (28), the behavior of the solution depends on the values of μ. We study each of the following cases: μ = 0.1 , μ = 0.04 , μ = 0.015 and μ = 0.01 . For μ = 0.1 , only a single soliton is formed as shown in Figure 6a. When μ = 0.04 and μ = 0.015 , two and three stable solitons are formed, respectively, as shown in Figure 6b, c. For μ = 0.01 , the Maxwellian initial condition has decayed into four solitary waves as shown in Figure 6d. All figures were drawn up at time t = 14.5 . The peaks of the well-developed wave lie on a straight line, so that their velocities are linearly dependent on their amplitudes. We also observe a small oscillating tail appearing behind the last wave in all Maxwellian figures. The obtained numerical values of the invariants are given in Table 6.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-27/MediaObjects/13661_2012_Article_441_Fig6_HTML.jpg
Figure 6

Maxwellian initial condition at t = 14.5 with a) μ = 0.1 , b) μ = 0.04 , c) μ = 0.015 , d) μ = 0.01 .

Table 6

Invariants of the MRLW equation using the Maxwellian initial condition

t

μ

I 1

I 2

I 3

μ

I 1

I 2

I 3

0

0.1

1.7724809

1.3786633

0.7609104

0.015

1.7724809

1.2721327

0.8674410

3

1.7721927

1.4728222

0.6667515

1.7436793

1.4180550

0.7215188

6

1.7717480

1.4720618

0.6675119

1.7244620

1.4036726

0.7359011

9

1.7713064

1.4715473

0.6680264

1.7116986

1.3940073

0.7455664

12

1.7708674

1.4711193

0.6684544

1.7021509

1.3870669

0.7525068

15

1.7704309

1.4707290

0.6688447

1.6945087

1.3816736

0.7579001

0

0.01

1.7724809

1.2658662

0.8737075

0.04

1.7724809

1.3034651

0.8361087

3

1.7264258

1.4001430

0.7394307

1.7685556

1.4501960

0.6893777

6

1.7031386

1.3850514

0.7545223

1.7637294

1.4456393

0.6939344

9

1.6885258

1.3755380

0.7640357

1.7592882

1.4415131

0.6980606

12

1.6777156

1.3686563

0.7709174

1.7551780

1.4378815

0.7016922

15

1.6691476

1.3635304

0.7760434

1.7513552

1.4345989

0.7049748

4 Conclusions

A numerical solution of the MRLW equation based on the quintic B-spline finite element has been successfully presented. The nonlinear term of the equation is linearized by using a form given in the paper [36]. Four test problems are studied to examine the performance of the scheme. To show how good and accurate the numerical solutions of the test problems are, the error norms L 2 and L and the invariant quantities I 1 , I 2 and I 3 have been used. It is seen that the error norms are sufficiently small and the invariants are well conserved. The method successfully models the motion and interaction of solitary waves. The computed results indicate that the present method is more accurate than some earlier results found in the literature. So, it can be said that the method is a reliable one for obtaining the numerical solutions of a wider range of physically important non-linear partial differential equations.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the reviewers for their careful reading and making some useful comments which improved the presentation of the paper.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Art, Nevsehir University
(2)
Department of Mathematics, Faculty of Science and Art, Inönü University

References

  1. Peregrine DH: Calculations of the development of an undular bore. J. Fluid Mech. 1966, 25: 321-330. 10.1017/S0022112066001678View ArticleGoogle Scholar
  2. Bona JL, Pryant PJ: A mathematical model for long wave generated by wave makers in nonlinear dispersive systems. Proc. Camb. Philos. Soc. 1973, 73: 391-405. 10.1017/S0305004100076945View ArticleGoogle Scholar
  3. Benjamin TB, Bona JL, Mahoney JL: Model equations for long waves in nonlinear dispersive media. Philos. Trans. R. Soc. Lond. 1972, 272: 47-78. 10.1098/rsta.1972.0032View ArticleGoogle Scholar
  4. Morrison PJ, Meiss JD, Cary JA: Scattering of regularized long wave solitary waves. Physica 1984, 11D: 324-336.MathSciNetGoogle Scholar
  5. Eilbeck JC, McGuire GR: Numerical study of the regularized long wave equation, II: Interaction of solitary wave. J. Comput. Phys. 1977, 23: 63-73. 10.1016/0021-9991(77)90088-2MathSciNetView ArticleGoogle Scholar
  6. Jain PC, Shankar R, Singh TV: Numerical solution of regularized long wave equation. Commun. Numer. Methods Eng. 1993, 9: 579-586. 10.1002/cnm.1640090705MathSciNetView ArticleGoogle Scholar
  7. Bhardwaj D, Shankar R: A computational method for regularized long wave equation. Comput. Math. Appl. 2000, 40: 1397-1404.MathSciNetView ArticleGoogle Scholar
  8. Chang Q, Wang G, Guo B: Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion. J. Comput. Phys. 1995, 93: 360-375.MathSciNetView ArticleGoogle Scholar
  9. Gardner LRT, Gardner GA: Solitary waves of the regularized long wave equation. J. Comput. Phys. 1990, 91: 441-459. 10.1016/0021-9991(90)90047-5MathSciNetView ArticleGoogle Scholar
  10. Gardner LRT, Gardner GA, Dogan A: A least-squares finite element scheme for the RLW equation. Commun. Numer. Methods Eng. 1996, 12: 795-804. 10.1002/(SICI)1099-0887(199611)12:11<795::AID-CNM22>3.0.CO;2-OMathSciNetView ArticleGoogle Scholar
  11. Gardner LRT, Gardner GA, Dag I: A B-spline finite element method for the regularized long wave equation. Commun. Numer. Methods Eng. 1995, 11: 59-68. 10.1002/cnm.1640110109MathSciNetView ArticleGoogle Scholar
  12. Alexander ME, Morris JL: Galerkin method applied to some model equations for nonlinear dispersive waves. J. Comput. Phys. 1979, 30: 428-451. 10.1016/0021-9991(79)90124-4MathSciNetView ArticleGoogle Scholar
  13. Serna JMS, Christie I: Petrov Galerkin methods for nonlinear dispersive wave. J. Comput. Phys. 1981, 39: 94-102. 10.1016/0021-9991(81)90138-8MathSciNetView ArticleGoogle Scholar
  14. Dogan A: Numerical solution of RLW equation using linear finite elements within Galerkin’s method. Appl. Math. Model. 2002, 26: 771-783. 10.1016/S0307-904X(01)00084-1View ArticleGoogle Scholar
  15. Esen A, Kutluay S: Application of lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 2006, 174(2):833-845. 10.1016/j.amc.2005.05.032MathSciNetView ArticleGoogle Scholar
  16. Soliman AA, Raslan KR: Collocation method using quadratic b-spline for the RLW equation. Int. J. Comput. Math. 2001, 78: 399-412. 10.1080/00207160108805119MathSciNetView ArticleGoogle Scholar
  17. Soliman AA, Hussien MH: Collocation solution for RLW equation with septic spline. Appl. Math. Comput. 2005, 161: 623-636. 10.1016/j.amc.2003.12.053MathSciNetView ArticleGoogle Scholar
  18. Raslan KR: A computational method for the regularized long wave (RLW) equation. Appl. Math. Comput. 2005, 167: 1101-1118. 10.1016/j.amc.2004.06.130MathSciNetView ArticleGoogle Scholar
  19. Saka B, Dag I, Dogan A: Galerkin method for the numerical solution of the RLW equation using quadratic B-splines. Int. J. Comput. Math. 2004, 81(6):727-739. 10.1080/00207160310001650043MathSciNetView ArticleGoogle Scholar
  20. Dag I, Saka B, Irk D: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 2004, 159: 373-389. 10.1016/j.amc.2003.10.020MathSciNetView ArticleGoogle Scholar
  21. Dag I, Ozer MN: Approximation of RLW equation by least-square cubic B-spline finite element method. Appl. Math. Model. 2001, 25: 221-231. 10.1016/S0307-904X(00)00030-5View ArticleGoogle Scholar
  22. Zaki SI: Solitary waves of the splitted RLW equation. Comput. Phys. Commun. 2001, 138: 80-91. 10.1016/S0010-4655(01)00200-4View ArticleGoogle Scholar
  23. Gou BY, Cao WM: The Fourier pseudo-spectral method with a restrain operator for the RLW equation. J. Comput. Phys. 1988, 74: 110-126. 10.1016/0021-9991(88)90072-1MathSciNetView ArticleGoogle Scholar
  24. Islam S, Haq F, Ali A: A meshfree method for the numerical solution of the RLW equation. J. Comput. Appl. Math. 2009, 223: 997-1012. 10.1016/j.cam.2008.03.039MathSciNetView ArticleGoogle Scholar
  25. Zhang L: A finite difference scheme for generalized long wave equation. Appl. Math. Comput. 2005, 168(2):962-972. 10.1016/j.amc.2004.09.027MathSciNetView ArticleGoogle Scholar
  26. Ramos JI: Solitary wave interactions of the GRLW equation. Chaos Solitons Fractals 2007, 33: 479-491. 10.1016/j.chaos.2006.01.016View ArticleGoogle Scholar
  27. Kaya D, El-Sayed SM: An application of the decomposition method for the generalized KdV and RLW equations. Chaos Solitons Fractals 2003, 17: 869-877. 10.1016/S0960-0779(02)00569-6MathSciNetView ArticleGoogle Scholar
  28. Roshan T: A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation. Comput. Math. Appl. 2012, 63: 943-956.MathSciNetView ArticleGoogle Scholar
  29. Gardner LRT, Gardner GA, Ayoup FA, Amein NK: Simulations of solitary waves of the MRLW equation by B-spline finite element. Arab. J. Sci. Eng. 1997, 22: 183-193.Google Scholar
  30. Khalifa AK, Raslan KR, Alzubaidi HM: A collocation method with cubic B-splines for solving the MRLW equation. J. Comput. Appl. Math. 2008, 212: 406-418. 10.1016/j.cam.2006.12.029MathSciNetView ArticleGoogle Scholar
  31. Khalifa AK, Raslan KR, Alzubaidi HM: A finite difference scheme for the MRLW and solitary wave interactions. Appl. Math. Comput. 2007, 189: 346-354. 10.1016/j.amc.2006.11.104MathSciNetView ArticleGoogle Scholar
  32. Raslan KR: Numerical study of the modified regularized long wave equation. Chaos Solitons Fractals 2009, 42: 1845-1853. 10.1016/j.chaos.2009.03.098MathSciNetView ArticleGoogle Scholar
  33. Raslan KR, Hassan SM: Solitary waves for the MRLW equation. Appl. Math. Lett. 2009, 22: 984-989. 10.1016/j.aml.2009.01.020MathSciNetView ArticleGoogle Scholar
  34. Haq F, Islam S, Tirmizi IA: A numerical technique for solution of the MRLW equation using quartic B-splines. Appl. Math. Model. 2010, 34: 4151-4160. 10.1016/j.apm.2010.04.012MathSciNetView ArticleGoogle Scholar
  35. Ali, A: Mesh free collocation method for numerical solution of initial-boundary value problems using radial basis functions. Dissertation, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology (2009)Google Scholar
  36. Rubin, SG, Graves, RA: Cubic spline approximation for problems in fluid mechanics. Nasa TR R-436, Washington DC (1975)Google Scholar
  37. Prenter PM: Splines and Variational Methods. Wiley, New York; 1975.Google Scholar
  38. Olver PJ: Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Philos. Soc. 1979, 85: 143-159. 10.1017/S0305004100055572MathSciNetView ArticleGoogle Scholar

Copyright

© Karakoc et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.