Application of Septic B-Spline Collocation Method for Solving the Coupled-BBM System

In the present paper, a numerical method is proposed for the numerical solution of a coupled-BBM system with appropriate initial and boundary conditions by using collocation method with septic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms L2, L∞ are computed. Furthermore, interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves after the interaction. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term. Citation: Raslan KR, EL-Danaf TS, Ali KK (2016) Application of Septic B-Spline Collocation Method for Solving the Coupled-BBM System. J Appl Computat Math 5: 327. doi: 10.4172/2168-9679.1000327


Introduction
In this paper, we consider the Coupled-BBM system, which belongs to the class of Boussinesq systems, modeling two-way propagation of long waves of small amplitude on the surface of water in a channel. The system is a good candidate for modeling long waves of small to moderate amplitude. The Coupled-BBM system is given by Bona and Chen [1], , 0 (1) , 0 6 (2) Where subscripts x and t denote differentiation x distance and t time, is considered, v(x,t) is a dimensionless deviation of the water surface from its undisturbed position and u(x,t) is the dimensionless horizontal velocity above the bottom of the channel.
And initial conditions. . ) , One of the advantages that equation (1) has over alternative Boussinesq-type systems is the easiness with which it may be integrated numerically [2]. Furthermore, it was proved in [2,3] that the initial value problem either for x ∈ ℜ or with boundary conditions (x ∈ [a,b]) for (1) is well posed in certain natural function classes. The initialboundary value problem of the form (1) posed on a bounded smooth plane domain with homogenous Dirichlet or Neumann or reflective (mixed) boundary conditions which is locally well-posed [4]. The existence and uniqueness of the system have been proved in Bona et al. [3]. They investigated the solution of the system as integral equation, while Chen [5] in his article established the existence of solitary waves for several Boussinesq types, including the Coupled-BBM system. Various numerical techniques including the finite element method have been used for the solution of Bona-Smith system of Boussinesq type in Antonopoulos et al. [6]. SS Behzadi and A Yildirim, using Quintic B-Spline Collocation Method for Solving the Coupled-BBM System [7]. ES Al-ℜawi and MAM Sallal, using finite element method to fiend the Numerical solution of Coupled-BBM system [8]. Chen fined the exact traveling-wave solutions to bidirectional wave equations [9]. The numerical solutions of coupled nonlinear systems are very important in applied science, for example, the hirota-satsuma coupled KDV equation which admits soliton solution and it has many applications in communication and optical fibers; this system has been discussed numerically by ℜaslan et al. finite element methods [10]. Also, the Hirota equation has been solving by ℜaslan et al. using finite element methods [11]. A finite element algorithm based on the collocation method with trial functions taken as septic B-spline functions over the elements will be constructed. The septic B-spline basis together with finite element methods are shown to provide very accurate solutions in solving some partial differential equations and have been used before by several authors. In this article we are going to derive a numerical solution of the coupled BBM-system. The brief outline of this paper is as follows. In Section 2, septic B-spline collocation scheme is explained. In Sections 3 and 4, the method is described and applied to the coupled BBM-system. In Section 5, stability of the method is discussed. In Section 6, numerical examples are included to establish the applicability and accuracy of the proposed method computationally. Conclusion is given in Section 7 that briefly summarizes the numerical outcomes.

Septic B-spline Functions
To construct numerical solution, consider nodal points ( The septic B-spline basis functions at knots are given by: Using septic B-spline basis function (5) the values of B j (x) and its derivatives at the knots points can be calculated, which is tabulated in Table 1.

Solution of Coupled-BBM System
To apply the proposed method, we rewrite (1) and (2) as , then from famous Cranck-Nicolson scheme and forward finite difference approximation for the derivative t, [12]. We get Where k=∆t is the time step (Table 1).
In the Crank-Nicolson scheme, the time stepping process is half explicit and half implicit. So the method is better than simple finite difference method.

Initial Values
To find the initial parameters 0 j c and 0 j δ , the initial conditions and the derivatives at the boundaries are used in the following way

Stability Analysis of the Method
The stability analysis of nonlinear partial differential equations is not easy task to undertake. Most researchers copy with the problem by linearizing the partial differential equation. Our stability analysis will be based on the Von-Neumann concept in which the growth factor of a typical Fourier mode defined as and g is the amplification factor of the schemes. We will be applied the stability of the septic schemes by assuming the nonlinear term as a constants λ 1 , λ 2 . This is equivalent to assuming that all the n j c and n j δ as a local constants λ 1 , λ 2 respectively. At x=x j systems (11) and (12) can be written as 1  1  1  1  1  1  1  1  1  3  2  2  3  1  4  5  1  6  2  7  3  8  3   1  1  1  1  1  9  2  10  1  10  1  9  2  8  3  7  3   6  2  5  1  4  3  1  2  2  1  3  8 The exact solution is Now, for comparison, we consider a test problem where,  Table 2. The motion of solitary wave using our scheme is plotted at times t=0,10,20 in Figure 1. These results illustrate that the scheme has a highest accuracy (Table 2 and Figure 1).
In Table 3 we show that our results are better than the results in [7] ( Table 3).

Interaction of two solitary waves
The interaction of two solitary waves having different amplitudes and traveling in the same direction is illustrated. We consider Coupled-BBM system with initial conditions given by the linear sum of two well separated solitary waves of various amplitudes. Similar substituting (13)

Numerical Tests and Results of Coupled-BBM system
In this section, we present some numerical examples to test validity of our scheme for solving coupled-BBM system.
The norms L 2 -norm and L ∞ -norm are used to compare the numerical solution with the analytical solution [14].
Where u E is the exact solution u and u N is the approximation solution U N . Now we can study our scheme from this problem.

Single soliton
Consider the coupled-BBM system (1) and (2) with the following initial and boundary conditions: .
Where j=1,2,g j ,x j and c j are arbitrary constants. In our computational work. Now, we choose  Figure 2, the interactions of these solitary waves are plotted at different time levels (Figure 2).

Interaction of three solitary waves
The interaction of three solitary waves having different amplitudes and traveling in the same direction is illustrated. We consider Coupled-BBM system with initial conditions given by the linear sum of three well separated solitary waves of various amplitudes