Error Analysis of the Numerical Solution of the

abstract: In this paper, the B-spline collocation scheme is implemented to find numerical solution of the nonlinear Benjamin-Bona-Mahony-Burgers equation. The method is based on collocation of quintic B-spline. 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 the numerical results have been compared with the analytical solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.


Introduction
In [1], a generalized Benjamin-Bona-Mahony-Burgers equation has been consider as follows equation where α is a positive constant, β ∈ R and g(u) is a C 2 -smooth nonlinear function.
In recent years, many different methods have been used to estimate the solution of the BBMB equation, for example, we note, Galerkin methods [3], The jacobi elliptic function solutions [4], Approximate wave solutions [5] and see [6,7].Also a class of Benjamin-Bona-Mahony-initial value problems are studied in [8].The layout of this article is as follows.In Section 2, we present a finite difference approximation to discretize the (1.2) in time variable and we applied quintic Bspline collocation method to solve the problem.In Section 3, the stability analysis of the method is given.In Section 4 we derive convergence of the B-spline collocation method.In Section 5, some examples have been conducted in order to validate the theoretical results.A summary is given at the end of the paper in Section 6.

Solution of BBMB equations via Quintic B-spline
The region a ≤ x ≤ b partitioned into a mesh of uniform length h = b−a N , by the knots x i = a + ih where i = 0, 1, 2, ...,N and a = x 0 < x 1 < . . .< x N −1 < x N = b.We use the following finite difference approximation to discretize the time variable where ∆t is the time setp, u n (x) := u(x, n∆t) and δtu n := u n+1 − u n .This finite difference scheme is used in [9].Rearranging the term and simplifying we get where To linearized the non-linear term (uu x ) n+1 in (2.2) we can use the Taylor expansions.We can get thus we have 2) can be rewritten as where We define the quintic B-spline basis functions at knots, by the following relationships [10,11] (2.7) To continue we define the approximation for u(x,t) as where B i (x) are the quintic B-spline basis functions, and c i (t) are time-dependent quantities.We can determine c i (t) from boundary conditions and collocation form of the differential equations.We calculate U , U ′ and U ′′ at node points as Substituting the approximate solution U for u and using (2.5) and (2.9)-(2.11)at the knots we get where The system (2.12) consists of N + 1 linear equations in N + 5 unknowns {c n+1 −2 , . . ., c n+1 N +2 }.To obtain a unique solution for {c n+1 −2 , . . ., c n+1 N +2 }, we must use the boundary conditions.From (1.4)-(1.5)and (2.8), we can write Then we write the last system in the matrix form where where, and Error Analysis of the Numerical Solution of the BBMB Equation 181 (2.16) The above system of equations given in (2.13) has been solved using the computer algebra system Mathematica-9.
To start any calculate, we must know U 1 (x).We assume that By using (2.2) and (2.9)-(2.11)we can write (2.17) The nonlinear system (2.17) consists of N + 1 equations in N + 5 unknowns To obtain a unique solution for C 1 , similar to the above discussion, we use the boundary conditions.From (1.4)-(1.5),we can write Then we obtain the nonlinear system consists of N + 5 equations in N + 5 unknowns.This system is solved by the computer algebra system Mathematica-9.

Stability analysis
In this section, we discuss the stability of the quintic B-spline approximation (2.1) using the Von Numann method [12,13].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 equation (2.1).We obtain the equation: With substituting c n i = ξ n exp(λkhi) into linearized form (3.2) and simplifying , we obtain where From (3.3), we can write (3.4) We note that X 1 ≤ X 2 , so |ξ| ≤ 1.This implies | ξ |≤ 1. Therefore the linearized numerical scheme for BBMB equation is unconditionally stable.

Convergence analysis
In this section we study the convergence of the quintic B-spline collocation method has been given in Section 2.
where ω 5 (h) denotes the modulus of continuity of f (5) and the coefficients λ j are independent of f and h.
Proof: For the proof see [14].✷ Remark 4.2.By using Theorem 4.1 and definition of the modulus of continuity, we can say that if |f (5) (x)| ≤ L, we can write (4.1) as we have the following inequality: Proof: From the real analysis we have and if x i−1 ≤ x ≤ x i , then, we can write Proof: At the (n + 1)th time step, we assume that S * be the unique spline interpolate to the exact solution u of (1.2)-(1.6)given by We note that matrix A is strictly diagonally dominant matrix.Let η i , (1 ≤ i ≤ N + 1) be the summation of the ith row of the matrix A. From the theory of matrices we know that where a −1 ki are the elements of A −1 .As a result we can write where G is is constant.We substituting S * (x) in (2.5), we get Subtracting (2.13), (4.8) and taking the infinity norm, we can write By using (4.2), we get the result as From (4.10), we get where Thus by taking norm and using Lemma 4.3, (4.7), (4.9), (4.11) we obtain Error Analysis of the Numerical Solution of the BBMB Equation 185and therefore with helping (4.12) and (4.13), we get where γ = λ 0 Lh 2 + 186M 2 .
In the next step, suppose that ε i = u(x, t i ) − U (t i ) be the local truncation error for (2.1) at the ith level of time.By using the truncation error, we get We assume that E n+1 be the global error in time discretizing process and ̺ = max{̺ 1 , ..., ̺ n }.We can write the following global error estimate at n + 1 level with the help of (4.15) we can write where ρ = ̺T .Which completes the proof.✷

Numerical examples
In this section to illustrate the performance of the B-spline collocation method in solving BBMB equation and the efficiency of the method, the following examples are considered.We defined L 2 and L ∞ as Note that we have computed the numerical results by Mathematica-9 programming.
For comparison, we consider the our results with methods [15,16,17].We assume that ∆t = 0.1 and N = 1000.Table 1 exhibits the compared results.Also Table 2 and Table 3 give a comparison between numerical and analytical solutions for different partitions.From Table 2, we see that the L 2 and error decrease as ∆t decreases or N increases.Also numerical results in Table 3, are in accordance with the order of convergence of our presented scheme.From Figure 1 we can see that numerical solutions show the same behavior as analytical solution.Also Figure 2 shows that the solution obtained by our method is close to the analytical solution.Figure 3 shows absolute errors.Example 2. In this example we consider α = 0 and β = 1 in the interval [−10, 30], with the initial condition u(x, 0) =sech 2 (x/4).The analytical solution is u(x, t) =sech 2 (x/4 − t/3) [18].Table 4 and Table 5 show L 2 error in different partitions.We can say that the numerical solution graph shows the same behavior as the analytical solution in the Figure 4.In addition Figure 5 shows absolute errors in different times.Also numerical results in Table 6 are in accordance with the order of convergence of our presented scheme.Error Analysis of the Numerical Solution of the BBMB Equation 189Example 3. We consider here a numerical solution of the BBMB equation with α = 1 and β = 1 in the interval [−10, 10], with the initial condition u(x, 0) = exp(−x 2 ).The behavior of the approximated solution with ∆t = 0.01 and N = 300 is presented in Figure 4.The graph shows the same behavior as in [19].Also the numerical results are tabulated in Table 7 for ∆t = 0.01 and N = 300.where G(x, t) = exp(−t)[cos(x) − sin(x) + 12 exp(−t) sin(2x)].The exact solution for this problem is given as u(x, t) = exp(−t) sin(x).The boundary and initial conditions can be found from exact solution.In Table 8, present method has been compared with method in [19].In this table we consider T = 10, N = N ′ + 1 and ∆t = T /M .

Conclusion
In this work, the Quintic B-spline collocation method is used to solve the Benjamin-Bona-Mahony-Burgers(BBMB) equation.The stability analysis and convergence analysis of the method are shown.In addition, approximate numerical results given in the previous section.Also, obtained results showed that this approach can solve the problem effectively.

5 Ab E t 4 Ab E t 3 Ab E t 2 Ab E t 1 Figure 5 :
Figure 5: Absolute errors for Example 2 with ∆t = 0.01 and N = 300.

Table 2 :
L 2 errors for Example 1 at different times.

Table 3 :
L∞ errors for Example 1 at different partitions.

Table 4 :
L 2 errors for Example 2 at different times.

Table 5 :
L 2 errors for Example 2 at different times.

Table 6 :
L∞ errors for Example 2 at different partitions.

Table 8 :
Numerical results for Example 4 with different partitions in t = 10.