Abstract
The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave () equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms and to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.
1. Introduction
The regularized long wave () equation is defined by the following nonlinear partial differential equation [1]: where and are positive parameters. This equation was first introduced by Peregrine [1] and after that by Benjamin et al. [2] to describe the behavior of the undular bore. It has also a great role in physics science, especially in physics media since it is useful in describing a phenomenon in different disciplines, such as the nonlinear transverse waves in magneto hydrodynamics waves in plasma, ion-acoustic waves in plasma, shallow water, longitudinal dispersive waves in elastic rods, phonon packets in nonlinear crystals, and pressure waves in liquids gas bubbles.
There are many analytical methods to obtain the solution of the equation for certain boundary and initial conditions; for example, see [2, 3]. Also, the numerical solutions of the equation has been studied by many researchers via various methods, such as finite difference methods [4, 5], Fourier pseudospectral methods [6], various models of finite element methods including least square, collocation, and Galerkin methods [7–9], mesh-free method [10], and Galerkin finite element methods [11–13].
The generalized form of the equation is known as the equation which is given by where is a positive parameter. In the extension of nonlinear dispersive waves, this equation has an important role. There are many numerical methods to investigate its solution. For more details, we advise the reader to visit [14–17]. In the current attempt, we consider a special case of the (namely, the ) equation, given by subject to the boundary conditions (s): and the initial condition () is taken as where is assumed to be localized disturbance inside the given interval. There are many authors who obtained the numerical solution of the equation; for example, Gardner et al. [18] used the cubic B-spline finite element method, Prenter [19] used variational and spline methods, and Khalifa et al. [20] used finite difference method; in [21], they used the Adomian decomposition method, they also in [22] used the collocation method, and Fazal-i-Haq et al. [23] used the quartic B-Spline collocation method to get an approximate solution of the equation.
In this study, inspired by the abovementioned studies, we use the sextic B-spline collocation methods to approximate the solution of the equations (3)–(5). The rest of the paper is organized as follows. In Sections 2.1 and 2.2, we discuss the B-spline collocation methods I and II and their stability analysis on the proposed equation. Section 3 is dedicated to the numerical implementations and comparison of our obtained results with those obtained in the literature: Section 3.1 is for single solitary wave, and Sections 3.2 and 3.3 are for interactions of multiple solitary waves. A conclusion is subsequently given in Section 3.
2. The Methods of B-Spline Collocation
Let us partition the finite interval into a uniform mesh by points such a way that , where . Then, we state the sextic B-spline collocation methods I (SBCM1) and II (SBCM2).
2.1. The SBCM1
We know that the sextic B-splines are usually defined on nodes over a given interval with additional nodes outside the interval . The additional nodes may be given as follows:
The sextic B-splines at the knots are defined as [1] for , and the set of sextic B-splines can be a basis over the interval .
The approximate solution of the equation to the exact solution will be determined as follows: where the time dependent parameters will be determined from the sextic B-spline collocation formula of equation (3).
In view of equation (8) and Table 1, the nodal values at the knots can be found as
Now, we implement the collocation method at the nodes . Also, we substitute the nodal variables and its derivatives at the knots in equation (9) into equation (3); then, we get the following system of nonlinear ordinary differential equations: where and denote derivative with respect to time.
The unknown parameters and are linearly interpolated between and ( and are two time levels) via the the Crank-Nicolson formula and the usual forward difference formula, respectively, as follows: where denotes the parameters at time . Then, by making use of equation (11), it follows that where
The system (12) consisting of the equations with unknown parameters. It can be solved uniquely if we eliminate the parameters by using the five s ; that is,
Consequently, we get a matrix system of dimension , and one can solve it easily by using a variant of the Thomas algorithm.
To deal with nonlinearity in (12) at each time step, we carry out the following corrector methods: (1)Approximating by using the following simple corrector: (2)As an approximation for , we use (3)Repeating this procedure twice at time step to refine (4)Repeating this procedure at each time step along the execution of the program
The iterative procedure in (4) can start by determining the initial parameters , and it can be determined by making use of the s (4), (5), and the following requirements:
Again, one can solve it by a variant of the Thomas algorithm and the approximate solution that can be obtained from equation (9).
Now, we can apply the von Neumann stability method to establish the stability of the scheme (12), but the von Neumann stability method is applicable to linear schemes; so, we shall line arise the nonlinear term by taking as a constant value , and thus the nonlinear term becomes . Then, by substitution the Fourier mode into our linearized form of equation (12) with writing , we get where is growth factor and
Thanks to Python software, we obtain same expressions for and in the following form: in order for the magnitude of the growth factor that is, and thus the linearized numerical algorithm for the equation will be unconditionally stable.
2.2. The SBCM2
One can split equation (3) as a system of partial differential equation:
To applied the collocation approach for system (20) and (21), we identify the collocation points with the nodes . If we put the approximation (7) into equations (20) and (21), we can obtain the following system of st order ordinary differential equations: where is defined in (10), and nonlinearity term is . Approximating the parameters between and by using the Crank-Nicolson formula and by using the finite difference rule as follows:
Thus, equation (22) becomes where
Analogously, in view of the Crank-Nicolson and forward finite difference approaches in time, both parameters and are linearly interpolated between two time levels and , respectively, as follows:
Thus, equation (23) becomes where
Equations (25) and (28) constitute the numerical algorithms for the equation. We can remove the nonlinearity terms occurring in equation (25) by replacing by in , and thus the equation (25) will be linearized.
Therefore, we see that the iterative systems of equations (25) and (28) consist the equations in the unknown parameters, and one can solve it by eliminating the parameters by making use of the five s ; then, we can obtain
Thus, we get a matrix system of dimension , and we can easily solve it by using a variant of the Thomas algorithm.
To deal with nonlinearity in (28) at each time step, we carry out the following corrector procedure:
This iterative scheme is executed two times by determining for , where .
We start the time evolution of the using (25) and (28) by calculating initial parameters . Therefore, the approximate solution (8) must agree with the at the knots, and this leads to equations. Also, the further five equations can be obtained by using the derivatives of in (8) at the ends:
Consequently, the parameters will be determined as solution of a matrix equation.
To establish the stability of the scheme (25), we carry out the von Neumann stability scheme by linearizing the nonlinear term by taking as a constant so that becomes . By substitution the Fourier mode of into our linearized form of equation (25) and by writing in the resulting iterative equation, we can deduce where
Here, note that the von Neumann condition is fulfilled; that is, . This affects the difference scheme (25) to be unconditionally stable. In the same way, we can show that the difference equation (28) can be unconditionally stable as well.
3. Numerical Calculations
Here, numerical tests are presented to demonstrate the performance of our proposed algorithm for single and interactions of multiple solitary waves. Also, the modified Maxwellian s are pointed out to generate a train of solitary waves. Furthermore, the accuracy of the presented schemes is measured in terms of the following discrete error norms and :
The conservation properties of the equation related to energy, mass, and momentum can be determined by finding the three basic invariants [24, 25]:
3.1. Single Solitary Wave
Let be any arbitrary constant. Then, the exact solution of the solitary wave of the equation is given as follows [20]:
The modified Maxwellian is defined by and the s can be concluded from the exact equation.
We choose so that we can compare our results with results in [20, 23]. The program is executed up to times to find error norms and the invariants at different times, and the results are given in Table 2. From Table 2, one can observe that the predicted error norms and are smaller than those obtained in [20, 23], and also the invariants , and are sanely in good agreement with their exact values. The solutions at and the motion of the solitary wave along with the interval with to the right are illustrated in Figure 1. Moreover, the error variations are demonstrated for the proposed algorithms SBCM1 and SBCM2 in Figure 2 at time . Consequently, we can observe from Table 2 that the results obtained by the SBCM2 are more accurate than those obtained by the SBCM1.
3.2. Two Solitary Waves
Now, we study the interaction of two solitary waves having different amplitudes, which is the sum of two modified Maxwellian :
Here, we take the parameters to concur with those used in [22, 23, 26]. The program is executed up to time , and the values of invariants are shown in Table 3 and compared with those obtained in [22, 23, 26] at time . On the other hand, Figure 3 illustrates the interaction of solitary waves at the times and , respectively.
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(b)
3.3. Three Solitary Waves
Here, we study the equation with the modified Maxwellian and different amplitudes:
Numerical experiments are executed for the parameters in the region in order to see an interaction of three solitary waves takes place. The program is executed up to time . Table 4 compares our obtained values of the invariants of the three solitary waves by SBCM2 with those obtained by [23]. It is clear from the table that our obtained results of the invariants remain almost the same during the computer run, and they are found to be very close to the results given in [23]. In addition, these are all in good agreement with their analytical results.
Also, numerical experiments are carried out for the parameters . The computation is done until time to find numerical results of the invariants . The result values of the invariants of the proposed SBCM2 algorithm together with the values of the invariants obtained in [26] are documented in Table 5.
In addition, we demonstrate the interaction of three solitary waves at times and , respectively, in Figure 4 and consequently, we can see that at time , the three solitary waves interact and then at times , the three solitary waves separate and emerging unchanged.
(a)
(b)
(c)
(d)
4. Conclusion
The main results of the article can be summarized as follows: (i)The sextic B-spline collocation methods are presented to approximate a new solution of the equation(ii)The unconditionally stability of the methods is derived(iii)The operations are established by calculating both error norms and (iv)The numerical applications are demonstrated through examples of equations with the modified Maxwellian s
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts interests.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by the Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia, and the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.