Peregrine puts forward to describing the nonlinear dispersive waves with small amplitude on the surface of water in a channel.^[1] Since then, some physical phenomena such as nonlinear transverse waves in shallow water, ion-acoustic and magnetohydrodynamic waves in plasma, and phonon packets in nonlinear crystals are modelled by RLW equations. Since it has no analytical solutions generally, good numerical methods are needed to reveal physical events modelled by the regularized long wave equation (RLW) equation.

Many studies exist for the numerical solutions of the differential equations using splines. Splines are piecewise functions which have certain continuity at the joint points given to set up the splines. The spline related numerical techniques mainly offer the economical computer code and easy computational calculations. Thus they are preferable in forming the numerical methods. Until now, polynomial splines have been extensively developed and used for approximation of curve and surfaces and finding solutions of the differential equations. The polynomial spline based algorithms have been found to be quite advantageous for finding solutions of the differential equations. Base of splines known as the B-splines is also widely used to build up the trial functions for numerical methods. The exponential spline is proposed to be more general form of these splines. In the approximation theory, the exponential B-spline approximation is shown to model the data which have sudden growth and decay whereas polynomials are not appropriate due to having osculatory behavior. Since some differential equations have steep solutions, the use of the exponential B-splines in the numerical methods may exhibit good solutions for differential equations. McCartin^[2] has introduced the exponential B-spline as a basis for the space of exponential splines. The exponential B-spline properties accord with those of polynomial B-splines such as smoothness, compact support, positivity, recursion for derivatives. Thus the exponential B-splines can be used as the trial function for the variational methods such as Galerkin and collocation methods.

The exponential B-spline based methods have been started to solve some differential equations: Numerical solution of the singular perturbation problem is solved with a variant of exponential B-spline collocation method in Ref. [3], the cardinal exponential B-splines is used for solving the singularly perturbed problems,^[4] the exponential B-spline collocation method is built up for finding the numerical solutions of the self-adjoint singularly perturbed boundary value problems in Ref. [5], the numerical solutions of the convection–diffusion equation is obtained by using the exponential B-spline collocation method.^[6] Recently, a collocation method based on the exponential B-splines has been set up to find numerical solutions of the Fisher equation, generalized Burgers–Fisher equation and RLW equation.^[7–9] In this study, the combination of the exponential B-splines are used to form the trial function for the Galerkin method which is known to produce the smaller error for solutions of the differential equations. The RLW equation will be solved with the proposed exponential B-spline Galerkin method over the problem domain. Newly, a study in which the exponential B-splines are adapted to the Galerkin method to have solutions of the Burgers equation has appeared.^[10] Thus we want to see the efficiency of the exponential B-spline Galerkin method for the RLW equation.

The RLW equation describes a large number of important physical phenomena, such as shallow waters and plasma waves. Therefore it plays a major role in the study of nonlinear dispersive waves. Because of having limited analytical solutions, numerical analysis of the RLW equation has an importance in its study. Various techniques have been developed to obtain the numerical solution of the RLW equation, some of which are finite difference methods,^[1,11–14] finite element methods,^[15–25] and spectral methods.^[26–28] Spline functions are adapted to some of these standard methods to construct spline-related algorithms.

The paper is outlined as follows. In Section 2, exponential B-splines and their some basic relations are introduced. In Section 3, the application of the numerical method is given. The efficiency and the accuracy of the present method are investigated by using three numerical experiments related to propagation of single solitary wave, interaction of two solitary waves and wave generation. Finally some remarks are concluded in the last section.

In this study, we will consider the RLW equation 1 where x is space coordinate, t is time, u is the wave amplitude, and ε and μ are positive parameters. Boundary and initial conditions of Eq. (1) are 2

The equation (1) was first introduced by Peregrine^[1] for modelling the propagation of unidirectional weakly nonlinear and weakly dispersive water waves. Since the RLW equation obviates the certain problematical aspects of KdV equation and it generally has more expedient mathematical properties, Benjamin et al.^[29] proposed the use of the RLW equation as a preferable model to the KdV equation.

Let us consider a uniform mesh with the knots on [a, b] such that where and .

Let be the B-splines at the points of together with knots , i = −3, −2, −1, N + 1, N + 2, N + 3 outside the interval [a, b] and having a finite support on the four consecutive intervals , . The can be defined as 3 where Each basis function is twice continuously differentiable. The values of , , and at the knots ’s are obtained from Table 1.

Table 1.

Table 1.

Table 1. Exponential B-spline values. . 0 1 0 0 0 0 0 0

Table 1. Exponential B-spline values. .

The , form a basis for functions defined on the interval [a, b]. We seek an approximation U(x, t) to the analytical solution u(x, t) in terms for the exponential B-splines 4 where (t) are time-dependent unknown to be determined from the boundary conditions and Galerkin approach to Eq. (1). The approximate solution and their derivatives at the knots can be found from Eqs. (3) and (4) as 5 where

Applying the Galerkin method to the RLW equation with the exponential B-splines as weight function over the element [a, b] gives 6

The approximate solution U over the element can be written as 7 where quantities (t), are element parameters and , are known as the element shape functions.

The contribution of the integral equation (6) over the sample interval is given by 8 Applying the Galerkin discretization scheme by replacing approximate solution (7) and its derivatives into the exact solution u and its derivatives respectively, we obtain a system of equations in the unknown parameters 9 where i, j and k take only the values m − 1, m, m + 1, m + 2 for and · denotes time derivative.

In the above system of differential equations, when , and are denoted by 10 where , , and are the element matrices of which dimensions are 4 × 4 and (δ) is the element matrix with the dimension 4 × 4 × 4, the matrix form of Eq. (9) can be written as 11 where

Gathering the systems (11) over all elements, we obtain global system 12 where A, B, C(δ), D are derived from the corresponding element matrices , , (δ), , respectively and contain all elements parameters.

The unknown parameters δ are interpolated between two time levels n and n + 1 with the Crank–Nicolson method we obtain iterative formula for the time parameters 13 The set of equations consist of (N + 3) equations with (N + 3) unknown parameters. Before starting the iteration procedure, boundary conditions must be adapted into the system and initial vector must also be determined.

We delete first and last equations from system (13) and eliminate the terms and from system (13) by using boundary conditions in Eq. (2), which give the following equations:

We obtain a septa-diagonal matrix with the dimension . Then we can solve this matrix system through Thomas algorithm. Since system (13) is an implicit system due to the term , we have used the following inner iteration: 14 In this iteration, before moving the calculation of the next time step approximation for time parameter, we calculate the new vectors using formula (14) from previous vectors finding form the system (13) and then repeat three times at all time steps.

To start evolution of the vector of initial parameters , it must be determined by using the initial condition and boundary conditions 15 The solution of matrix equation (15) with the dimensions is obtained by the way of Thomas algorithm. Once is determined, we can start the iteration of the system to find the parameters at time . Approximate solutions at the knots is found from Eq. (5) and solution over the intervals is determined from Eq. (7).

To investigate the stability of system of the difference scheme (13), we apply the von Neumann stability analysis. A typical member of the linearized equation corresponding to Eq. (13) is given by where the parameters , are determined from system (13), in which these values are not preferred to document here due to being too long.

Substituting the Fourier mode into the linearized form of the difference equation becomes Here, the growth factor is determined as where

Since the magnitude of the growth factor is , the difference scheme (13) is unconditionally stable.

We have carried out three test problems to demonstrate the given algorithm. Accuracy of the method is measured by the error norm: The RLW equation satisfies the following conservation laws which are corresponding to mass, momentum, and energy:^[30]

In numerical calculations, the conservation laws are calculated by use of the trapezoidal rule and the determination of p in the exponential B-spline is found by scanning the predetermined interval with a small increment experimentally in a way that the best numerical solutions are obtained for the test problems.

3.1. Propagation of single solitary wave

The exact solution of RLW equation is given in Ref. [1] as follows: 16 where This form of the solution is known as a single solitary wave with the amplitude 3c and the velocity 1 + εc. The initial condition is obtained by taking t = 0 in Eq. (16). We have used boundary conditions . The values of the parameters seen in the above equations are With these parameters and the mentioned initial condition, the solitary wave moves across the interval in time period . Similar with some early studies, space step h = 0.125 and time step are used in numerical calculations. In this test problem, the p = 0.01262 is determined by scanning the interval [0, 80] with the increment 0.1 first, then according to the results scanning the interval [0, 1] with the increment 10⁻⁵. The solution profiles are illustrated in Fig. 1 at selected times. It is clear from this figure that the peak of the solitary wave remain kept during the running time.

Fig. 1.

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	Fig. 1. (color online) Solitary wave solutions.

The distribution of the absolute error at t = 20 for amplitudes 0.3 and 0.09 is given in Figs. 2(a) and 2(b), respectively. The maximum error for the EBSGM occurs at the right hand boundary as shown in Fig. 2(b). We believe that this error arises due to magnitude of the wave and the physical boundary conditions to fit and . If we extend the solution interval from [−40, 60] to [−80, 120], error norm is seen to reduce from to at time t = 20.

Fig. 2.

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	Fig. 2. (color online) The distribution of absolute error at t = 20 for (a) amplitude = 0.3, (b) amplitude = 0.09.

The absolute error norms and the values of the conservation invariants , , and are recorded in Tables 2 and 3 for different amplitudes, respectively. To make a comparison with some early studies, the maximum errors with the conservation invariants are presented in Table 4 and Table 5. According to these tables, the EBSGM gives more accurate results than or similar to some others except for the method of collocation accompanied with the exponential B-splines. The values of the conservation invariants , , and at different times remain fairly the same when compared with the analytical invariants , , for amplitude 0.3. When we take the amplitude as 0.09, the value of the conservation invariant has some minor difference than the analytical value of it, whereas and are fairly same at different times.

Table 2.

Table 2.

Table 2. Errors and Invariants for amplitude 3c = 0.3, h = 0.125, , p = 0.01262. . Time ×10 0 0.000 3.9799267 0.8104625 2.5790074 4 1.268 3.9799299 0.8104625 2.5790075 8 2.445 3.9799281 0.8104625 2.5790076 12 3.418 3.9799255 0.8104626 2.5790077 16 4.197 3.9799168 0.8104626 2.5790078 20 4.820 3.9798827 0.8104626 2.5790079

Table 2. Errors and Invariants for amplitude 3c = 0.3, h = 0.125, , p = 0.01262. .

Table 3.

Table 3.

Table 3. Errors and Invariants for amplitude 3c = 0.09, h = 0.125, , p = 0.01262. . Time ×10 0 0.000 2.1070467 0.1273013 0.3888047 4 2.302 2.1070975 0.1273011 0.3888041 8 2.214 2.1068946 0.1273011 0.3888041 12 2.129 2.1065454 0.1273011 0.3888040 16 2.139 2.1053719 0.1273012 0.3888032 20 4.315 2.1045885 0.1273012 0.3888023

Table 3. Errors and Invariants for amplitude 3c = 0.09, h = 0.125, , p = 0.01262. .

Table 4.

Table 4.

Table 4. Errors and Invariants for amplitude 3c = 0.3, h = 0.125, , p = 0.01262. . Method Time ×10 Analytical 3.97995 0.8104624 2.579007 EBSGM 20 0.04820 3.979883 0.8104626 2.579008 Ref. [9] 20 0.007743 3.979908 0.8104625 2.5790074 Ref. [24] 20 0.02643 3.979909 0.8104625 2.579007 Ref. [22] 20 0.07337 3.979883 0.8104612 2.579003 Ref. [21] 20 1.56640 3.961597 0.804185 2.558292 Ref. [23] 20 0.10299 3.979858 0.8104596 2.578999 Ref. [18] 20 0.116 3.979883 0.8102762 2.578393 Ref. [25] 20 0.91465 3.97972 0.81026 2.57873 Ref. [19] 20 0.07344 3.979888 0.8104622 2.579006 Ref. [20] 20 0.073 3.97989 0.81046 2.57901

Table 4. Errors and Invariants for amplitude 3c = 0.3, h = 0.125, , p = 0.01262. .

Table 5.

Table 5.

Table 5. Errors and Invariants for amplitude 3c = 0.09, h = 0.125, , p = 0.01262. . Method Time ×10 Analytical 2.109407 0.12730171 0.3888059 EBSGM 20 0.431512 2.104589 0.12730121 0.3888023 Ref. [9] 20 0.004174 2.107035 0.12730152 0.3888043 Ref. [21] 20 1.5506 2.128869 0.127228 0.388571 Ref. [18] 20 0.432 2.104584 0.12729366 0.3887776 Ref. [25] 20 0.439145 2.10902 0.12730 0.38880 Ref. [19] 20 0.19806 2.104708 0.1273006 0.3888025 Ref. [20] 20 0.199 2.10467 0.12730 0.38880

Table 5. Errors and Invariants for amplitude 3c = 0.09, h = 0.125, , p = 0.01262. .

3.2. Interaction of two solitary waves

In this section, we will study the interaction of two solitary waves having different amplitudes and moving in the same direction. The initial condition is where the following parameters are chosen to coincide values in the literature Initially, these parameters yield the solitary waves with the amplitudes 5.333375 and 1.687502 positioning around points x = 15 and x = 35 respectively. The computation is carried out up to time t = 30 with time step , N = 400 over the finite interval [0, 120] and p is selected as 1 for EBSGM.

Numerical solutions of u(x, t) at various times are depicted in Fig. 3, and the initial solution has been propagated rightward. It is seen from Fig. 3 that the solitary waves are subjected to a collision about time t = 15 and after the interaction they propagate with their original amplitudes to the right seeing at time t = 30.

Fig. 3.

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	Fig. 3. (color online) Interaction of two positive solitary waves at t = 0, 15 and 30.

The conservation invariants are presented at some selected times in Table 6. According to this table, during the interaction there are some changes at the conservation constants and whereas the constant remain nearly the same.

Table 6.

Table 6.

Table 6. Invariants for h = 0.3, , p = 1. . Time 0 37.916502 120.522247 744.081209 5 37.861378 119.772286 737.819263 10 37.810403 119.129874 732.319072 15 37.790851 119.012803 730.931470 20 37.752779 118.312087 725.698903 25 37.701304 117.600957 719.868731 30 37.650120 116.909863 714.174760

Table 6. Invariants for h = 0.3, , p = 1. .

3.3. Wave generation

An applied force like an introduction of fluid mass, an action of some mechanical device, to a free surface, will induce waves. In this numerical experiment, we take following boundary condition to generate waves with the RLW equation. and is studied to generate waves. This forced boundary condition known as a wave maker at one end.

The parameters , , , , are chosen to make a comparison with earlier works over the region . p is selected as 1 for EBSGM. During the run time of the algorithm, five solitary waves are produced. Although first four waves have reached amplitudes larger than forcing amplitudes, the last one is less than that of the forcing one. When forcing is switched off, the last wave has not enough time to evolve. Subsequently, no new wave is born. A view of travelling solitary wave is presented at time t = 100 in Fig. 4 Amplitudes of solitary waves versus time are depicted in Fig. 5 At various time, amplitudes of the solitary waves and the conservation constants are demonstrated in Table 7 for EBSGM. In addition to this, other amplitudes of the solitary waves which are reduced the other studies are shown in Table 7 for time t = 100. Our results are in conformity with that of studies^[20,31,32]

Fig. 4.

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	Fig. 4. (color online) Solitary wave produced by boundary forcing of duration and amplitude at time t = 100, h = 0.4, , p = 1.

Fig. 5.

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	Fig. 5. (color online) The evolution of wave amplitudes.

Table 7.

Table 7.

Table 7. Solitary wave amplitude with , h = 0.4, , p = 1 and period of forcing , . . Time 1 2 3 4 5 2.5 2.26 8.9397 16.1942 74.8986 5 2.78 19.3958 41.3363 206.5459 7.5 3.05 2.24 29.3109 64.3064 324.6697 10 3.23 2.53 2.02 39.2124 87.2152 442.2752 15 3.44 2.91 2.43 2.06 59.2209 133.8372 682.1665 20 3.53 3.14 2.73 2.38 1.90 76.8502 175.1059 882.7008 40 3.67 3.46 3.08 2.35 1.08 78.5060 174.3022 882.6289 60 3.70 3.50 3.05 2.33 1.07 78.4275 173.5305 877.6376 80 3.70 3.48 3.03 2.33 1.07 78.3394 172.7643 872.6609 100 3.68 3.47 3.03 2.32 1.07 78.2589 172.0094 867.7610 100[31] 3.76 3.52 3.06 2.33 1.07 100[20] 3.77 3.52 3.08 2.38 1.18 100[32] 3.76 3.51 3.07 2.31 0.98

Table 7. Solitary wave amplitude with , h = 0.4, , p = 1 and period of forcing , . .

In this paper, we investigated the utility of the exponential B-spline algorithm for solving the RLW equation. The efficiency of the method is tested on the propagation of the single solitary wave, the interaction of two solitary waves and wave generation. To see the accuracy of the method, error norm and conservation quantities , and are documented based on the obtained results. Exponential B-spline based method gives accurate and reliable results for solving the RLW equation. For the first test problem, generally EBSGM leads to more accurate results than the collocation-based methods but similar results with the some Galerkin methods. In the second test problem, there is no exact solution therefore simulation is shown graphically and the conservation quantities are tabulated. Generation of waves by using variable boundary conditions at left has been achieved and wave profiles and their amplitudes are documented. In conclude, the numerical algorithm in which the exponential B-spline functions are used, performs well compared with other existing numerical methods for the solution of RLW equation.