Numerical study of the Benjamin-Bona-Mahony-Burgers equation

In this paper, the quadratic B-spline collocation method is implemented to find numerical solution of the Benjamin-Bona-Mahony-Burgers (BBMB) equation. Applying the Von-Neumann stability analysis technique, we show that the method is unconditionally stable. Also the convergence of the method is proved. The method is applied on some test examples, and numerical results have been compared with the exact solution. The numerical solutions show the efficiency of the method computationally.


Introduction
In this paper we consider the solution of the Benjamin-Bona-Mahony-Burgers (BBMB) equation with the initial condition and boundary conditions u(a, t) = u(b, t) = 0, where α and β are positive constants.For α = 0, Eq. on this topic, see [1,2].In recent years, many different methods have been used to estimate the solution of the Benjamin-Bona-Mahony-Burgers equation and the BBM equation, for example, see [3,4,5,7,11].
The paper is organized as follows.In Section 2, quadratic B-spline collocation method is explained.In Section 3, is devoted to stability analysis and convergence analysis of the method.In Section 4, examples are presented.A summary is given at the end of the paper in Section 5.

Quadratic B-spline collocation method
Our numerical treatment for BBMB equation using the collocation method with quadratic B-spline is to find an approximate solution U (x, t) to the exact solution u(x, t) in the form where c i (t) are time-dependent quantities to be determined from the boundary conditions and collocation form of the differential equations.Also B i (x) are the quadratic B-spline basis functions at knots, given by [6,12] otherwise. (2. 2) The solution domain a ≤ z ≤ b partitioned into a mesh of uniform length h = b−a N , by the knots z j where j = 0, 1, 2, . . ., N such that a = z 0 < z 1 ...z N −1 < z N = b and z j = z 0 + jh.The values of B i (z) and its first and second derivatives at the mid knots points are given in Table 1.Also numerical solutions are given at mid points.We note that the mid points are By using approximate function (2.1) and Table 1, we have ) where c n i := c i (t n ).In this step by using the finite difference method, we can write (2.6) The nonlinear term in (2.6) can be approximated by using the following formula [9]: Substituting the approximate solution U for u and putting the values of the mid values U , its derivatives using (2.3),(2.4)and (2.5) at the knots in (2.6) yield the following difference equation with the variables c i , i = 0, 1, . . ., N, where The system (2.8) consists of N linear equations in N + 2 unknowns {c −1 , c 0 , ..., c N −1 , c N }.To obtain a unique solution for c = {c −1 , c 0 , ..., c N −1 , c N }, we must use the boundary conditions.From the boundary conditions and Table 2, we can write Associating (2.10) and (2.11) with (2.8), we obtain a (N + 2) × (N + 2) system of equations in the following form where (2.15)

Stability and convergence analysis
We present the stability of the quadratic B-spline approximation (2.8) using the Von Numann method [8,10].According to the Von-Neumann method, we have where k is the mode number and h is the element size.To apply this method, we have linearized the nonlinear term uu x by consider u as a constant ̟ in term (2.6).
We obtain the equation where Dividing both sides of (3.2) by exp (iλkh), we can obtain Numerical study of the BBMB equation 131 (3.3) can be rewritten in a simple form as where X and X 1 can be rewritten in the form: We note that X ≤ X 1 , so 1 +Y 2 ≤ 1.Therefore, the linearized numerical scheme for the BBMB equation is unconditionally stable.Now we discuss the convergence of the collocation method.
b} be the equally spaced partition of [a, b] with step size h.If S(x) be the unique spline function interpolate f (x) at knots x 0 , x 1 , • • • , x N ∈ ∆ then there exist a constant λ j such that ∞ ≤ λ j Lh 4−j , j = 0, 1, 2. (3.5) Proof: For the proof see [14].
Proof: At any nodal point x = x i , we can write which completes the proof.✷ Theorem 3.3.Suppose that u(x, t) be the exact solution of (1.1) and assume that | ∂ 4 u(x,t) ∂x 4 | ≤ L and U (x, t) be the approximate solution of BBMB (1.1) given by our approach, then u(x, t) where ̺ is a constant and independent of h.
Proof: At the (n + 1)th time step, we assume that S be the unique spline interpolate to the exact solution u of (1.1)-(1.3)given by To continue, we note that matrix A is strictly diagonally dominant matrix, and from [13] we can find Ḿ independent of h, such that ||A −1 || ∞ ≤ Ḿ .Also from Theorem 3.1, we can write we substituting S(x) in (2.6), we have the following result Subtracting (2.12), (3.10) and taking the infinity norm, we can write From (2.9) and using Theorem 3.1, we get where Applying (3.13) and Lemma 3.2, we get the result as where M = 7 M Ḿ /6.From (3.9) and (3.14),we have In the next step, suppose that ε i = u(t i ) − U (t i ) be the local truncation error for (2.6) at the ith level of time.By using the truncation error , we get Numerical study of the BBMB equation 133 where v i is some finite constant.We can write the following global error estimate at n + 1 level thus with the help of (3.16), we can write where ρ = υT and v = max{v 1 , ..., v n }.Which completes the proof.✷

Numerical examples
In order to illustrate the performance of the quadratic B-spline collocation method in solving BBMB equation and justify the accuracy and efficiency of the present method, we consider the following examples.To show the efficiency of the present method for our problem in comparison with the solution, we report L ∞ and L 2 using formulae where U is numerical solution and u denotes exact solution.Note that we have computed the numerical results by Mathematica-9 programming.4v .The initial condition is taken from the solution.Also the solution satisfy three conservation laws: 3 and Table 6 give C 1 , C 2 , C 3 , L ∞ and L 2 found by our method in different times for c = 0.1, 0.03 and Table 4 and Table 7 give numerical results from method in [4,5].Figure. 1 and Figure. 2 show that solution obtained by our method is closed to the solutions.In addition, in Table 5 and Table 8 we see that L 2 decreases with decreasing in ∆t or increasing in N .Also from Figure 3 we can see that numerical solutions show the same behavior as solutions.

Conclusion
The quadratic B-spline collocation method is used to solve the Benjamin-Bona-Mahony-Burgers(BBMB) equation with initial and boundary conditions.We study the stability analysis and the convergence analysis of the method.The numerical results given in the previous section demonstrate the good accuracy and stability of the proposed scheme in this research.

( 1 . 1 )
is called the Benjamin-Bona-Mahony (BBM) equation.The BBMB equation has been proposed as a model for propagation of long waves.This equation incorporates dispersive and dissipative effects.The dissipative term can be found in −αu xx .For more details 2000 Mathematics Subject Classification: 65M10, 78A48 128 M. Zarebnia and R. Parvaz

Table 2 :
B i and B ′i at node points.

Table 5 :
Comparison of L 2 for Example 4.

Table 9 and
[7]le10give numerical results found by our method in different times.Also Figure.4showsapproximatesolutiongraphs.In addition, we can see that the graph shows the same behavior as in[7].

Table 8 :
Comparison of L 2 for Example 4.1 at different N with c = 0.1.

Table 9 :
Numerical results for Example 2 with ∆t = 0.1 and N = 200.Figure 4: Approximate solution graphs of Example 4.2 in different times with ∆t = 0.1 and N = 200.