On numerical soliton and convergence analysis of Benjamin-Bona-Mahony-Burger equation via octic B-spline collocation

Abstract In this work, we employ an octic B-spline function to construct a collocation technique for obtaining solutions to soliton on Benjamin-Bona-Mahony-Burgers (BBMB) equation. The BBMB equation is fully-discretized in two forums such as: spatial and time discretization using octic B-spline function of the unknown variable and the Crank-Nicolson procedure, respectively. The robustness of the proposed method is determined by examining four test problems. On Neumann (Fourier) method is employed to obtain unconditional stability. A convergence analysis for the current scheme is also performed, resulting in O(h9+(Δt)2). The accuracy and efficiency of the proposed method are justified with error norms, invariants and the current result is also compared with existing results and found to be better as well as a better agreement to analytical solution. The proposed scheme is found to be more computationally efficient with (2N+16) operations for whole process.

Here, the novelty of our paper is the application of a higher-order banded sparse matrix as well as higher-order derivative to improve the numerical results of BBMB equation.Nonlinear terms are handled by quasi-linearization.A convergence analysis is established with the help of three difference operators and Taylor series.An unconditional stability analysis is based on Von Neumann approach.Numerical examples are soughted and found to be better than existing results and agree with the analytical solution.The proposed scheme may be employed for other nonlinear PDEs in the physical sciences for future research.
The system of equations with a coefficient matrix is solved using the octa-diagonal algorithm, which involves (2N þ 16) simple arithmetic operations.Hence, the method can be used with simple and elementary operations rather than complex calculations.The method is inexpensive, simple to implement, dependable, and computationally efficient.It can be used as an alternative for a wide range of nonlinear problems for computational complexity.
This work is organized as follows: The introductory part is given in Section 1. Section 2 models an octic B-spline scheme and BBMB equation is implemented in Section 3. The Von Neumann method is employed in Section 4 to obtain stability in unconditional mode.Section 5 describes the convergence of this method.In section 6, four test problems are taken to justify the efficiency, robustness and computational complexity of the proposed method.Section 7 ends with some conclusions.

Octic B-Spline collocation (OBSC)
Let us consider uniform partition such that a x 0 < x 1 < ::: The zero th degree B-spline is expressed as; The m th basis function of g th degree by (Boor, 1968) Employing equation (2.1) and substituting g ¼ 1 in equation (2.2), B-spline of degree one is found in equation (2.3) Similarly, applying septic B-spline and taking g ¼ 8 in equation (2.2), our desired octic B-spline is as follows (Table 1): (2.5) where The approximate solution wðx, tÞ using octic B-spline n m ðxÞ is defined as follows; where, a m ðtÞ is time parameter. (2.6) (2.7)

Application of method
In the governing equation (1.1), the Octic B-spline function of the unknown variable and the Crank-Nicolson technique (NLT) are used for spatial and temporal discretization sense.
The non-linear term ww x can be transformed to linear form by using quasi-linearization where Associating equation (2.7) in equation (3.4) and arranging the coefficient where m ¼ 0ð1ÞN The boundary conditions and higher-order derivatives of the octic B-spline introduce seven more terms to ensure a unique solution.The system of equations forms a matrix In this case, both D and G are ðN þ 8Þ Â ðN þ 8Þ banded matrices.MATLAB is used to solve the system of equations (3.6) using the octa-diagonal algorithm Initial state Applying the initial conditions in equation (3.6) To determine the initial state a 0 À4 , a 0 À3 , a 0 À2 :::::::a 0 Nþ2 , a 0 Nþ3 È É , the approximate solution wðx, tÞ satisfy the boundary conditions as follows: Initial matrix is of the form AC 0 ¼ c Where C 0 ¼ a 0 À4 , a 0 À3 , a 0 À2 :::::::: To solve the system, we apply the octa diagonal algorithm to obtain the initial vector General solution at x ¼ x m can be written as:

Stability analysis
To investigate the stability of equation (3.4), employ Von-Neumann procedure where, i ¼ ffiffiffiffiffiffi ffi À1 p , an imaginary unit, also h and h is the mode number and the element size respectively.Associating equation (4.1) into equation (3.5) and further simplification Applying the Euler's formula e 6ihh ¼ cosðhhÞ6isinðhhÞ in (4.2) where Applying the stability condition The unconditionally stable condition is followed from equation (4.4).

Conclusions
Higher-order B-spline collocation (BSC) is coupled with the Crank-Nicolson scheme to obtain numerical solition of BBMB equation.The higher-order sparse banded matrix and derivatives of higher degree act as catalyst for better improvement approximate soliton of BBMB in the present situation.The robustness of the current method is justified by employing four tests for numerical simulation of L 2 and L 1 error norms and conservative constants like I 1 , I 2 and I 3 invariants.The obtained results are also compared and found to be better to existing and closer to exact result.The unconditional stability is established by Fourier mode and convergence order of Oðh 9 þ DtÞ 2 accuracy in space and time is well derived for present scheme with the application of three difference operators and Taylor series method.The proposed algorithm may be utilized for other nonlinear PDEs in physical sciences.

Figure 3 .
Figure 3. Individual study of exact and approximate solution Example 1.

Figure 2 .
Figure 2. Comparison of numerical solution with analytic solution Example 1.
Figure 1(a) and (b) surf approximate and analytical solutions for time level t ¼ 4, N ¼ 100, Dt ¼ 0.01, respectively.Figure 2 depicts a comparison of the approximate and analytical solutions at various time points for Dt ¼ 0.01, N ¼ 100.For clarity, we compare the approximate and exact solutions for Dt ¼ 0.1 and N ¼ 100 at time t ¼ 2, 4, 6, and 8 in Figures 3(a), 3(b), 3(c) and 3(d), respectively.Figure 4 represents the absolute error at different times for Dt ¼ 0.01 and N ¼ 900.Example 2. Employ k ¼ 1 and b ¼ 0 in the governing equation (1.1) in the space domain [À10] and time domain [1, 5] with w x, 0 ð Þ ¼ e Àx 2

Figure 4 .
Figure 4. Numerical simulation of absolute error of Example 1.

Figure 6 .
Figure 6.3-D solution profile of individual numerical solution of Example 2.

Figure 7 .
Figure 7. 3D surf plot for numerical and analytic solution of Example 3.

Figure 8 .
Figure 8. Comparative study of exact and numerical results Example 3.

Figure 9 .
Figure 9. Brief analysis of approximate and exact soliton of Example 4.

Table 1 .
Basis functions of octic B-spline and it's derivatives.

Table 3 .
Computation of I 1 , I 2 and I 3 invariants of Example 1 for Dt ¼ 0.01.

Table 4 .
Comparison of I 1 , I 2 and I 3 invariants for Example 1 at N ¼ 400 and Dt ¼ 0:01:

Table 5 .
Computation for L 2 , L 1 error norms of Example-1 for N ¼500 and 1000 and Dt ¼ 0.05.

Table 6 .
Numerical computation of soliton of Example 2 with

Table 9 .
Error norms for Example 3 with different spatial and time domain.

Table 8 .
Computational approach of Example 2.

Table 12 .
Computational error norms of present method with others of Example 3 at Dt ¼ 0:01 in [0, p].