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

- Seydi Battal Gazi Karakoc
^{1}, - Nuri Murat Yagmurlu
^{2}Email author and - Yusuf Ucar
^{2}

**2013**:27

https://doi.org/10.1186/1687-2770-2013-27

© Karakoc et al.; licensee Springer. 2013

**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 ${\mathit{L}}_{2}$ and ${\mathit{L}}_{\mathrm{\infty}}$ 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

*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 [5–7], finite element [8–22], 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

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

*h*by the knots ${\mathit{x}}_{m}$ such that ${\mathit{a}=\mathit{x}}_{0}{<\mathit{x}}_{1}{<\cdots <\mathit{x}}_{\mathit{N}}=\mathit{b}$. The set of quintic B-spline functions $\{{\varphi}_{-2}(x),{\varphi}_{-1}(x),\dots ,{\varphi}_{N+1}(x),{\varphi}_{N+2}(x)\}$ forms a basis over the problem domain $[\mathit{a},\mathit{b}]$. We seek the numerical solution ${\mathit{U}}_{\mathit{N}}(\mathit{x},\mathit{t})$ to the exact solution $\mathit{U}(\mathit{x},\mathit{t})$ in the form of

where ${\delta}_{j}(\mathit{t})$ are time dependent parameters to be determined from the boundary and collocation conditions.

*U*, ${U}^{\mathrm{\prime}}$, ${U}^{\mathrm{\prime}\mathrm{\prime}}$ at the knots ${x}_{m}$ are obtained in terms of the element parameters ${\delta}_{m}$ by

where the symbols ′ and
represent first and second differentiation with respect to *x*, respectively. The splines ${\varphi}_{m}(x)$ and their four principle derivatives vanish outside the interval $[{\mathit{x}}_{\mathit{m}-3}{,\mathit{x}}_{\mathit{m}+3}]$.

*U*and Crank-Nicolson approximation for the space derivatives ${\mathit{U}}_{\mathit{x}}$ and ${\mathit{U}}_{\mathit{xx}}$ in Eq. (2) leads to

*U*, ${\mathit{U}}_{\mathit{x}}$ and ${\mathit{U}}_{\mathit{xx}}$ given by (8) into (10), we obtain the following iterative system:

**A**and

**B**are pentagonal $(\mathit{N}+1)\times (\mathit{N}+1)$ matrices given as

### 2.1 A linear stability analysis

*k*is a mode number and

*h*is the element size. The non-linear term ${\mathit{U}}^{2}{\mathit{U}}_{\mathit{x}}$ 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

where *g* is the growth factor.

*n*and $\mathit{n}+1$ relating two unknown parameters ${\delta}_{i}^{n+1}$, ${\delta}_{i}^{n}$ for $\mathit{i}=m-2,m-1,\dots ,m+1,m+2$

*g*of the form

The modulus of $|\mathit{g}|$ is 1, therefore the linearized scheme is unconditionally stable.

## 3 Results and discussion

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

*c*are arbitrary constants. The constants of motion, for a solitary wave of amplitude $\sqrt{c}$ and width depending on

*p*may be evaluated analytically as [29]

**Invariants and error norms for a single solitary wave with**
$\mathit{c}\mathbf{=}\mathbf{1}$
**,**
$\mathit{h}\mathbf{=}\mathbf{0.2}$
**,**
$\mathit{k}\mathbf{=}\mathbf{0.025}$
**,**
$\mathbf{0}\mathbf{\le}\mathit{x}\mathbf{\le}\mathbf{100}$

t | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | ${\mathit{L}}_{\mathbf{2}}\times {\mathbf{10}}^{\mathbf{3}}$ | ${\mathit{L}}_{\mathbf{\infty}}\times {\mathbf{10}}^{\mathbf{3}}$ |
---|---|---|---|---|---|

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 |

**Errors and invariants for a single solitary wave with**
$\mathit{c}\mathbf{=}\mathbf{1}$
**,**
$\mathit{h}\mathbf{=}\mathbf{0.2}$
**,**
$\mathit{k}\mathbf{=}\mathbf{0.025}$
**,**
$\mathbf{0}\mathbf{\le}\mathit{x}\mathbf{\le}\mathbf{100}$
**, at**
$\mathit{t}\mathbf{=}\mathbf{10}$

Method | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | ${\mathit{L}}_{\mathbf{2}}\times {\mathbf{10}}^{\mathbf{3}}$ | ${\mathit{L}}_{\mathbf{\infty}}\times {\mathbf{10}}^{\mathbf{3}}$ |
---|---|---|---|---|---|

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 |

**Invariants and error norms for a single solitary wave with**
$\mathit{c}\mathbf{=}\mathbf{0.3}$
**,**
$\mathit{h}\mathbf{=}\mathbf{0.1}$
**,**
$\mathit{k}\mathbf{=}\mathbf{0.01}$
**,**
$\mathbf{0}\mathbf{\le}\mathit{x}\mathbf{\le}\mathbf{100}$

t | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | ${\mathit{L}}_{\mathbf{2}}\times {\mathbf{10}}^{\mathbf{4}}$ | ${\mathit{L}}_{\mathbf{\infty}}\times {\mathbf{10}}^{\mathbf{4}}$ |
---|---|---|---|---|---|

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

**Comparison of invariants for the interaction of two solitary waves with results from** [34] **with** $\mathit{h}\mathbf{=}\mathbf{0.2}$**,** $\mathit{k}\mathbf{=}\mathbf{0.025}$ **in the region** $\mathbf{0}\mathbf{\le}\mathit{x}\mathbf{\le}\mathbf{250}$

Present method | [34] | |||||
---|---|---|---|---|---|---|

t | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ |

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

**Comparison of invariants for the interaction of three solitary waves with results from** [34] **with** $\mathit{h}\mathbf{=}\mathbf{0.2}$**,** $\mathit{k}\mathbf{=}\mathbf{0.025}$ **in the region** $\mathbf{0}\mathbf{\le}\mathit{x}\mathbf{\le}\mathbf{250}$

Present method | [34] | |||||
---|---|---|---|---|---|---|

t | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ |

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

*μ*. We study each of the following cases: $\mu =0.1$, $\mu =0.04$, $\mu =0.015$ and $\mu =0.01$. For $\mu =0.1$, only a single soliton is formed as shown in Figure 6a. When $\mu =0.04$ and $\mu =0.015$, two and three stable solitons are formed, respectively, as shown in Figure 6b, c. For $\mu =0.01$, the Maxwellian initial condition has decayed into four solitary waves as shown in Figure 6d. All figures were drawn up at time $\mathit{t}=14.5$. 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.

**Invariants of the MRLW equation using the Maxwellian initial condition**

t | μ | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ | μ | ${\mathit{I}}_{\mathbf{1}}$ | ${\mathit{I}}_{\mathbf{2}}$ | ${\mathit{I}}_{\mathbf{3}}$ |
---|---|---|---|---|---|---|---|---|

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 ${L}_{\mathrm{\infty}}$ 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

## References

- Peregrine DH:
**Calculations of the development of an undular bore.***J. Fluid Mech.*1966, 25: 321-330. 10.1017/S0022112066001678View ArticleGoogle Scholar - 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 - 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 - Morrison PJ, Meiss JD, Cary JA:
**Scattering of regularized long wave solitary waves.***Physica*1984, 11D: 324-336.MathSciNetGoogle Scholar - 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 - 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 - Bhardwaj D, Shankar R:
**A computational method for regularized long wave equation.***Comput. Math. Appl.*2000, 40: 1397-1404.MathSciNetView ArticleGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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
- Rubin, SG, Graves, RA: Cubic spline approximation for problems in fluid mechanics. Nasa TR R-436, Washington DC (1975)Google Scholar
- Prenter PM:
*Splines and Variational Methods*. Wiley, New York; 1975.Google Scholar - 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

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.