An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method

and Applied Analysis 3 Table 1: Invariants and error norms for 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 4.4428661 3.2998227 1.4142046 0.00000000 0.00000000 1 4.4428661 3.2998085 1.4142188 0.28537793 0.16594258 2 4.4428661 3.2997808 1.4142465 0.56248008 0.31854916 3 4.4428661 3.2997573 1.4142700 0.82836630 0.45535369 4 4.4428661 3.2997415 1.4142858 1.08566992 0.58528925 5 4.44286601 3.2997313 1.4142960 1.33772774 0.71261445 6 4.4428661 3.2997248 1.4143025 1.58675627 0.83879372 7 4.4428661 3.2997207 1.4143067 1.83403948 0.96441682 8 4.4428661 3.2997180 1.4143093 2.08032250 1.08975930 9 4.4428661 3.2997162 1.4143111 2.32602024 1.21494581 10 4.4428661 3.2997151 1.4143122 2.57148152 1.34021078 Table 2: Errors and invariants for 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 3.2997151 1.4143122 2.57148 1.334021 [6] 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 Quintic B-splines coll [36] 4.4428661 3.2997108 1.4143165 2.58891 1.35164 Table 3: Invariants and error norms for 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 3.5819531 1.3450721 0.1537217 0.0000000 0.0000000 2 3.5819531 1.3450719 0.1537219 0.0373696 0.0211791 4 3.5819531 1.3450715 0.1537223 0.0711480 0.0387624 6 3.5819531 1.3450711 0.1537227 0.1001141 0.0515117 8 3.5819531 1.3450708 0.1537231 0.1249329 0.0614203 10 3.5819531 1.3450705 0.1537234 0.1466243 0.0700260 12 3.5819531 1.3450702 0.1537236 0.1659668 0.0775889 14 3.5819531 1.3450700 0.1537238 0.1833628 0.0844911 16 3.5819531 1.3450698 0.1537240 0.2015361 0.0909663 18 3.5819531 1.3450697 0.1537241 0.2560750 0.0993420 20 3.5819531 1.3450696 0.1537243 0.3585031 0.1702101 20 [6] 3.58197 1.34508 0.153723 0.645295 0.301923 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 20 [36] 3.5820204 1.3450974 0.1537250 0.8112594 0.3569076 4 Abstract and Applied Analysis Table 4: Invariants and error norms for single solitary wave with c = 0.6, h = 0.1, k = 0.1, −40 ≤ x ≤ 60. t I 1 I 2 I 3 L 2 × 10 4 L ∞ × 10 4 CPU Time 4 8.070902 4.100549 14.361115 2.716636 1.323585 0.437 s 8 8.070925 4.100534 14.361111 5.240855 2.458097 0.908 s 12 8.070943 4.100519 14.361103 7.543043 3.372256 1.435 s 16 8.070949 4.100503 14.361089 9.662561 4.154427 1.774 s 20 8.070921 4.100489 14.361074 11.660256 4.862191 2.319 s Table 5: Invariants and error norms for single solitary wave with c = 0.18, h = 0.1, k = 0.1, −80 ≤ x ≤ 120. t I 1 I 2 I 3 L 2 × 10 5 L ∞ × 10 5 CPU Time 4 7.809873 2.129887 7.031111 3.077451 1.138290 0.801 s 8 7.809875 2.129887 7.031112 6.124574 2.313797 1.571 s 12 7.809877 2.129887 7.031112 9.106266 3.455915 2.470 s 16 7.809879 2.129887 7.031113 11.967534 4.530312 3.124 s 20 7.809880 2.129887 7.031115 14.731556 5.538799 3.789 s order forward difference formula for the time derivative of the U in (3) have been used, which lead to


Introduction
The generalized regularized long wave equation is given by where is a positive integer and and are positive constants. This equation is one of the most important nonlinear wave equation used a model for small amplitude long waves on the surface of water in a channel [1,2]. A few authors solved the equation numerically: among others, Zhang [3] used a finite difference method for a Cauchy problem and Kaya [4] applied the Adomian decomposition method and a quasilinearization method based on finite differences was used by Ramos [5]. Roshan [6] implemented the Petrov-Galerkin method using a linear hat function as the trial function and a quintic B-spline function as the test function. A mesh-free technique for the numerical solution of the equation has been presented by Mokhtari and Mohammadi [7]. For = 1, Equation (2) is known as regularized long wave equation, originally introduced to describe the behavior of the undular bore by Peregrine [1] and later widely studied by Benjamin et al. [8]. The RLW equation has been solved numerically by finite element methods [9][10][11][12][13][14][15][16][17][18][19][20][21][22], finite difference methods [23][24][25][26], Fourier pseudospectral [27], and mesh-free method [28]. For = 2, Another particular case of (2) is called modified regularized long wave (MRLW) equation. Like RLW equation, the MRLW equation has been solved numerically by various methods. Among many others, a collocation solution to the equation using quintic B-spline finite element method is developed by Gardner et al. [29]. Khalifa et al. [30,31] obtained the numerical solutions of the equation using finite difference method and cubic B-spline collocation finite element method. Solutions based on collocation method with quadratic B-spline finite elements and the central finite difference method for time are investigated by Raslan [32]. The equation was solved with a collocation finite element method using quadratic, cubic, quartic, and quintic B-splines to obtain the numerical solutions of the single solitary wave by Raslan and Hassan [33]. Haq et al. [34] have designed a numerical scheme based on quartic B-spline collocation method for the numerical solution of the equation. Ali [35] has formulated a classical radial basis functions (RBFs) collocation method for solving the equation. Karakoc et al. [36] have obtained a type of the quintic B-spline collocation procedure in which nonlinear term in the equation is linearized by using the form introduced by the Rubin and Graves [37] to solve the equation. A Petrov-Galerkin method using cubic B-spline function as trial function and a quadratic B-spline function as the test function is set up to solve the equation by Karakoc and Geyikli [38]. A homotopy analysis method has been employed to obtain approximate numerical solution of the modified regularized long wave (MRLW) equation with some specified initial conditions by Khan et al. [39].
In the present paper, a numerical scheme based on the septic B-spline collocation method has been set up for solving the MRLW equation with a variant of both initial and boundary conditions. This paper is set out as follows. In Section 2, septic B-spline collocation scheme is presented. Also stability analysis is considered. In Section 3, test problems including single, two, and three solitary waves and Maxwellian initial condition are discussed. Finally in Section 4, a summary is given at the end of the paper.

Septic B-Spline Finite Element Solution
Consider the MRLW Equation (3) given with the following boundary conditions, otherwise.
Using expansion (6) and trial function (7), the nodal values ( ) and their first, second, and third derivatives ( ) , ( ) , ( ) can be calculated at the nodal points in terms of nodal parameters by the following set of equations: The splines ( ) and their six principle derivatives vanish outside the interval [ −4 , +4 ].
To apply the proposed method, Crank-Nicolson approximation for the space derivatives and and usual first Abstract and Applied Analysis 3 In order to linearize the nonlinear term ( 2 ) +1 , we can write the term as follows, and apply the linearization form introduced by Rubin and Graves [37] ( to (9), and we get Substituting the approximate solution and putting the nodal values of and its derivatives given by (8) into (12) one obtains the following iterative system for = 0, 1, . . . , : The newly obtained iterative system (13) consists of To obtain a unique solution of this system, six additional constraints are required. Applying the boundary conditions (4) and using the values of (8), these constraints are used and this enables us to eliminate the unknowns −3 , −2 , −1 , 0 , . . . , , +1 , +2 , +3 from system (13). So system (13) is reduced to a septa-diagonal system of ( + 1) linear equations in ( + 1) unknowns given by . The coefficient matrixes are given by Abstract and Applied Analysis 5 Before starting the solution process, initial parameters 0 must be determined by using the initial condition and the following derivatives at the boundaries: So we have the following matrix form for the initial vector 0 :

A Linear Stability Analysis.
We have investigated stability analysis by applying the von-Neumann approach in which the growth factor of typical Fourier mode is given by where is a mode number and ℎ is the element size. To apply this method, we have linearized the nonlinear term 2 by considering 2 as a constant such as in (9). If we substitute (20) into the iterative system (13) we obtain the following equation: where is the growth factor. We have identified the collocation points with the nodes and used (8) to evaluate and its space derivatives in (3). This leads to a set of ordinary differential equations in the following form: where = ( −3 +120 −2 +1191 −1 +2416 +1191 +1 +120 +2 + +3 ) 2 . Here ⋅ denotes derivative with respect to time. If the parameters 's and their time derivatives in (22) are discretized by the Crank-Nicolson formula and usual forward finite difference approximation, respectively, we obtain a recurrence relationship between two time levels and + 1 relating two unknown parameters +1 , for = − 3, − 2, . . . , + 2, + 3, Substituting the Fourier mode (20) into (24) leads to the growth factor of the form Abstract and Applied Analysis | | 2 = 1; therefore the linearized scheme is unconditionally stable.

Numerical Examples and Results
In this section, we have obtained numerical solution of the MRLW equation for motion of single solitary wave, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves. Accuracy of the method is measured by using the following error norms: The discrete conservation properties of the MRLW equation corresponding to mass, momentum, and energy are determined by finding the following three invariants [41]: 3.1. The Motion of Single Solitary Wave. As a first problem, MRLW equation (3) is considered with the boundary conditions → 0 as → ±∞ and the initial condition Single solitary wave solution of the MRLW equation has an analytical solution of the form where = √ / ( + 1) and 0 and are arbitrary constants. This solution corresponds to motion of single solitary wave with amplitude √ , initially centered at 0 and with wave velocity 1 + . For this problem the analytical values of the invariants are [29] For the computational work, two sets of parameters have been chosen and discussed. First of all, we have taken the parameters = 1, = 1, ℎ = 0.2, 0 = 40, = 0.025 over the interval [0, 100] to compare our results with [6,29,30,35,36]. Thus, the solitary wave has an amplitude 1.0 and the computations are done up to time = 10 to obtain the invariants and error norms 2 and ∞ . Values of the three invariants and error norms are reported in Table 1  at different time levels. It is observed that the solitary wave moves to the right with constant velocity and amplitude. At = 0, the amplitude is 1.0 which is located at = 40, while it is 0.999950 at = 10 located at = 60. The absolute difference in amplitudes at times = 0 and = 10 is found to be 5×10 −5 , so there is a little change between the amplitudes.

Interaction of Two Solitary Waves.
As a second problem, interaction of two well separated solitary waves having different amplitudes and traveling in the same direction is considered by using the initial condition We have studied the interaction of two positive solitary waves having the parameters = 1, ℎ = 0.2, = 0.025, 1 = 4, 2 = 1, 1 = 25, 2 = 55 through the interval 0 ≤ ≤ 250 to coincide with those used by [6,30,[34][35][36]. The simulations are maintained up to = 20. Constant values 1 , 2 , and 3 at various time steps together with equivalent results for the previous methods are shown in Table 6. It is seen that the numerical values of the invariants remain almost constant during the computer run. The interaction of two solitary waves is shown in Figure 4. It can be seen from the figure that, at = 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 = 8 and then moves away from the shorter one as time increases. At = 20, the amplitude of larger waves is 2.001102 at the point = 127.4, whereas the amplitude of the smaller one is 0.996403 at the point = 92. It is found that the absolute difference in amplitude is 3.59×10 −3 for the smaller wave and 1.10 × 10 −3 for the larger wave for this case.

Interaction of Three Solitary Waves.
As a third problem, interaction of three solitary waves having different amplitudes and travelling in the same direction is studied. We consider (3) with initial conditions given by Abstract and Applied Analysis In order to be able to compare with the previous works, computations are performed for the parameters = 1, ℎ = 0.2, = 0.025, 1 = 4, 2 = 1, 3 = 0.25, 1 = 15, 2 = 45, 3 = 60 over the interval 0 ≤ ≤ 250. The simulation is run from = 0 to = 45. Table 7 shows a comparison of the values of the invariants obtained by the present method with those obtained in [30,[34][35][36]. It is evident 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 (36). Figure 5 depicts the interaction of three solitary waves at different times. It is observed from Figure 5 that interaction started about time = 10, overlapping processes occurred between time = 15 and = 40, and waves started to resume their original shapes after the time = 40.
3.4. The Maxwellian Initial Condition. As our last problem, the development of the Maxwellian initial condition, into a train of solitary waves is considered. For the Maxwellian initial condition, behavior of the solution depends on the values of . So we take = 0.1, = 0.04, = 0.015, and = 0.01. The numerical values of the invariants quantities during the simulations are given in Table 8. For = 0.1, only a single soliton occurred as depicted in Figure 6(a). When = 0.015 and = 0.01, two and three stable solitons occurred, respectively, as depicted in Figures  6(b) and 6(c). For = 0.04, Maxwellian initial condition has decayed into four solitary waves as depicted in Figure 6(d).
As is seen, when is reduced, more solitary waves occurred. All figures are drawn up at time = 14.5. The peaks of the well developed wave lie on a straight line so that their velocities are linearly dependent on their amplitudes and also we observe a small oscillating tail appearing behind the last wave in all Maxwellian figures.

Conclusion
In this paper, a numerical treatment of the MRLW equation has been introduced using septic B-spline collocation finite element method. The nonlinear term in the equation has been linearized by using a form given in the paper [37]. To examine the performance of the scheme, four test problems have been studied. The performance and accuracy of the method have been tested by calculating the error norms 2 and ∞ and the invariant quantities 1 , 2 , and 3 . Linear stability analysis proved that the present scheme is unconditionally stable. The experimental results of the algorithm are much satisfactory in comparison with the previous results. Our method can successfully be used to model the motion and interaction of the solitary waves. Thus, we can assert that our scheme is efficient and reliable for obtaining the numerical solutions of the other physically important nonlinear partial differential equations.