Solving theModified Regularized LongWave Equations via Higher Degree B-Spline Algorithm

Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O.Box 84428, Riyadh 11671, Saudi Arabia Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania


Introduction
The regularized long wave (RLW) equation is defined by the following nonlinear partial differential equation [1]: where μ and ζ 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 RLW equation for certain boundary and initial conditions; for example, see [2,3]. Also, the numerical solutions of the RLW 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][8][9], mesh-free method [10], and Galerkin finite element methods [11][12][13].
The generalized form of the RLW equation is known as the GMRLW equation which is given by more details, we advise the reader to visit [14][15][16][17]. In the current attempt, we consider a special case of the GMRLW (namely, the MRLW) equation, given by subject to the boundary conditions (B:Cs): and the initial condition (I:C:) is taken as where f ðηÞ is assumed to be localized disturbance inside the given interval. There are many authors who obtained the numerical solution of the MRLW 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 MRLW equation.
In this study, inspired by the abovementioned studies, we use the sextic B-spline collocation methods to approximate the solution of the MRLW 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 MRLW 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.
The approximate solution σ J ðη, τÞ of the GMRLW equation to the exact solution σðη, τÞ will be determined as follows: where the time dependent parameters ϑ ℓ ðτÞ will be determined from the sextic B-spline collocation formula of equation (3). In view of equation (8) and Table 1, the nodal values σ ðiÞ , i = 0, 1, ⋯, 5 at the knots η ℓ can be found as Now, we implement the collocation method at the nodes η i , i = 0, 1, ⋯, J. Also, we substitute the nodal variables σ ℓ and its derivatives at the knots η i in equation (9) into equation 2 Journal of Function Spaces (3); then, we get the following system of nonlinear ordinary differential equations: where d = 1 + ζz ℓ = 1 + ζðϑ ℓ−3 + 57ϑ ℓ−2 + 302ϑ ℓ−1 + 302ϑ ℓ + 57ϑ ℓ+1 + ϑ ℓ+2 Þ p and ∘ denote derivative with respect to time. The unknown parameters ϑ ℓ and ϑ ∘ ℓ are linearly interpolated between n and n + 1 (n and n + 1 are two time levels) via the the Crank-Nicolson formula and the usual forward difference formula, respectively, as follows: where ϑ n ℓ denotes the parameters at time nΔτ. Then, by making use of equation (11), it follows that where The system (12) consisting of the ðJ + 1Þ equations with ðJ + 6Þ unknown parameters. It can be solved uniquely if we eliminate the parameters ϑ n+1 J+2 by using the five B:Cs σðρ 1 , τÞ = β 1 , σðρ 2 , τÞ = β 2 , σ η ðρ 1 , τÞ = σ η ðρ 2 , τÞ = σ ηη ðρ 1 , τÞ = 0; that is, Consequently, we get a matrix system of dimension ðJ + 1Þ × ðJ + 1Þ, 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 ϑ n+1 by using the following simple corrector: (2) As an approximation for ϑ n+1 , we use ðϑ * Þ n+1 (3) Repeating this procedure twice at time step n + 1 to refine ϑ n+1 (4) Repeating this procedure at each time step along the execution of the program The iterative procedure ϑ n ℓ in (4) can start by determining the initial parameters ϑ 0 ℓ , and it can be determined by making use of the B:Cs (4), I:C: (5), and the following requirements: Again, one can solve it by a variant of the Thomas algorithm and the approximate solution σ ðiÞ J ðη, τÞ, i = 0, 1, ⋯, 5 that can be obtained from equation (9).

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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 σ p σ η by taking σ as a constant value k, and thus the nonlinear term becomes σ p = z ℓ = k p . Then, by substitution the Fourier mode ϑ n ℓ =p n e iℓφ into our linearized form of equation (12) with writinĝ p n+1 = qp n , we get where q is growth factor and Thanks to Python software, we obtain same expressions for A 2 1 + A 2 2 and A 2 3 + A 2 4 in the following form: in order for the magnitude of the growth factor that is|q| = 1, and thus the linearized numerical algorithm for the GMRLW 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 η i , i = 0, 1, ⋯, J. If we put the approximation (7) into equations (20) and (21), we can obtain the following system of 1st order ordinary differential equations: where ∘ is defined in (10), and nonlinearity term is z ℓ = ðϑ ℓ−3 + 57ϑ ℓ−2 + 302ϑ ℓ−1 + 302ϑ ℓ + 57ϑ ℓ+1 + ϑ ℓ+2 Þ p . Approximating the parameters ϑ ℓ between n and n + 1/2 by using the Crank-Nicolson formula and ϑ ∘ ℓ 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 ϑ ℓ and ϑ ∘ ℓ are linearly interpolated between two time levels n + 1/2 and n + 1, respectively, as follows: Journal of Function Spaces Thus, equation (23) becomes where α 10 = 302ℏ 2 + 120ℏΔτ + 300μ, Equations (25) and (28) constitute the numerical algorithms for the GMRLW equation. We can remove the nonlinearity terms occurring in equation (25) by replacing ϑ ℓ by ϑ n ℓ in z ℓ , and thus the equation (25) will be linearized. Therefore, we see that the iterative systems of equations (25) and (28) Thus, we get a matrix system of dimension ðJ + 1Þ × ðJ + 1Þ, 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 ðϑ * Þ j i for ϑ j i , where j = n + 1/2, n + 1.
We start the time evolution of the ϑ j ℓ , j = n + 1/2, n + 1 using (25) and (28) by calculating initial parameters ϑ 0 ℓ . Therefore, the approximate solution (8) must agree with the I:C: at the knots, and this leads to J + 1 equations. Also, the further five equations can be obtained by using the derivatives of σ J in (8) at the ends: Consequently, the parameters ϑ 0 i , i = −3, −2, ⋯, J + 2 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 σ p σ η by taking σ as a constant k so that σ p becomes σ p = z ℓ = k p . By substitution the Fourier mode of ϑ n ℓ =p n e iℓφ into our linearized form of equation (25) and by writingp n+1 = qp n in the resulting iterative equation, we can deduce where Here, note that the von Neumann condition is fulfilled; that is, |q | ≤1. 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.

Numerical Calculations
Here, numerical tests are presented to demonstrate the performance of our proposed algorithm for single and 5 Journal of Function Spaces interactions of multiple solitary waves. Also, the modified Maxwellian I:C: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 L 2 and L ∞ : The conservation properties of the MRLW equation related to energy, mass, and momentum can be determined by finding the three basic invariants [24,25]: 3.1. Single Solitary Wave. Let η 0 be any arbitrary constant. Then, the exact solution of the solitary wave of the MRLW equation is given as follows [20]: The modified Maxwellian I:C: is defined by and the B:Cs can be concluded from the exact equation. We choose ζ = μ = 1, λ = 0:1, ℏ = 0:125, Δτ = 0:1, η 0 = 0, −40 ≤ η ≤ 60 so that we can compare our results with results in [20,23]. The program is executed up to times τ = 10 to find error norms and the invariants χ 1 , χ 2 , χ 3 at different times, and the results are given in Table 2. From Table 2, one can observe that the predicted error norms L 2 and L ∞ are smaller than those obtained in [20,23], and also the invariants χ 1 , χ 2 , and χ 3 are sanely in good agreement with their   Figure 1. Moreover, the error variations are demonstrated for the proposed algorithms SBCM1 and SBCM2 in Figure 2 at time τ = 10. Consequently, we can observe from Table 2 that the results obtained by the SBCM2 are more accurate than those obtained by the SBCM1.

Three Solitary Waves.
Here, we study the MRLW equation with the modified Maxwellian I:C: and different amplitudes: Numerical experiments are executed for the parameters ζ = 6, y = 0:2, μ = 1, Δτ = 0:025, λ 1 = 0:03, λ 2 = 0:02, λ 3 = 0:01, η 1 = 18, η 2 = 48, η 3 = 88 in the region −40 ≤ η ≤ 180 in order to see an interaction of three solitary waves takes place. The program is executed up to time τ = 45. 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    Journal of Function Spaces 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.    Journal of Function Spaces done until time τ = 45 to find numerical results of the invariants χ 1 , χ 2 , χ 3 . 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 τ = 1, 5 and τ = 10, respectively, in Figure 4 and consequently, we can see that at time τ = 0, 5, the three solitary waves interact and then at times τ = 20, 35 , the three solitary waves separate and emerging unchanged.

Conclusion
The main results of the article can be summarized as follows:

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts interests.