Chebyshev Collocation Spectral Method for Solving the RLW Equation

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.7(2009) No.2,pp Chebyshev Collocation Spectral Method for Solving the RLW Equation A. H. A. Ali Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom, Egypt. Mathematics Department, Faculty of Education (Muzahmia), Riyadh University for Women,Saudi Arabia. (Received 24 January 2008, accepted 1 November 2008) Abstract: A spectral solution of the RLW equation based on collocation method using Chebyshev polynomials as a basis for the approximate solution is proposed. Test problems, including the motion of a single solitary wave with different amplitudes are used to validate this algorithm which is found to be more accurate than previous ones. The interaction of solitary waves is used to discuss the effect of the behavior of the solitary waves after the interaction. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied. Comparisons are made with the most recent results both of the error norms and the invariant values. Keywords: RLW equation; Collocation method; Chebyshev polynomials; Spectral method 1 Introduction The regularised long wave equation (RLW) is an important nonlinear wave equation. Solitary waves are wave packets or pulses which propagate in nonlinear dispersive media. The dynamical balance between the nonlinear and the dispersive effects of these waves retain a stable wave which also keeps its wave form after collision with other solitons. The RLW equation is an alternative description of nonlinear dispersive waves to the more usual Korteweg-de Vries (KdV) equation [1]. It has been shown to have solitary wave solutions and to govern a large number of important physical phenomena such as shallow water waves and plasma wave [1,2]. Few analytic solutions are known. Approximate solutions based on finite difference techniques [3, 4], Runge-Kutta and predictor corrector methods [5] and Galerkin s method [6], are also popular. Wahlbin [7] using a trial function composed of Hermite cubic polynomials, while Alexander and Morris presented a global trial function constructed mainly from cubic splines. Finite element methods based on both quadratic and cubic B-spline finite elements with Galerkin s method [8-10] have been used for getting the solutions of the RLW equation and have provided smaller error than previous methods. Soliman and Raslan [11] solved RLW equation by collocation method using quadratic B-splines as element shape function. Recently, efficient numerical schemes have been given to the numerical solution of the RLW equation by employing the least square technique [12-14]. Raslan [15] solved RLW equation using collocation method with cubic B-spline shape function. Sloan [16] solved RLW equation using Fourier pseudospectral method. Bona et al [17] studied the stability and unstability solutions for the general RLW equation. Araujo and Duran [18] studied the error propagation of time integrators of solitary wave solutions for the RLW equation. Duran and Lopez-Marcos [19] analyzed the behavior in time of the numerical approximations to solitary wave solutions of the generalized Benjamin-Bona-Mahony equation. Ali [20] solved the equal width (EW) equation using the spectral collocation method based on Chebyshev polynomials. In this paper we setup the spectral method Corresponding author. address: ahaali 49@yahoo.com Copyright c World Academic Press, World Academic Union IJNS /210

2 132 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp based on collocation method with Chebyshev polynomials as bases. The resulting system will be a system of ordinary differential equations which can be solved using the fourth order Runge-Kutta algorithm. 2 Governing equation and numerical method The RLW equation for the long waves propagating in the positive X-direction can take the form: with the Dirichlet boundary conditions and the initial condition u t + u X + εuu X νu XXt = 0, a X b, t > 0, (1) u(a, t) = u(b, t) = 0, t > 0 (2) u(x, 0) = f(x), (3) where ε and ν are positive parameters, subscripts X and t denote differentiation, and f(x) is a localized disturbance inside the considered interval. Using a linear transformation to transfer the interval [a, b] into the interval [ 1, 1], the equations (1)-(3) become: α 2 u t + 2αu x + 2εαu u x 4νu xxt = 0, 1 x 1, t > 0, (4) where x is the new variable and α = b a, the Dirichlet boundary conditions and the initial condition u( 1, t) = u(1, t) = 0, t > 0 (5) u(x, 0) = f(x), (6) The approximate solution u N (x, t) to the exact solution u(x, t) can be written in the form: u N (x, t) = N δ n (t)t n (x), (7) where δ n are the time dependent quantities to be determined, T n (x) are the Chebyshev polynomials defined as: T n (x) = cos(n cos 1 (x)) (8) α 2 From equation (7) and the property of the Chebyshev polynomials, we get the following [21]: u x = u xxt = N 1 N 2 u t = N δ ń (t)t n (x), (9) δ n (t)t ń (x) = N 1 N 2 δ n (t)t n (x) = Using equations (9)-(11) and substituting in equation (4) we obtain N N 1 δ ń (t)t n (x)+2α δ (1) n (t)t n (x)+2εα N N 1 δ n (t)t n (x) δ (1) n (t)t n (x), (10) δ (2) n (t)t n (x) (11) k=0 δ (1) N 2 k (t)t k(x) 4ν δ (2) n (t)t n (x) = 0 (12) IJNS for contribution: editor@nonlinearscience.org.uk

3 A. H. A. Ali : Chebyshev Collocation Cpectral Method for Solving the RLW Equation 133 where δ ń denotes the derivative of δ n with respect to t and δ (1), δ (2) are defined by δ (1) = 2 n+2j 1 N (n + 2j 1)δ n+2j 1, δ (2) = 2 n+2j N j(n + j)(n + 2j)δ n+2j (13) c n c j=1 n j=1 The nonlinear term T n (x)t k (x) can be expressed by a linear combination which is defined as: T n (x)t k (x) = 1 2 [ Tn+k (x) + T n k (x) ] (14) We substitute equation (14) into equation (12) and hence we use the inner product with the weight function ψ j (x). If we choose the weight function ψ j (x) = δ(x x i ), where δ(x x i ) is the Dirac Delta function, then this equation can be deduced using the property of the Dirac Delta function we obtain [22] εα N 1 α 2 N,N 1 N l,k=0 l+k=n 4ν N 2 δ n(t)t n (x i ) + 2α N 1 δ (1) n (t)t n (x i )+ δ l (t)δ (1) N,N 1 k (t)t l+k(x i ) + δ l (t)δ (1) l,k=0 l k =n k (t)t l k (x i ) δ (2) n (t)t n (x i ) = 0, i = 1, 2, N 1 (15) where the collocation points of the Chebyshev polynomials are calculated from equation (8) at x = x i. Equation (15) can be written in the recurrence relation as: α 2 δ ń + 2αδ (1) n N,N 1 + εα l,k=0 l+k=n δ l (t)δ (1) k N,N 1 (t) + l,k=0 l k =n δ l (t)δ (1) k (t) 4νδ (2) n = 0 (16) Hence, we can write equation (16) in the matrix form as a system of ordinary differential equations ( α 2 I 4νA 1 ) δ = ( 2αA 2 εαa 3 ( δ)) δ, (17) where A 1, A 2 and A 3 are the coefficient matrices for the second derivative, the first derivative, and the nonlinear term respectively. Equation (17) can be written in the simple form: where A 4 = α 2 I 4νA 1, A 4 δ (t)=b 1 δ(t), (18) B 1 = 2αA 2 εαa 3 ( δ) are two matrices of order (N + 1) (N + 1). Multiplying both sides of equation (18) by the matrix S, we get SA 4 δ (t)=sb 1 δ(t), (19) where S = (S in ) = (T n (x i )), i = 1, 2,..., N 1, n = 0, 1,..., N (20) The matrix S is of order (N 1) (N + 1). The system given in the equation (19) consists of N 1 equations in N + 1 unknowns and to obtain a unique solution for this system we need two further equations. For this, we add the two Dirichlet boundary conditions: u( 1, t) = N a n T n ( 1) = a 0 a 1 + a 2 a a N = 0 IJNS homepage:

4 134 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp u(1, t) = N a n T n (1) = a 0 + a 1 + a 2 + a a N = 0 to equation (20) to obtain a new system: A δ (t) = F (t, δ(t)), (21) where δ(t) and F are N + 1 vectors with N + 1 components and A is a matrix of order (N + 1) (N + 1). This system of first order of ordinary differential equations can be solved numerically using the fourth order Runge-Kutta algorithm to get the numerical solution F(t). 3 The initial state From the initial condition u(x, 0) on the function u(x, t), we must determine the initial vector δ (0) (t) so that the time evolution of δ, using equation (21), can be started. We rewrite equation (7) for the initial condition as u N (x i, 0) = δ (0) n T n (x i ), i = 0, 1,..., N (22) Equation (22) gives a system of N + 1 equations which can be written in the matrix form as: Sδ (0) = b (23) where S is a matrix of order (N + 1) (N + 1) b = (f(x 0 ), f(x 1 ),..., f(x N )) T. and is defined by equation (20) and the vector is 4 Test problems A computer program using FORTRAN language with an algorithm to calculate the nonlinear term is written to obtain soliton solutions and modeling the undular bore to the RLW equation. Accuracy of the method is measured by the L 2 and L error norms L 2 2 = [ u exact u N 2 2 = h N u exact i u N i i=0 L = u exact u N = max i u exact i u N i ] 2, (24), and the conservation properties of the proposed algorithm are examined by calculating the invariants which was shown by Olver [23] and that correspond to mass, momentum, and energy, respectively are I 1 = b a udx, I 2 = 4.1 The solitary wave motion b a (u 2 + νu 2 x)dx, I 3 = b a (u 3 + 3u 2 )dx (25) We now validate our algorithm by studying the motion of solitary waves. It is well known that equation (1) has a two parameter analytic solution of the form [5]: u(x, t) = 3c sec h 2 [k (x νt x 0 )], (26) where ν = 1 + εc is the wave velocity, and k 2 = εc/(4ν(c + 1)). This equation represents a single soliton of magnitude 3c with the constant speed c + 1. In the following simulation of the motion of a single solitary wave ε = ν = 1. To compare our results with those of previous simulations, equation (26) is taken as the initial condition where 40 x 60, t = 0, x = 1, t = 0.1 and x 0 = 0, with c = 0.03, and 0.1, so that the solitary waves have amplitudes IJNS for contribution: editor@nonlinearscience.org.uk

5 A. H. A. Ali : Chebyshev Collocation Cpectral Method for Solving the RLW Equation , and 0.3 respectively. The simulations are run up to a time t = 20. In Tables 1 and 2, the errors L 2 and L norms and the invariant values I 1, I 2, and I 3 for the experiment are taken at various time steps together with corresponding results from previous methods. Both L 2 and L error norms for these simulation are found as small as L 2 = , L = , L 2 = , and L = , for amplitudes 0.09 and 0.3, respectively, at time t = 20. Tables 1 and 2 illustrate a comparison between our L 2, L error norms results and those previously obtained at the same time t = 20. We note that our L 2 value is about one-half of the smallest one of previous result [13]. Moreover, the obtained L value is one-fourth of the smallest previous result [10], regarding that N = 100 in the present algorithm while N = 800 in the previous algorithms. The traveling solitons are graphed at t = 0 and 20 in Figs. 1 and 2. At time t = 20, both of the numerical and exact solutions are plotted, and the profiles of those solutions are indistinguishable. When the initial wave profile and that at time t = 20 are compared, they appear to be no degradation of wave amplitudes. The amplitudes errors 0.01% and 0.00% corresponding to initial wave amplitudes 0.09 and 0.3, respectively. For the solitary wave of amplitude 0.09, the invariant values I 1, I 2, and I 3 change by less than 0.008%, %, and % respectively, at time t = 20 relative to the initial invariants. Changes are also found with large amplitude of 0.3 for the invariant values I 1, I 2, and I 3 by less than %, %, and % respectively. So we can regard the conservative quantities as a satisfactory constant during the computer run. In the third numerical experiment with an amplitude 0.9 of the solitary wave. The simulations are run up to time t = 20 and the computed values of L 2 and L norms and the invariant values I 1, I 2, and I 3 are given in Tables 3 and 4 compared with the results obtained by [15]. From Table 3, we observe that the L 2 and L error norms obtained by our scheme is smaller by 0.03 than that obtained by [15]. Table 4 shows us that the invariant values I 1, I 2, and I 3 computed by our scheme are satisfactorily constant, each change less than from the initial to time t = 20 while in [15], the changes in the invariant values are less than by in each. Table 1: Invariants and error norms for a single solitary wave: amplitude 0.09, t = 0.1, 40 x 60 Method t N L L 10 3 I 1 I 2 I 3 Collocation spectral Least square cubic[13] Galerkin cubic [12] Least square quadratic[11] Least square linear[10] Galerkin quadratic[8] Collocation cubic [15] Finite element cubic[9] The interaction of solitary waves Consider the initial condition of two solitary waves: u(x, 0) = u 1 + u 2 (27) where u i = 3c i sec h 2 (p i (x x i )), i = 1, 2 (28) IJNS homepage:

6 136 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp Table 2: Invariants and error norms for a single solitary wave: amplitude 0.3, t = 0.1, 40 x 60 Method t N L L 10 3 I 1 I 2 I 3 Collocation spectral Least square cubic[13] Galerkin cubic [12] Least square quadratic[11] Least square linear[10] Galerkin quadratic[8] Collocation cubic [15] Finite element cubic[9] Figure 1: Graphs of initial solution and solution soliton at time=20 and amplitude 0.09 Figure 2: Graphs of initial solution and solution soliton at time=20 and amplitude 0.3 Table 3: Error norms for a single solitary wave: amplitude 0.9, t = 0.1, 40 x 60 t N L L 10 3 N L L Present [15] method Table 4: Invariant values for a single solitary wave: amplitude 0.9, t = 0.1, 40 x 60 t N I 1 I 2 I 3 N I 1 I 2 I Present [15] method IJNS for contribution: editor@nonlinearscience.org.uk

7 A. H. A. Ali : Chebyshev Collocation Cpectral Method for Solving the RLW Equation 137 with ε = ν = 1, and appropriate Dirichlet boundary conditions The interaction of a positive and a negative solitary wave c i = 4p 2 i /(1 4p 2 i ) (29) Santarelli [24] has simulated the interaction of a positive and a negative solitary wave and observed that the collision produced additional pairs of daughter solitary waves emanating from the point of initial contact, an observation confirmed by Lewis and Tjon [25]. We have repeated these experiments using the appropriate initial condition equations (27) - (29) and solved the RLW equation over the region 0 x 80 taking ν = 1, p 1 = 0.4, x 1 = 23, p 2 = 0.6, x 2 = 38, x = 0.4 and t = 0.1. In Fig. 3 we show the initial of two solitary waves before and after the interaction at time t = 20. After the interaction of two solitary waves is completed at time t = 20. The smallest of the original pair of waves now lies at x = 70 and the larger (negative) wave is passing through the left-hand end. The interval between x = 6 and x = 26 is the undisturbed part of the region, away from the pulse, where the solution remains zero. The waves lying between x = 26.5 and x = 63 have resulted from the interaction. We observe that the collision produces additional pair of daughter solitary waves emanating from the point of initial contact. The values of I 1, I 2, and I 3 throughout the simulation are shown in Table 5 compared with Raslan [15] results, we note that the changes in the computed quantities I 1, I 2, and I 3 using the present approach scheme are less than 0.009%, 0.005%, and 0.003% respectively while in [15] were 0.04%, 0.2%, and 0.04% respectively, which shows again the accuracy of the present algorithm. Table 5: Invariant values for the interaction of a positive and a negative solitary wave, 0 x 80 t N I 1 I 2 I 3 N I 1 I 2 I 3 0 = = Present [15] method t = t = Figure 3: The motion of a positive and a negative solitary wave before and after the interaction The interaction of two negative solitary waves The interaction of two negative solitary waves with initial condition equations (27)-(29) has studied over the region 0 x 120 with p 1 = 0.6, x 1 = 82, p 2 = 0.8, x 2 = 67, x = 0.5 and t = 0.2. The simulations are run to time t = 20, and the invariants are I 1, I 2, and I 3 recorded in Table 6. The computed IJNS homepage:

8 138 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp values of the invariants I 1, I 2, and I 3 are satisfactorily constant compared with the corresponding invariant values obtained by [15]. The two negative solitary waves before and after the interaction is plotted in Fig. 4. The two waves have apparently passed through one another and emerged unchanged by the encounter, which shows the initial of two solitary waves with amplitude centered at x = 67 and amplitude centered at x = 82. After the interaction of two solitary waves is completed we have the situation shown in Fig. 4b which shows quite a bit of radiation between x = 60 and the downstream boundaries. It is found that the radiation is very sensitive to the locations of the upstream and downstream boundaries. The invariant values I 1, I 2, and I 3 computed by our scheme are satisfactory constant, each changing are less than %, 1%, and 2.1% respectively from the initial to time t = 20 while in [15] the changes are less than by %, 0.8%, and 2.8% respectively. Table 6: Invariant values for the interaction of two negative solitary waves, 0 x 120 t N I 1 I 2 I 3 N I 1 I 2 I 3 0 = = Present [15] method t = t = Figure 4: The motion of two negative solitary waves before and after the interaction The interaction of two positive solitary waves Abdulloev et al. [2], Gardner [8] and Raslan [15] they have studied the interaction of two positive solitary waves for the RLW equation with initial condition observed the two waves have apparently passed through one another and emerged unchanged by the encounter. We chose to study a similar situation using the initial condition given by equations (27)-(29). The RLW equation is solved over the region 0 x 120 with p 1 = 0.4, x 1 = 15, p 2 = 0.3, x 2 = 35, x = 1 and t = 0.1. The simulations are run to time t = 25, and the invariants I 1, I 2, and I 3 are recorded in Table 7. The changes in the computed invariant values I 1, I 2, and I 3 are less than 0.002%, 0.002%, and 0.011% respectively which are smaller than the corresponding changes of the invariants obtained by [15]( 0.4%, 0.6%, and 2.24%). In Fig. 5, we show the initial of two solitary waves with amplitude centered at x = 15, and amplitude centered at x = 35. After IJNS for contribution: editor@nonlinearscience.org.uk

9 A. H. A. Ali : Chebyshev Collocation Cpectral Method for Solving the RLW Equation 139 the interaction of two solitary waves is completed with complete separation of clear two solitary waves at t = 25, the changes in the two positive solitary waves are amplitude centered at x = 70 and the other with amplitude centered at. The invariant values I 1, I 2, and I 3, computed by the present scheme are satisfactory constant, each change is less than 0.002%, 0.009% and 0.02% respectively from the initial to time while in [15] the changes are less than by 0.36%, 0.58%, and 2.24% in each. Table 7: Invariant values for the interaction of two positive solitary waves, 0 x 120 t N I 1 I 2 I 3 N I 1 I 2 I 3 0 = = Present [15] method t = t = Figure 5: The motion of two positive solitary waves before and after the interaction 4.3 Maxwellian initial condition Consider the Maxwellian initial condition u(x, 0) = exp( (x 7) 2 ) (30) We have solved the RLW equation with the initial condition of equation (30) and various values of the parameter ν. We discuss the numerical solution in the cases: ν = 0.04, 0.01, For ν = 0.04 the Maxwellian develops into a single solitary wave plus a small developed oscillating tail as shown in Fig. 6a at time t = 12. A bit of radiation observed between the upstream boundary and the main peak wave. The values of the quantities I 1, I 2, and I 3 are given in Table 8; each is satisfactorily constant. The maximum changes are less than % while in [15] is less than 5.7% in each. For ν = 0.01 the Maxwellian pulse breaks up into a train of at least three solitary waves as shown in Fig.6b at time. A bit of radiation observed between the upstream boundary and the main peak wave. The IJNS homepage:

10 140 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp Figure 6: Maxwellian initial condition at different values of v and t values of the quantities I 1, I 2, and I 3 are given in Table 8; each is satisfactorily constant. The maximum changes are less than 0.02% which is better than the corresponding computed values by [15] which is less than 5%. For ν = the Maxwellian pulse breaks up into a train of solitary waves as shown in Fig.6c at time t = 3. It is noted from the comparison between Fig 6c and Fig. 6d that decreasing the step leads to the significance of the nonlinearity which is more illustrated in Fig 6c and Fig. 6d. So, for small value of the effect of the nonlinearity becomes the dominant and hence we obtain a sharp wave. The values of the quantities are given in Table 8; the changes in the values of I 1, I 2, and I 3 are 0.38%, 0.55%, and 1.13%, respectively which are better than the corresponding computed values by [15] (0.0004%, 0.66%, and 1.77%). Table 8: Invariant values for Maxwellian initial condition ν t I 1 [Present] I 2 [Present] I 3 [Present] I 1 [15] I 2 [15] I 3 [15] IJNS for contribution: editor@nonlinearscience.org.uk

11 A. H. A. Ali : Chebyshev Collocation Cpectral Method for Solving the RLW Equation Conclusion It has been shown that the numerical solution for solving the RLW equation using the spectral method based on Chebyshev polynomials within the collocation method is more accurate compared with the recent results during all run to the simulations. The error norms computed by the present algorithm with different amplitudes compared to the previous results found to be smaller. The three invariants of motion are satisfactory constant in all computer simulations described here, so that the algorithm can fairly be described as conservative. All the recent algorithms did not solve the RLW equation with initial amplitude 0.9 except [15], and when we compare the present algorithm with [15], we find that the errors are smaller and the three invariant values are constant. So we deduce that this algorithm is more accurate than the previous algorithms and we believe that this approach will also be useful for solving similar nonlinear partial differential equations. It is worthwhile noticing that all our computations have been conducted on a 32 bit machine, which means that the accuracy could have been much better if one uses a mainframe with 256 or 512 bit processor. References [1] D.H. Peregrine: Calculations of the development of an undular bore. J. Fluid Mech. 25, (1966) [2] Kh. O. Abdulloev, H. Bogolubsky and V. G. Makhankov: One more example of inelastic soliton interaction. Phys. Lett. A.56, (1976) [3] J. C. Eilbeck and G. R. McGuire: Numerical study of the regularized long wave equation II. J. Comput. Phys. 23, (1977) [4] P.J. Jain, L. Iskandar: Numerical solutions of the regularized long wave equation. Comput. Methods Appl. Mech. Engrg. 20, (1979) [5] J. L. Bona, W. G. Pritchard, L. R. Scott: Numerical schemes for a model of nonlinear dispersive waves. J. Comput. Phys. 60, (1985) [6] M. E. Alexander and J. Ll. Morris: Galerkin methods applied to some model equations for nonlinear dispersive waves. J. Comput. Phys. 30, (1979) [7] L. Wahlbin: Error estimates for a Galerkin method for a class of model equations for long waves. Numer. Math. 23, (1975) [8] L. R. T. Gardner and G. A. Gardner: Solitary wave of the regularised long wave equation. J. Comput. Phys. 91, (1990) [9] L. R. T. Gardner, G. A. Gardner and I. Dag: A B-spline finite element method for the regularised long wave equation. Commun. Numer. Meth. Eng. 12, (1995) [10] L. R. T. Gardner and I. Dag: The boundary-forced regularised long wave equation. Nuova Cimento B. 110(12), (1995) [11] A. A. Soliman, K. R. Raslan: Collocation method using quadratic B-spline for the RLW equation. Int. J. Comput. Math. 78, (2001) [12] L. R. T. Gardner, G. A. Gardner and A. Dogan: A least squares finite element scheme for the RLW equation. Commun. Numer. Meth. Eng. 11, 59-68(1995) [13] I. Dag: Least square quadratic B-spline finite element method for the regularised long wave equation. Comp. Meth. Appl. Mech. Eng. 182, (2000) [14] I. Dag, M. Naci Ozer: Approximation of the RLW equation by the least square cubic B-spline finite element method. Appl. Math. Mod. 25, (2001) IJNS homepage:

12 142 International Journal of Nonlinear Science,Vol.7(2009),No.2,pp [15] K. R. Raslan: A computational method for the regularized long wave (RLW) equation. App. Math. Comp. 167, (2005) [16] D.M. Sloan: Fourier pseudospectral solution of the regularized long wave equation. J. Comput. Appl. Math. 36, (1991) [17] J.I. Bona, W.R. McKinney and J.M. Restrepo: Stable and unstable solitary wave solutions of the generalized regularized long wave equation. J. Nonlinear Sci. 10, (2000). [18] A. Araujo and A. Duran: Error propagation in the numerical integration of solitary waves the regularized long wave equation. Appl. Numer. Math. 36, (2001) [19] A. Duran and M.A. Lopez-Marcos: Numerical behaviour of stable and unstable solitary waves. Appl. Numer. Math.42, (2002) [20] A.H.A. Ali: Spectral method for solving the equal width equation based on Chebyshev polynomials. Nonlinear Dyn. 51, 59-70(2008) [21] E. H. Doha: The coefficients of differentiated expansion and derivatives of Ultraspherical polynomials. J. Comput. Math. Appl. 21, (1991) [22] B. Fornberg: A practical guide to pseudospectral methods. Cambridge University Press, (1995) [23] P. J. Olver: Euler operators and conservation laws of the BBM equation. Math. Proc. Cambridge Philso. Soc. 85, (1979) [24] A. R. Santarelli: Numerical analysis of the regularized long wave equation. Nuovo Cim. 39, (1981) [25] J. C. Lewis, J. A. Tjon: Resonant production of solitons in the RLW equation. Phys. Lett. A. 73, (1979) IJNS for contribution: editor@nonlinearscience.org.uk