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 $O(h^9+(\Delta {t)}^2).$ 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.

Keywords:

• BBM-Burger equation
• collocation
• convergence
• finite difference
• stability

1. Introduction

BBMB is an important tool in the physical sciences to study heat transfer (Chen & Gurtin, 1968), seepage theory of homogeneous fluids (Barenblatt, Zheltov, & Kochina, 1960), design of an undular bore (Peregrine, 1966), and non-steady fluid flow (Ting, 1963).

Generalized Benjamin-Bona-Mohany-Burger’s equation is (1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) with, initial and boundary conditions (1.2) $$w(x,0)=f\left(x\right),\mathrm{ }x\in [a,b]$$(1.2) (1.3) $$w\left(a,t\right)=g_0\left(t\right),\mathrm{ }w(b,t)=g_1\left(t\right)$$(1.3) (1.4) $$w_x(a,t)=w_x(b,t)=0.$$(1.4)

There are various analytical and numerical techniques available in the literature to obtain the soliton of BBMB equation, such as: modified BBM equation by first integral method (Abbasbandy & Shirzadi, 2010), odd degree spline interpolation convergence (Boor, 1968), homotopy technique for soliton of explicit BBMB (Fakhari & Domairry, 2007), decomposition method for soliton solution (Al-Khaled, Momani, & Alawneh, 2005), improvised B-spline collocation method of fourth order (Kukreja, 2022), collocation method (Zarebnia & Parvaz, 2016), nonlinear BBMB equations by He’s methods (Tari & Ganji, 2007), finite difference method (FDM) through Crank-Nicolson Type (CNT) (Omrani & Ayadi, 2008), exp function technique for nonlinear BBMB explicitly (Ganji, Ganji, & Bararnia, 2009), Painlevé analysis and lie Symmetries (Kumar, Gupta, & Jiwari, 2013), quartic B-spline collocation method (Arora, Mittal, & Singh, 2014), B-spline collocation (Fyfe, 1969; Jena & Gebremedhin, 2023; Senapati & Jena, 2023; Shallal, Ali, Raslan, & Taqi, 2019), block method (Jena & Gebremedhin, 2020; Gebremedhin & Jena, 2020; Jena, Mohanty, & Misra, 2018), association of radial basis functions (RBF) and meshless method (Dehghan, Abbaszadeh, & Mohebbi, 2014), Galerkin method (Dehghan & Abbaszadeh, 2015), B-spline quasi-interpolation (Kumar & Baskar, 2016), hybridization of Lucas with Fibonacci polynomials (Oruç, 2017), finite element method (FEM) coupled the regularized long wave equation (RLW) (Gardner, Gardner, & Dag, 1995), time-space pseudo-spectral method (Mittal, Balyan, & Tiger, 2018), solitary waves and roguwaves with interaction phenomena (Hossen, Roshid, & Ali, 2018).

Crank-Nicolson finite difference method based on a midpoint upwind scheme (Kadalbajoo & Awasthi, 2008), finite difference technique (Kutluay & Esen, 2006), finite difference type operators on cubic spline interpolation (Lucas, 1974), improvised collocation algorithm (Shallu & Kukreja, 2021), Hermite splines with quintic collocation (Arora, Jain, & Kukreja, 2020), error analysis for approximate solution of BBMB equation (Zarebna & Parvaz, 2020), numerical treatment and analysis of BBMB equation (Zarebnia & Parvaz, 2016), Hermite collocation of sixth order (Kumari & Kukreja, 2022), fourth-order accurate difference scheme (Bayarassou, 2019). Legendre spectral element method (Dehghan, Shafieeabyaneh, & Abbaszadeh, 2021), bilinear formalism and ansatze’s function towards solution of BBMB (Hossen, Roshid, & Ali, 2019) and lie algebra (Casas, 1996). The other methods related with the schemes are B-spline collocation for Kuramoto Sivashinsky equation (Lakestani & Dehghan, 2012), MRLW equation in B-Spline environment (Jena, Senapati, & Gebremedhin, 2020a), study of solitions in BFRK scheme (Jena, Senapati, & Gebremedhin, 2020b), Burgers soliton by decatic B-spline Function for collocation (Jena & Gebremedhin, 2021a), computational technique for heat and advection diffusion equations (Jena & Gebremedhin, 2021b), fifth-order boundary value problems by septic collocation (Senapati & Jena, 2022), soliton of Kuramoto-Sivashinsky equation(KSE) (Jena & Gebremedhin, 2021c), boundary value problem of fifth order (Jena & Gebremedhin, 2022), integral transform (Jena, Naya, & Acharya, 2017; Mohanty & Jena, 2018; Mohanty, Jena, & Misra, 2021a, 2021b).

Recently, various fractional differential equations are introduced to describe different engineering real situations. So many authors are working on fractional differential equations based on the reproducing kernel (Arqub, 2019), singular Fredholm time-fractional partial integro differential equations (Arqub, 2019), Hirota bilinear method (Geng, Mou & Dai, 2023), time-fractional Ablowitz-Ladik model using the fractional exponential function (Fang, Mou, Zhang, & Wang, 2021), PT-symmetric potential method (Bo, Wang, Fang, Wang, & Dai, 2023; Wen et al., 2021), Projecting transformation and the Darboux method (Wu & Jiang, 2022), MPS-PINN method (Wen, Wu, Liu, & Dai, 2022), an energy-conservation deep-learning (ECDL) method (Fang et al., 2022) are constructed to study a coupled nonlinear Schrödinger equation (CNLSE).

Some authors are suggested Lie symmetric methods to find the solution of partial differential equations like two-dimensional Burgers–Huxley equation (Hussain, Bano, Khan, Baleanu, & Sooppy Nisar, 2020), (2 + 1) and (3 + 1) dimensional sine-Gordon equation (Wang, Yang, Gu, Guan, & Kara, 2020; Wang, 2021b), (3 + 1)-dimensional Schrödinger equation (Wang, 2021a), Kdv and mKdv equation (Wang & Wazwaz, 2022a), Kdv-Burgers-Kuramoto equation in Wang (2021c), Study of modified Gardner-type equation and its time fractional form (Wang & Wazwaz, 2022b), Cauchy problem (Casas, 1996), Daftardar-Jafari method (Al-Jawary, Radhi, & Ravnik, 2018), differential transformation method (Mohanty & Jena, 2018).

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 ($2N+16$) 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.

2. Octic B-Spline collocation (OBSC)

Let us consider uniform partition such that $a\le x_0<x_1<\dots .<x_{\mathrm{N}}\le b$ for the domain [a, b] with step size $h=x_{m+1}-x_m,\mathrm{ }m\mathrm{ }=\mathrm{ }0,1,2\cdots \cdots N-1.$

The zero^th degree B-spline is expressed as; (2.1) $${\xi }_{m,0}=\{\begin{array}{l}1, x\in [x_m,x_{m+1}]\\ 0 \mathit{otherwise} \end{array}\mathrm{ }$$(2.1)

The $m^{\mathrm{th}}$ basis function of $g^{\mathrm{th}}$ degree by (Boor, 1968) (2.2) $${\xi }_{m,g}\left(x\right)=\frac{x-x_m}{x_{m+g}-x_m}{\xi }_{m,g-1}\left(x\right)+\frac{x_{m+g+1}-x}{x_{m+g+1}-x_{m+1}}{\xi }_{m+1,g-1}\left(x\right),\mathrm{ }g\ge 1\mathrm{ }$$(2.2)

Employing equation (2.1)(2.1) $${\xi }_{m,0}=\{\begin{array}{l}1, x\in [x_m,x_{m+1}]\\ 0 \mathit{otherwise} \end{array}\mathrm{ }$$(2.1) and substituting $g=1$ in equation (2.2)(2.2) $${\xi }_{m,g}\left(x\right)=\frac{x-x_m}{x_{m+g}-x_m}{\xi }_{m,g-1}\left(x\right)+\frac{x_{m+g+1}-x}{x_{m+g+1}-x_{m+1}}{\xi }_{m+1,g-1}\left(x\right),\mathrm{ }g\ge 1\mathrm{ }$$(2.2) , B-spline of degree one is found in equation (2.3)(2.3) $${\xi }_{m,1}=\frac{1}{\mathrm{h}}\{\begin{array}{l}x-x_m, x\in [x_m,x_{m+1}] \\ x_{m+2}-x, x\in [x_{m+1},x_{m+2}] \\ 0 \mathit{otherwise} \end{array}\mathrm{ }$$(2.3) (2.3) $${\xi }_{m,1}=\frac{1}{\mathrm{h}}\{\begin{array}{l}x-x_m, x\in [x_m,x_{m+1}] \\ x_{m+2}-x, x\in [x_{m+1},x_{m+2}] \\ 0 \mathit{otherwise} \end{array}\mathrm{ }$$(2.3)

Similarly, applying septic B-spline and taking $g=8$ in equation (2.2)(2.2) $${\xi }_{m,g}\left(x\right)=\frac{x-x_m}{x_{m+g}-x_m}{\xi }_{m,g-1}\left(x\right)+\frac{x_{m+g+1}-x}{x_{m+g+1}-x_{m+1}}{\xi }_{m+1,g-1}\left(x\right),\mathrm{ }g\ge 1\mathrm{ }$$(2.2) , our desired octic B-spline is as follows (Table 1): (2.4) $${\xi }_{m,8}\left(x\right)=\frac{x-x_m}{8h}{\xi }_{g,7}\left(x\right)+\frac{x_{m+9}-x}{8h}{\xi }_{m+1,7}\left(x\right)\mathrm{ }$$(2.4) (2.5) $${\mathrm{\xi }}_{\mathrm{m},8}=\frac{1}{h^8}\{\begin{array}{l}{a}_1^8 x\in I_{m-4} \\ a_1^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_2^8 x\in I_{m-3}\\ a_1^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_2^8 +\left(\genfrac{}{}{0pt}{}{9}{2}\right)a_3^8 x\in I_{m-2} \\ a_1^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_2^8 +\left(\genfrac{}{}{0pt}{}{9}{2}\right)a_3^8- \left(\genfrac{}{}{0pt}{}{9}{3}\right)a_4^8 x\in I_{m-1}\\ a_1^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_2^8 +\left(\genfrac{}{}{0pt}{}{9}{2}\right)a_3^8- \left(\genfrac{}{}{0pt}{}{9}{3}\right)a_4^8+\left(\genfrac{}{}{0pt}{}{9}{4}\right)a_5^8 x\in I_m\\ a_6^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_7^8 +\left(\genfrac{}{}{0pt}{}{9}{2}\right)a_8^8- \left(\genfrac{}{}{0pt}{}{9}{3}\right)a_9^8 x\in I_{m+1}\\ a_6^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_7^8 +\left(\genfrac{}{}{0pt}{}{9}{2}\right)a_8^8 x\in I_{m+2}\\ a_6^8-\left(\genfrac{}{}{0pt}{}{9}{1}\right)a_7^8 x\in I_{m+3}\\ a_6^8 x\in I_{m+4}\\ 0 \mathit{otherwise}\end{array}\mathrm{ }$$(2.5) where $$x-x_{m-4}=\mathrm{ }{a}_{1}x-x_{m-3}=a_{2}x-x_{m-2}=a_3$$ $$x-x_{m-1}=a_{4}x-{x_m}_{\mathrm{ }}=a_5{x_{m+5}-x}_{\mathrm{ }}=a_6$$ $$x_{m+4}-x_{\mathrm{ }}=a_7{x}_{m+3}-x_{\mathrm{ }}=a_8{x}_{m+2}-x_{\mathrm{ }}=a_9$$

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

$x$	$x_{m-4}$	$x_{m-3}$	$x_{m-2}$	$x_{m-1}$	$x_m$	$x_{m+1}$	$x_{m+2}$	$x_{m+3}$	$x_{m+4}$	$x_{m+5}$
${\xi }_m(x)$	0	1	247	4293	15619	15619	4293	247	$1$	0
$h{{\xi }_m}^{\prime }(x)$	0	$8$	$952$	$8568$	9800	$-9800$	$-8568$	$-952$	$-8$	0
${{h^2\xi }_m}^{″}(x)$	0	$56$	$3080$	$10584$	$-13720$	$-13720$	$10584$	$3080$	$56$	0
${h^3\xi ″\prime }_m(x)$	0	$336$	$7728$	$-3024$	$-31920$	$31920$	$3024$	$-7728$	$-336$	0
${{h^4\xi }_m}^{(4)}(x)$	0	$1680$	$11760$	$-4536$0	$31920$	$31920$	$-45360$	$11760$	$1680$	0
${{h^5\xi }_m}^{(5)}(x)$	0	$6720$	$-6720$	$-60480$	$168000$	$-168000$	$60480$	$6720$	$-6720$	0
${{h^6\xi }_m}^{(6)}(x)$	0	$20160$	$100800$	$181440$	$-100800$	$-100800$	$181440$	$-100800$	$20160$	0
${\mathrm{h}}^7{{\xi }_m}^{(7)}(x)$	0	$40320$	$-282240$	$846720$	$-1411200$	$1411200$	$-846720$	$282240$	$-40320$	0

The approximate solution $w(x,t)$ using octic B-spline ${\xi }_m(x)$ is defined as follows; $$w_m(x,t)=\sum _{m-4}^{N+3}{\alpha }_m\left(t\right){\xi }_m\left(x\right);$$ where, ${\alpha }_m(t)$ is time parameter. (2.6) (2.7) $$\left.\begin{array}{l}w\left(x_m\right)={\mathrm{\alpha }}_{\mathrm{m}-4}+247{\mathrm{\alpha }}_{\mathrm{m}-3}{{+4293\mathrm{\alpha }}_{\mathrm{m}-2}+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}-1}{+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}}{{+4293\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}+1}+247\mathrm{\alpha }}_{\mathrm{m}+2}{+\mathrm{\alpha }}_{\mathrm{m}+3} \\ {\mathrm{ }w}^{\left(1\right)}\left(x_m\right)=\frac{8}{\mathrm{h}}\left(-{\alpha }_{m-4}-119{\alpha }_{m-3}-1071{\alpha }_{m-2}-1225{\alpha }_{m-1}+1225{\alpha }_m+1071{\alpha }_{m+1}+119{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(2\right)}\left(x_m\right)=\frac{56}{{\mathrm{h}}^2}\left({\alpha }_{m-4}+\mathrm{ }55{\alpha }_{m-3}+\mathrm{ }189\mathrm{ }{\alpha }_{m-2}-245{\alpha }_{m-1}-245{\alpha }_m+189{\alpha }_{m+1}+55{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(3\right)}\left(x_m\right)=\frac{336}{{\mathrm{h}}^3} \left(-{\alpha }_{m-4}-23{\alpha }_{m-3}+\mathrm{ }9{\alpha }_{m-2}+\mathrm{ }95{\alpha }_{m-1}-95{\alpha }_m-9{\alpha }_{m+1}+23{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(4\right)}\left(x_m\right)=\frac{1680}{{\mathrm{h}}^4}\left({\alpha }_{m-4}+7{\alpha }_{m-3}\mathrm{ }-27{\alpha }_{m-2}+19{\alpha }_{m-1}+\mathrm{ }19\mathrm{ }{\alpha }_m-27{\alpha }_{m+1}+7{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(5\right)}\left(x_m\right)=\frac{6720}{{\mathrm{h}}^5}\left(-{\alpha }_{m-4}+{\alpha }_{m-3}+\mathrm{ }9\mathrm{ }{\alpha }_{m-2}-25{\alpha }_{m-1}+25{\alpha }_m-9{\alpha }_{m+1}-{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(6\right)}\left(x_m\right)=\frac{20160}{{\mathrm{h}}^6}\left({\alpha }_{m-4}-5{\alpha }_{m-3}+\mathrm{ }9{\alpha }_{m-2}-5{\alpha }_{m-1}-5{\alpha }_m+\mathrm{ }9{\alpha }_{m+1}-5{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(7\right)}\left(x_m\right)=\frac{40320}{{\mathrm{h}}^7}\left(-{\alpha }_{m-4}+7{\alpha }_{m-3}-21{\alpha }_{m-2}+\mathrm{ }35{\alpha }_{m-1}-35{\alpha }_m+21{\alpha }_{m+1}-7{\alpha }_{m+2}+{\alpha }_{m+3}\right)\end{array}\right\}$$(2.7)

3. Application of method

In the governing equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(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. (3.1) $${(w-w_{xx})}_t-\beta w_{xx}+w{w}_x+\lambda w_x=0$$(3.1) (3.2) $$\left\{\frac{{(w-w_{xx})}^{n+1}-{(w-w_{xx})}^n}{k}\right\}-\beta \left\{\frac{{w_{xx}}^{n+1}+{w_{xx}}^n}{2}\right\}+\left\{\frac{(w{w_x)}^{n+1}+(w{w_x)}^n}{2}\right\}+\lambda \left\{\frac{{w_x}^{n+1}+{w_x}^n}{2}\right\}\mathrm{ }=0\mathrm{ }$$(3.2)

The non-linear term $w{w}_x$ can be transformed to linear form by using quasi-linearization (3.3) $${\left(w{w}_x\right)}^{n+1}=w^{n+1}{w}_x^n+w^n{w}_x^{n+1}-(w{w_x)}^n\mathrm{ }$$(3.3)

Employing equation (3.3)(3.3) $${\left(w{w}_x\right)}^{n+1}=w^{n+1}{w}_x^n+w^n{w}_x^{n+1}-(w{w_x)}^n\mathrm{ }$$(3.3) in equation (3.2)(3.2) $$\left\{\frac{{(w-w_{xx})}^{n+1}-{(w-w_{xx})}^n}{k}\right\}-\beta \left\{\frac{{w_{xx}}^{n+1}+{w_{xx}}^n}{2}\right\}+\left\{\frac{(w{w_x)}^{n+1}+(w{w_x)}^n}{2}\right\}+\lambda \left\{\frac{{w_x}^{n+1}+{w_x}^n}{2}\right\}\mathrm{ }=0\mathrm{ }$$(3.2) (3.4) $$A^{n+1}={\psi }^n\mathrm{ }$$(3.4) where $A^{n+1}=\left[w^{n+1}-w_{xx}^{n+1}\right]+\frac{k}{2}\left[-\beta w_{xx}^{n+1}+w^{n+1}{w}_x^n+w^n{w}_x^{n+1}+\lambda w_x^{n+1}\right]$ $${\psi }^n=\left[w^n-w_{xx}^n\right]-\frac{k}{2}\left[-\beta w_{xx}^n+\lambda w_x^n\right]$$

Associating equation (2.7)(2.7) $$\left.\begin{array}{l}w\left(x_m\right)={\mathrm{\alpha }}_{\mathrm{m}-4}+247{\mathrm{\alpha }}_{\mathrm{m}-3}{{+4293\mathrm{\alpha }}_{\mathrm{m}-2}+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}-1}{+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}}{{+4293\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}+1}+247\mathrm{\alpha }}_{\mathrm{m}+2}{+\mathrm{\alpha }}_{\mathrm{m}+3} \\ {\mathrm{ }w}^{\left(1\right)}\left(x_m\right)=\frac{8}{\mathrm{h}}\left(-{\alpha }_{m-4}-119{\alpha }_{m-3}-1071{\alpha }_{m-2}-1225{\alpha }_{m-1}+1225{\alpha }_m+1071{\alpha }_{m+1}+119{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(2\right)}\left(x_m\right)=\frac{56}{{\mathrm{h}}^2}\left({\alpha }_{m-4}+\mathrm{ }55{\alpha }_{m-3}+\mathrm{ }189\mathrm{ }{\alpha }_{m-2}-245{\alpha }_{m-1}-245{\alpha }_m+189{\alpha }_{m+1}+55{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(3\right)}\left(x_m\right)=\frac{336}{{\mathrm{h}}^3} \left(-{\alpha }_{m-4}-23{\alpha }_{m-3}+\mathrm{ }9{\alpha }_{m-2}+\mathrm{ }95{\alpha }_{m-1}-95{\alpha }_m-9{\alpha }_{m+1}+23{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(4\right)}\left(x_m\right)=\frac{1680}{{\mathrm{h}}^4}\left({\alpha }_{m-4}+7{\alpha }_{m-3}\mathrm{ }-27{\alpha }_{m-2}+19{\alpha }_{m-1}+\mathrm{ }19\mathrm{ }{\alpha }_m-27{\alpha }_{m+1}+7{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(5\right)}\left(x_m\right)=\frac{6720}{{\mathrm{h}}^5}\left(-{\alpha }_{m-4}+{\alpha }_{m-3}+\mathrm{ }9\mathrm{ }{\alpha }_{m-2}-25{\alpha }_{m-1}+25{\alpha }_m-9{\alpha }_{m+1}-{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(6\right)}\left(x_m\right)=\frac{20160}{{\mathrm{h}}^6}\left({\alpha }_{m-4}-5{\alpha }_{m-3}+\mathrm{ }9{\alpha }_{m-2}-5{\alpha }_{m-1}-5{\alpha }_m+\mathrm{ }9{\alpha }_{m+1}-5{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(7\right)}\left(x_m\right)=\frac{40320}{{\mathrm{h}}^7}\left(-{\alpha }_{m-4}+7{\alpha }_{m-3}-21{\alpha }_{m-2}+\mathrm{ }35{\alpha }_{m-1}-35{\alpha }_m+21{\alpha }_{m+1}-7{\alpha }_{m+2}+{\alpha }_{m+3}\right)\end{array}\right\}$$(2.7) in equation (3.4)(3.4) $$A^{n+1}={\psi }^n\mathrm{ }$$(3.4) and arranging the coefficient ${\alpha }_{m-l}^{n+1},l=1(1)4\mathrm{ }{\alpha }_{m+e}^{n+1},e=0(1)3$ separately, (3.5) $$A_1{\alpha }_{m-4}^{n+1}+A_2{\alpha }_{m-3}^{n+1}+A_3{\alpha }_{m-2}^{n+1}+A_4{\alpha }_{m-1}^{n+1}+A_5{\alpha }_m^{n+1}+A_6{\alpha }_{m+1}^{n+1}+A_7{\alpha }_{m+2}^{n+1}+A_8{\alpha }_{m+3}^{n+1}=B_1{\alpha }_{m-4}^n+B_2{\alpha }_{m-3}^n+B_3{\alpha }_{m-2}^n+B_4{\alpha }_{m-1}^n+B_5{\alpha }_m^n+B_6{\alpha }_{m+1}^n+B_7{\alpha }_{m+2}^n+B_8{\alpha }_{m+3}^n$$(3.5) where $m=0(1)N$ $$A_1=\frac{h^2-4h(\lambda -P+Q)k-28(2+\beta k)}{h^2}\mathrm{ }{B}_1=\frac{-56+h^2+k(28\beta +4\lambda h)}{h^2}$$ $$A_2=247-\frac{4(119\lambda -247P+119Q)k}{h}-\frac{1540(2+\beta k)}{h^2}\mathrm{ }{B}_2=247+\frac{476\lambda k}{h}+\frac{1540(-2+\beta k)}{h^2}$$ $$A_3=\frac{-9(-477h^2+4h(119\lambda -477P+119Q)k+588(2+\beta k)}{h^2}\mathrm{ }{B}_3=\frac{9(477h^2+476\lambda hk+588(-2+\beta k\left)\right)}{h^2}$$ $$A_4=15619-\frac{4(1225\lambda -15619P+1225Q)k}{h}+\mathrm{ }\frac{6860(2+\beta k)}{h^2}$$ $$B_4=15619+\frac{4900\lambda k}{h}-\frac{6860(-2+\beta k)}{h^2}$$ $$A_5=15619+\frac{4(1225\lambda +15619P+1225Q)k}{h}+\mathrm{ }\frac{6860(2+\beta k)}{h^2}$$ $$B_5=15619-\frac{4900\lambda k}{h}-\frac{6860(-2+\beta k)}{h^2}$$ $$A_6=\frac{9(477h^2+4h(119\lambda +477P+119Q)k-588(2+\beta k)}{h^2}\mathrm{ }{B}_6=\frac{9(477h^2-476\lambda hk+588(-2+\beta k\left)\right)}{h^2}$$ $$A_7=247+\frac{4(119\lambda +247P+119Q)k}{h}-\frac{1540(2+\beta k)}{h^2}\mathrm{ }{B}_7=247-\frac{476\lambda k}{h}+\frac{1540(-2+\beta k)}{h^2}$$ $$A_8=\frac{h^2+4h(\lambda +P+Q)k-28(2+\beta k)}{h^2}\mathrm{ }{B}_8=\frac{-56+h^2+k(28\beta -4\lambda h)}{h^2}$$

Equation (3.5)(3.5) $$A_1{\alpha }_{m-4}^{n+1}+A_2{\alpha }_{m-3}^{n+1}+A_3{\alpha }_{m-2}^{n+1}+A_4{\alpha }_{m-1}^{n+1}+A_5{\alpha }_m^{n+1}+A_6{\alpha }_{m+1}^{n+1}+A_7{\alpha }_{m+2}^{n+1}+A_8{\alpha }_{m+3}^{n+1}=B_1{\alpha }_{m-4}^n+B_2{\alpha }_{m-3}^n+B_3{\alpha }_{m-2}^n+B_4{\alpha }_{m-1}^n+B_5{\alpha }_m^n+B_6{\alpha }_{m+1}^n+B_7{\alpha }_{m+2}^n+B_8{\alpha }_{m+3}^n$$(3.5) contains ($N+1$) unknow with ($N+8$) parameters (${\alpha }_{-4},{\alpha }_{-3},{\alpha }_{-2},\dots .{\alpha }_{N+2},{\alpha }_{N+3}$).

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 (3.6) $$\mathrm{D}{\left({\mathrm{\alpha }}^{\mathrm{n}}\right)\mathrm{C}}^{\mathrm{n}+1}=G{C}^n$$(3.6)

In this case, both $D$ and $G$ are $(N+8)\times (N+8)$ banded matrices. MATLAB is used to solve the system of equations (3.6)(3.6) $$\mathrm{D}{\left({\mathrm{\alpha }}^{\mathrm{n}}\right)\mathrm{C}}^{\mathrm{n}+1}=G{C}^n$$(3.6) using the octa-diagonal algorithm

Initial state

Applying the initial conditions in equation (3.6)(3.6) $$\mathrm{D}{\left({\mathrm{\alpha }}^{\mathrm{n}}\right)\mathrm{C}}^{\mathrm{n}+1}=G{C}^n$$(3.6) (3.7) $${ w}_m(x,t)=\sum _{m-4}^{N+3}{\alpha }_m^0\left(t\right){\xi }_m\left(x\right)$$(3.7)

To determine the initial state $\left\{{\alpha }_{-4}^0,{\alpha }_{-3}^0,{\alpha }_{-2}^0\dots \dots .{\alpha }_{N+2}^0,{\alpha }_{N+3}^0\right\},$ the approximate solution $w(x,t)$ satisfy the boundary conditions as follows: $$w^3(x_0,0)=\mathrm{ }\left(-{\alpha }_{-4}^0-23{\alpha }_{-3}^0+\mathrm{ }9{\alpha }_{-2}^0+\mathrm{ }95{\alpha }_{-1}^0-95{\alpha }_0^0-9{\alpha }_1^0+23{\alpha }_2^0+{\alpha }_3^0\right)=0$$ $$w^4(x_0,0)=\left({\alpha }_{-4}^0+7{\alpha }_{-3}^0\mathrm{ }-27{\alpha }_{-2}^0+19{\alpha }_{-1}^0+\mathrm{ }19\mathrm{ }{\alpha }_0^0-27{\alpha }_1^0+7{\alpha }_2^0+{\alpha }_3^0\right)=0$$ $$w^5(x_0,0)=\left(-{\alpha }_{-4}^0+{\alpha }_{-3}^0+\mathrm{ }9\mathrm{ }{\alpha }_{-2}^0-25{\alpha }_{-1}^0+25{\alpha }_0^0-9{\alpha }_1^0-{\alpha }_2^0+{\alpha }_3^0\right)=0$$ $$w^7(x_0,0)=\left(-{\alpha }_{-4}^0+7{\alpha }_{-3}^0-21{\alpha }_{-2}^0+\mathrm{ }35{\alpha }_{-1}^0-35{\alpha }_0^0+21{\alpha }_1^0-7{\alpha }_2^0+{\alpha }_3^0\right)=0$$ $$w(x_m,0)={\mathrm{\alpha }}_{\mathrm{m}-4}^0+247{\mathrm{\alpha }}_{\mathrm{m}-3}^0+4293{\mathrm{\alpha }}_{\mathrm{m}-2}^0+15619{\mathrm{\alpha }}_{\mathrm{m}-1}^0+15619{\mathrm{\alpha }}_{\mathrm{m}}^0+4293{\alpha }_{m+1}^0+{247\alpha }_{m+2}^0+{\alpha }_{m+3}^0=f(x_m,0)$$ $$w^4(x_N,0)=\left({\alpha }_{N-4}^0+7{\alpha }_{N-3}^0\mathrm{ }-27{\alpha }_{N-2}^0+19{\alpha }_{N-1}^0+\mathrm{ }19\mathrm{ }{\alpha }_N^0-27{\alpha }_{N+1}^0+7{\alpha }_{N+2}^0+{\alpha }_{N+3}^0\right)=0$$ $$w^5(x_N,0)=\left(-{\alpha }_{N-4}^0+{\alpha }_{N-3}^0+\mathrm{ }9\mathrm{ }{\alpha }_{N-2}^0-25{\alpha }_{N-1}^0+25{\alpha }_N^0-9{\alpha }_{N+1}^0-{\alpha }_{N+2}^0+{\alpha }_{N+3}^0\right)=0$$ (3.8) $$w^6(x_N,0)=\left({\alpha }_{N-4}^0-5{\alpha }_{N-3}^0+\mathrm{ }9{\alpha }_{N-2}^0-5{\alpha }_{N-1}^0-5{\alpha }_N^0+\mathrm{ }9{\alpha }_{N+1}^0-5{\alpha }_{N+2}^0+{\alpha }_{N+3}^0\right)=0$$(3.8)

Initial matrix is of the form $\mathrm{A}{\mathrm{C}}^0=\mathrm{\gamma }$

Where $C^0={\left[{\alpha }_{-4}^0,{\alpha }_{-3}^0,{\alpha }_{-2}^0\dots \dots \dots .{\alpha }_{N+2}^0,{\alpha }_{N+3}^0\right]}^T,\mathrm{\gamma }={\left[0,0,0,0,f(x_m)0,0,0\right]}^T$

$m=0,1,\dots .N$ and (3.9) $$D=\left(\begin{array}{cccccccccccc}-1& -23& 9& 95& -95& -9& 23& 1& & & & \\ 1& 7& -27& 19& 19& -27& 7& 1& & & & \\ -1& 1& 9& -25& 25& -9& -1& 1& & & & \\ -1& 7& -21& 35& -35& 21& -7& 1& & & & \\ 1& 247& 4293& 15619& 15619& 4293& 247& 1& & & & \\ & 1& 247& 4293& 15619& 15619& 4293& 247& 1& & & \\ & & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & & \\ & & & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \\ & & & & 1& 247& 4293& 15619& 15619& 4293& 247& 1\\ & & & & 1& 7& -27& 19& 19& -27& 7& 1\\ & & & & -1& 1& 9& -25& 25& -9& -1& 1\\ & & & & 1& -5& 9& -5& -5& 9& -5& 1\end{array}\right)$$(3.9)

To solve the system, we apply the octa diagonal algorithm to obtain the initial vector $C^0={\left[{\alpha }_{-4}^0,{\alpha }_{-3}^0,{\alpha }_{-2}^0\dots \dots \dots .{\alpha }_{N+2}^0,{\alpha }_{N+3}^0\right]}^T.$ Numerical solution of equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) is obtained by iterating equation (3.6)(3.6) $$\mathrm{D}{\left({\mathrm{\alpha }}^{\mathrm{n}}\right)\mathrm{C}}^{\mathrm{n}+1}=G{C}^n$$(3.6) .

General solution at $x=x_m$ can be written as: (3.10) $$\mathrm{w}({\mathrm{x}}_{\mathrm{m}},\mathrm{t})={\mathrm{\alpha }}_{\mathrm{m}-4}+247{\mathrm{\alpha }}_{\mathrm{m}-3}{{+4293\mathrm{\alpha }}_{\mathrm{m}-2}+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}-1}{+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}}{{+4293\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}+1}+\mathrm{ }247\mathrm{\alpha }}_{\mathrm{m}+2}{+\mathrm{\alpha }}_{\mathrm{m}+3}\mathrm{ }$$(3.10)

4. Stability analysis

To investigate the stability of equation (3.4)(3.4) $$A^{n+1}={\psi }^n\mathrm{ }$$(3.4) , employ Von-Neumann procedure (4.1) $${\mathrm{ }\sigma }_m^n={\gamma }^n{e}^{\mathit{\text{iθmh}}}$$(4.1) where, $i=\sqrt{-1},\text{ an imaginary unit},$ also $\theta$ and h is the mode number and the element size respectively. Associating equation (4.1)(4.1) $${\mathrm{ }\sigma }_m^n={\gamma }^n{e}^{\mathit{\text{iθmh}}}$$(4.1) into equation (3.5)(3.5) $$A_1{\alpha }_{m-4}^{n+1}+A_2{\alpha }_{m-3}^{n+1}+A_3{\alpha }_{m-2}^{n+1}+A_4{\alpha }_{m-1}^{n+1}+A_5{\alpha }_m^{n+1}+A_6{\alpha }_{m+1}^{n+1}+A_7{\alpha }_{m+2}^{n+1}+A_8{\alpha }_{m+3}^{n+1}=B_1{\alpha }_{m-4}^n+B_2{\alpha }_{m-3}^n+B_3{\alpha }_{m-2}^n+B_4{\alpha }_{m-1}^n+B_5{\alpha }_m^n+B_6{\alpha }_{m+1}^n+B_7{\alpha }_{m+2}^n+B_8{\alpha }_{m+3}^n$$(3.5) and further simplification $${\gamma }_{n+1}[A_1{e}^{-4i\theta h}+A_2{e}^{-3i\theta h}+A_3{e}^{-2i\theta h}+A_4{e}^{-i\theta h}+A_5+A_6{e}^{i\theta h}+A_7{e}^{2i\theta h}+A_8{e}^{3i\theta h}]$$ (4.2) $$={\gamma }_n[B_1{e}^{-4i\theta h}+B_2{e}^{-3i\theta h}+B_3{e}^{-2i\theta h}+B_4{e}^{-i\theta h}+B_5+B_6{e}^{i\theta h}+B_7{e}^{2i\theta h}+B_8{e}^{3i\theta h}]$$(4.2)

Applying the Euler’s formula $e^{\pm i\theta h}=\cos (\theta h)\pm \mathit{isin}(\theta h)$ in (4.2)(4.2) $$={\gamma }_n[B_1{e}^{-4i\theta h}+B_2{e}^{-3i\theta h}+B_3{e}^{-2i\theta h}+B_4{e}^{-i\theta h}+B_5+B_6{e}^{i\theta h}+B_7{e}^{2i\theta h}+B_8{e}^{3i\theta h}]$$(4.2) (4.3) $$\frac{{\gamma }_{n+1}}{{\gamma }_n}=\frac{S_1+i{R}_1}{S_2+i{R}_2}\mathrm{ }$$(4.3) where $$S_1=B_1\hspace{0.17em}\cos \hspace{0.17em}\left(4\theta h\right)+(B_2+B_8)\cos \left(3\theta h\right)+(B_3+B_7)\cos \left(2\theta h\right)+(B_4+B_6)\cos \left(\theta h\right)+B_5$$ $$R_1=-B_1\hspace{0.17em}\sin \hspace{0.17em}\left(4\theta h\right)+(B_8-B_2)\sin \left(3\theta h\right)+(B_7-B_3)\sin \left(2\theta h\right)+(B_6-B_4)\sin \left(\theta h\right)$$ $$S_2=A_1\hspace{0.17em}\cos \hspace{0.17em}\left(4\theta h\right)+(A_2+A_8)\cos \left(3\theta h\right)+(A_3+A_7)\cos \left(2\theta h\right)+(A_4+A_6)\cos \left(\theta h\right)+A_5$$ $$R_2=-A_1\hspace{0.17em}\sin \hspace{0.17em}\left(4\theta h\right)+(A_8-A_2)\sin \left(3\theta h\right)+(A_7-A_3)\sin \left(2\theta h\right)+(A_6-A_4)\sin \left(\theta h\right)$$

Applying the stability condition $\left|\frac{{\gamma }_{n+1}}{{\gamma }_n}\right|\le 1$ (4.4) $$S_1^2+R_1^2-S_2^2-R_2^2\le 1\mathrm{ }$$(4.4)

The unconditionally stable condition is followed from equation (4.4)(4.4) $$S_1^2+R_1^2-S_2^2-R_2^2\le 1\mathrm{ }$$(4.4) .

5. Convergence of the scheme

This section contains study for the convergence of present topic.

Lemma 5.1.

Octic B-Spline {${\mathrm{\xi }}_{-4},{\mathrm{\xi }}_{-3},{\mathrm{\xi }}_{-2},\dots \dots {\mathrm{\xi }}_{\mathrm{N}+3}\}$ satisfy the following inequality (5.1) $$\left|\sum _{m=-4}^{N+3}{\mathrm{\xi }}_{\mathrm{m}}\left(\mathrm{x}\right)\right|\le 55938$$(5.1)

Proof.

From real analysis, we have (5.2) $$\left|\sum _{m=-4}^{N+3}{\mathrm{\xi }}_m\left(x\right)\right|\le \sum _{m=-4}^{N+3}\left|{\mathrm{\xi }}_m\left(x\right)\right|$$(5.2)

At any node $x_m,$ (5.3) $$\left|\sum _{m=-4}^{N+3}{\mathrm{\xi }}_m\left(x\right)\right|=\left|{\mathrm{\xi }}_{m-4}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m-3}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m-2}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m-1}\left(x\right)\right|+\left|{\mathrm{\xi }}_m\left(x\right)\right|+\left|{\mathrm{\xi }}_{m+1}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m+2}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m+3}\left(x\right)\right|=1+247+4293+15619+15619+4293+247+1\le 40320\mathrm{ }$$(5.3)

Also, in each sub interval $x_{m-1}\le x\le x_m,$ $${\mathrm{\xi }}_{m-4}\left(x_{m-1}\right)=247,\mathrm{ }{\mathrm{\xi }}_{m-3}\left(x_{m-1}\right)=4293,\mathrm{ }{\mathrm{\xi }}_{m-2}\left(x_{m-1}\right)=15619,$$ $${\text{ ξ}}_{m-1}\left(x_{m-1}\right)=15619,{\mathrm{\xi }}_m\left(x_m\right)=15619,{\mathrm{\xi }}_{m+1}\left(x_m\right)=4293,$$ $${\mathrm{\xi }}_{m+2}\left(x_m\right)=247,{\mathrm{\xi }}_{m+3}\left(x_m\right)=1$$

Hence, in each sub interval $x_{m-1}\le x\le x_m$ (5.4) $$\sum _{m=-4}^{N+3}\left|{\mathrm{\xi }}_m\left(x\right)\right|\le \left|{\mathrm{\xi }}_{m-4}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m-3}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m-2}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m-1}\left(x\right)\right|+\left|{\mathrm{\xi }}_m\left(x\right)\right|+\left|{\mathrm{\xi }}_{m+1}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m+2}\left(x\right)\right|+\left|{\mathrm{\xi }}_{m+3}\left(x\right)\right|\le 247+4293+15619+15619+15619+4293+247+1\le 55938$$(5.4)

Hence, $\left|{\sum }_{m=-4}^{N+3}{\mathrm{\xi }}_m(x)\right|\le 55938.$

Theorem 5.2.

Suppose that $w(x,t)$ and $W(x,t)$ be the analytic and approximate solution respectively of equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) and $w(x,t)\in {\mathrm{C}}^8[a,b]$ and also $\left|\frac{{\partial }^{8}w(x,t)}{\partial x^8}\right|\le L,$ then (5.5) $${\Vert w(x,t)-W(x,t)\Vert }_{\mathrm{\infty }}=O(h^9+(\Delta {t)}^2)$$(5.5)

We have a unique octic B-spline (3.7)(3.7) $${ w}_m(x,t)=\sum _{m-4}^{N+3}{\alpha }_m^0\left(t\right){\xi }_m\left(x\right)$$(3.7) that satisfies the interpolation condition $w(x_m)=H{(x}_m);\mathrm{ }m=0,1,2\dots .N.$ and $H(x)$ is also smooth.

Equation (2.7)(2.7) $$\left.\begin{array}{l}w\left(x_m\right)={\mathrm{\alpha }}_{\mathrm{m}-4}+247{\mathrm{\alpha }}_{\mathrm{m}-3}{{+4293\mathrm{\alpha }}_{\mathrm{m}-2}+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}-1}{+15619\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}}{{+4293\mathrm{ }\mathrm{\alpha }}_{\mathrm{m}+1}+247\mathrm{\alpha }}_{\mathrm{m}+2}{+\mathrm{\alpha }}_{\mathrm{m}+3} \\ {\mathrm{ }w}^{\left(1\right)}\left(x_m\right)=\frac{8}{\mathrm{h}}\left(-{\alpha }_{m-4}-119{\alpha }_{m-3}-1071{\alpha }_{m-2}-1225{\alpha }_{m-1}+1225{\alpha }_m+1071{\alpha }_{m+1}+119{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(2\right)}\left(x_m\right)=\frac{56}{{\mathrm{h}}^2}\left({\alpha }_{m-4}+\mathrm{ }55{\alpha }_{m-3}+\mathrm{ }189\mathrm{ }{\alpha }_{m-2}-245{\alpha }_{m-1}-245{\alpha }_m+189{\alpha }_{m+1}+55{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(3\right)}\left(x_m\right)=\frac{336}{{\mathrm{h}}^3} \left(-{\alpha }_{m-4}-23{\alpha }_{m-3}+\mathrm{ }9{\alpha }_{m-2}+\mathrm{ }95{\alpha }_{m-1}-95{\alpha }_m-9{\alpha }_{m+1}+23{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(4\right)}\left(x_m\right)=\frac{1680}{{\mathrm{h}}^4}\left({\alpha }_{m-4}+7{\alpha }_{m-3}\mathrm{ }-27{\alpha }_{m-2}+19{\alpha }_{m-1}+\mathrm{ }19\mathrm{ }{\alpha }_m-27{\alpha }_{m+1}+7{\alpha }_{m+2}+{\alpha }_{m+3}\right)\\ w^{\left(5\right)}\left(x_m\right)=\frac{6720}{{\mathrm{h}}^5}\left(-{\alpha }_{m-4}+{\alpha }_{m-3}+\mathrm{ }9\mathrm{ }{\alpha }_{m-2}-25{\alpha }_{m-1}+25{\alpha }_m-9{\alpha }_{m+1}-{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(6\right)}\left(x_m\right)=\frac{20160}{{\mathrm{h}}^6}\left({\alpha }_{m-4}-5{\alpha }_{m-3}+\mathrm{ }9{\alpha }_{m-2}-5{\alpha }_{m-1}-5{\alpha }_m+\mathrm{ }9{\alpha }_{m+1}-5{\alpha }_{m+2}+{\alpha }_{m+3}\right) \\ w^{\left(7\right)}\left(x_m\right)=\frac{40320}{{\mathrm{h}}^7}\left(-{\alpha }_{m-4}+7{\alpha }_{m-3}-21{\alpha }_{m-2}+\mathrm{ }35{\alpha }_{m-1}-35{\alpha }_m+21{\alpha }_{m+1}-7{\alpha }_{m+2}+{\alpha }_{m+3}\right)\end{array}\right\}$$(2.7) estimates the coefficients of ${\alpha }_m\mathrm{ }$ for $m=-4,-3,-2\dots .N+3$ (5.6) $$w^{\prime }\left(x_{m-4}\right)+247w^{\prime }\left(x_{m-3}\right)+4293w^{\prime }\left(x_{m-2}\right)+15619w^{\prime }\left(x_{m-1}\right)+15619w^{\prime }\left(x_m\right)+4293w^{\prime }\left(x_{m+1}\right)+247w^{\prime }\left(x_{m+2}\right)+w^{\prime }\left(x_{m+3}\right)=\frac{8}{h}(-H(x_{m-4})-119H(x_{m-3})-1071H(x_{m-2})-1225H(x_{m-1})+1225H(x_m)+1071H(x_{m+1})+119H(x_{m+2})+H(x_{m+3})\mathrm{ }$$(5.6) (5.7) $$w^{″}\left(x_{m-4}\right)+247w^{″}\left(x_{m-3}\right)+4293w^{″}\left(x_{m-2}\right)+15619w^{″}\left(x_{m-1}\right)+15619w^{″}\left(x_m\right)+4293w^{″}\left(x_{m+1}\right)+247w^{″}\left(x_{m+2}\right)+w^{″}\left(x_{m+3}\right)=\frac{56}{h^2}\left(H\right(x_{m-4})+55H(x_{m-3})+189H(x_{m-2})-245H(x_{m-1})-245H(x_m)+189H(x_{m+1})+55H(x_{m+2})+H(x_{m+3}\left)\right)$$(5.7) (5.8) $$w^{″\prime }\left(x_{m-4}\right)+247w^{″\prime }\left(x_{m-3}\right)+4293w^{″\prime }\left(x_{m-2}\right)+15619w^{″\prime }\left(x_{m-1}\right)+15619w^{″\prime }\left(x_m\right)+4293w^{″\prime }\left(x_{m+1}\right)+247w^{″\prime }\left(x_{m+2}\right)+w^{″\prime }\left(x_{m+3}\right)=\frac{336}{h^3}(-H(x_{m-4})-23H(x_{m-3})+9H(x_{m-2})+95H(x_{m-1})-95H(x_m\left)\text{-}\mathrm{ }9H\right(x_{m+1})+23H(x_{m+2})+H(x_{m+3}\left)\right)$$(5.8) (5.9) $$w^{iv}\left(x_{m-4}\right)+247w^{iv}\left(x_{m-3}\right)+4293w^{iv}\left(x_{m-2}\right)+15619w^{iv}\left(x_{m-1}\right)+15619w^{iv}\left(x_m\right)+4293w^{iv}\left(x_{m+1}\right)+247w^{iv}\left(x_{m+2}\right)+w^{iv}\left(x_{m+3}\right)=\frac{1680}{h^4}\left(H\right(x_{m-4})+7H(x_{m-3})-27H(x_{m-2})+\mathrm{ }19H(x_{m-1})+19H(x_m)-27H(x_{m+1})+7H(x_{m+2})+H(x_{m+3}\left)\right)$$(5.9) (5.10) $$w^{\left(v\right)}\left(x_{m-4}\right)+247w^{\left(v\right)}\left(x_{m-3}\right)+4293w^{\left(v\right)}\left(x_{m-2}\right)+15619w^{\left(v\right)}\left(x_{m-1}\right)+15619w^{\left(v\right)}\left(x_m\right)+4293w^{\left(v\right)}\left(x_{m+1}\right)+247w^{\left(v\right)}\left(x_{m+2}\right)+w^{\left(v\right)}\left(x_{m+3}\right)=\frac{6720}{h^5}(-H(x_{m-4})+H(x_{m-3})+9H(x_{m-2})-25H(x_{m-1})+25H(x_m)-9H(x_{m+1})-H(x_{m+2})+H(x_{m+3})$$(5.10) (5.11) $$w^{\left(vi\right)}\left(x_{m-4}\right)+247w^{\left(vi\right)}\left(x_{m-3}\right)+4293w^{\left(vi\right)}\left(x_{m-2}\right)+15619w^{\left(vi\right)}\left(x_{m-1}\right)+15619w^{\left(vi\right)}\left(x_m\right)+4293w^{\left(vi\right)}\left(x_{m+1}\right)+247w^{\left(vi\right)}\left(x_{m+2}\right)+w^{\left(vi\right)}\left(x_{m+3}\right)=\frac{20160}{h^6}\left(H\right(x_{m-4})-5H(x_{m-3})+9H(x_{m-2})-5H(x_{m-1})-5H(x_m)+9H(x_{m+1})-5H(x_{m+2})+H(x_{m+3}\left)\right)\mathrm{ }$$(5.11) (5.12) $$w^{\left(\mathit{vii}\right)}\left(x_{m-4}\right)+247w^{\left(\mathit{vii}\right)}\left(x_{m-3}\right)+4293w^{\left(\mathit{vii}\right)}\left(x_{m-2}\right)+15619w^{\left(\mathit{vii}\right)}\left(x_{m-1}\right)+15619w^{\left(\mathit{vii}\right)}\left(x_m\right)+4293w^{\left(\mathit{vii}\right)}\left(x_{m+1}\right)+247w^{\left(\mathit{vii}\right)}\left(x_{m+2}\right)+w^{\left(\mathit{vii}\right)}\left(x_{m+3}\right)=\frac{40320}{h^7}(-H(x_{m-4})+7H(x_{m-3})-21H(x_{m-2})+35H(x_{m-1})-35H(x_m)+21H(x_{m+1})-7H(x_{m+2})+H(x_{m+3}\left)\right)$$(5.12)

Employing, $H^{\prime }\left(x\right)=D(H(x)),$ $H\left(x\right)=I(H(x))$ and $H\left(x+h\right)=E(H(x)),$ refer (Lucas, 1974; Fyfe, 1969), equations (5.6)–(5.12)(5.12) $$w^{\left(\mathit{vii}\right)}\left(x_{m-4}\right)+247w^{\left(\mathit{vii}\right)}\left(x_{m-3}\right)+4293w^{\left(\mathit{vii}\right)}\left(x_{m-2}\right)+15619w^{\left(\mathit{vii}\right)}\left(x_{m-1}\right)+15619w^{\left(\mathit{vii}\right)}\left(x_m\right)+4293w^{\left(\mathit{vii}\right)}\left(x_{m+1}\right)+247w^{\left(\mathit{vii}\right)}\left(x_{m+2}\right)+w^{\left(\mathit{vii}\right)}\left(x_{m+3}\right)=\frac{40320}{h^7}(-H(x_{m-4})+7H(x_{m-3})-21H(x_{m-2})+35H(x_{m-1})-35H(x_m)+21H(x_{m+1})-7H(x_{m+2})+H(x_{m+3}\left)\right)$$(5.12) can be expressed as: $$W^{\prime }\left(x_m\right)=\frac{8}{h}\left(\frac{-E^{-4}-119E^{-3}-1071E^{-2}-1225E^{-1}+1225I+1071E+119E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$ $$W^{″}\left(x_m\right)=\frac{56}{h^2}\left(\frac{E^{-4}+55E^{-3}+189E^{-2}-245E^{-1}-245I+189E+55E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$ $$W^{″\prime }\left(x_m\right)=\frac{336}{h^3}\left(\frac{-E^{-4}-23E^{-3}+9E^{-2}+95E^{-1}-95I-9E+23E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$ $$W^{\left(iv\right)}\left(x_m\right)=\frac{1680}{h^4}\left(\frac{E^{-4}+7E^{-3}-27E^{-2}+19E^{-1}+19I-27E+7E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$ $$W^{\left(v\right)}\left(x_m\right)=\frac{6720}{h^5}\left(\frac{-E^{-4}+E^{-3}+9E^{-2}-25E^{-1}+25I-9E-E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$ $$W^{\left(vi\right)}\left(x_m\right)=\frac{20160}{h^6}\left(\frac{E^{-4}-5E^{-3}+9E^{-2}-5E^{-1}-5I+9E-5E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$ $$W^{\left(\mathit{vii}\right)}\left(x_m\right)=\frac{40320}{h^7}\left(\frac{-E^{-4}+7E^{-3}-21E^{-2}+35E^{-1}-35I+21E-7E^2+E^3}{E^{-4}+247E^{-3}+4293E^{-2}+15619E^{-1}+15619I+4293E+247E^2+E^3}\right)H\left(x_m\right)$$

Let $E=\exp (hD)$ and expand them in the power of $hD$ (5.13) $$W^{\prime }\left(x_m\right)=H^{\prime }{(x}_m)[phone]\frac{\mathrm{}}{\mathrm{}}(h^8{H}^9\left(x_m\right))-\frac{1}{1064448}(h^{10}{H}^{11}{(x}_m))[phone]\frac{\mathrm{}}{\mathrm{}}(h^{12}{H}^{13}\left(x_m\right))+O(h^{13})$$(5.13) (5.14) $$W^{″}{(x}_m)=H^{″}(x_m)-\frac{1}{172800}(h^8{H}^{10}{(x}_m))[phone]\frac{\mathrm{}}{\mathrm{}}(h^{10}{H}^{12}{(x}_m))-\frac{691}{4402944000}(h^{12}{H}^{14}{(x}_m))+O(h^{13})\mathrm{ }$$(5.14) (5.15) $$W^{″\prime }\left(x_m\right)=H^{″\prime }{(x}_m)-\frac{1}{30240}(h^6{H}^9{(x}_m))+\frac{1}{57600}(h^8{H}^{11}{(x}_m))-\frac{1}{354816}(h^{10}{H}^{13}\left(x_m\right))+\frac{13129}{39626496000}(h^{12}{H}^{15}{(x}_m\left)\right)+O\left(h^{13}\right)\mathrm{ }$$(5.15) $$W^{\left(iv\right)}\left(x_m\right)=H^{\left(4\right)}{(x}_m)+\frac{1}{6048}(h^6{H}^{10}{(x}_m))-\frac{1}{34560}(h^8{H}^{12}{(x}_m))[phone]\frac{\mathrm{}}{\mathrm{}}(h^{10}{H}^{14}{(x}_m))-\frac{691}{792529920}(h^{12}{H}^{16}{(x}_m\left)\right)+O\left(h^{13}\right)\mathrm{ }\left(5.16\right)$$ (5.17) $$W^{\left(v\right)}{(x}_m)=\mathrm{ }{H}^{\left(5\right)}{(x}_m)+\frac{1}{720}\left(h^4{H}^9{(x}_m)\right)-\frac{1}{3024}\left(h^6{H}^{11}{(x}_m)\right)+\frac{1}{34560}\left(h^8{H}^{13}{(x}_m)\right)-\frac{31}{15966720}\left(h^{10}{H}^{15}{(x}_m\right))[phone]\frac{\mathrm{}}{\mathrm{}}(h^{12}{H}^{17}{(x}_m\left)\right)+O\left(h^{13}\right)\mathrm{ }$$(5.17) (5.18) $$W^{\left(vi\right)}{(x}_m)=H^{\left(6\right)}{(x}_m)-\frac{1}{240}\left(h^4{H}^{10}{(x}_m)\right)+\frac{1}{3024}\left(h^6{H}^{12}{(x}_m)\right)-\frac{1}{57600}\left(h^8{H}^{14}{(x}_m\left)\right)\right)[phone]\frac{\mathrm{}}{\mathrm{}}\left(h^{10}{H}^{16}{(x}_m\right)\left)\right)[phone]\frac{\mathrm{}}{\mathrm{}}\left(h^{12}{H}^{18}{(x}_m\right))+O(h^{13})\mathrm{ }$$(5.18) (5.19) $$W^{\left(\mathit{vii}\right)}{(x}_m)=H^{\left(7\right)}{(x}_m)-\frac{1}{12}\left(h^2{H}^9{(x}_m)\right)+\frac{1}{240}\left(h^4{H}^{11}{(x}_m)\right)-\frac{1}{6048}\left(h^6{H}^{13}\left(x_m\right)\right)+\frac{1}{172800}\left(h^8{H}^{15}{(x}_m\right))-\frac{1}{2661120}(h^{10}{H}^{17}{(x}_m\left)\right)+\frac{142091}{1307674368000}\left(h^{12}{H}^{19}{(x}_m\right))+O(h^{13})\mathrm{ }$$(5.19)

Let us consider the error $\mathrm{ }\mathrm{e}(\mathrm{x})=\mathrm{W}(\mathrm{x})\text{-}\mathrm{H}(\mathrm{x})$ and associte equations (5.13)–(5.19)(5.19) $$W^{\left(\mathit{vii}\right)}{(x}_m)=H^{\left(7\right)}{(x}_m)-\frac{1}{12}\left(h^2{H}^9{(x}_m)\right)+\frac{1}{240}\left(h^4{H}^{11}{(x}_m)\right)-\frac{1}{6048}\left(h^6{H}^{13}\left(x_m\right)\right)+\frac{1}{172800}\left(h^8{H}^{15}{(x}_m\right))-\frac{1}{2661120}(h^{10}{H}^{17}{(x}_m\left)\right)+\frac{142091}{1307674368000}\left(h^{12}{H}^{19}{(x}_m\right))+O(h^{13})\mathrm{ }$$(5.19) in equation (5.20)(5.20) $$\mathrm{e}(x+\mathrm{\gamma }\mathrm{h})=\mathrm{e}\left(x_{\mathrm{m}}\right)+\mathrm{\gamma }\mathrm{h}{\mathrm{e}}^{\prime }\left(x_{\mathrm{m}}\right)+\frac{{\mathrm{\gamma }}^2{\mathrm{h}}^2}{2}{\mathrm{e}}^{″}\left(x_{\mathrm{m}}\right)+\frac{{\mathrm{\gamma }}^3{\mathrm{h}}^3}{6}{\mathrm{e}}^{″\prime }\left(x_{\mathrm{m}}\right)+\cdots \cdots \cdots \cdots \cdots \cdots$$(5.20) to obtain equation (5.21). Employing Taylor series expansion in equation (5.20)(5.20) $$\mathrm{e}(x+\mathrm{\gamma }\mathrm{h})=\mathrm{e}\left(x_{\mathrm{m}}\right)+\mathrm{\gamma }\mathrm{h}{\mathrm{e}}^{\prime }\left(x_{\mathrm{m}}\right)+\frac{{\mathrm{\gamma }}^2{\mathrm{h}}^2}{2}{\mathrm{e}}^{″}\left(x_{\mathrm{m}}\right)+\frac{{\mathrm{\gamma }}^3{\mathrm{h}}^3}{6}{\mathrm{e}}^{″\prime }\left(x_{\mathrm{m}}\right)+\cdots \cdots \cdots \cdots \cdots \cdots$$(5.20) (5.20) $$\mathrm{e}(x+\mathrm{\gamma }\mathrm{h})=\mathrm{e}\left(x_{\mathrm{m}}\right)+\mathrm{\gamma }\mathrm{h}{\mathrm{e}}^{\prime }\left(x_{\mathrm{m}}\right)+\frac{{\mathrm{\gamma }}^2{\mathrm{h}}^2}{2}{\mathrm{e}}^{″}\left(x_{\mathrm{m}}\right)+\frac{{\mathrm{\gamma }}^3{\mathrm{h}}^3}{6}{\mathrm{e}}^{″\prime }\left(x_{\mathrm{m}}\right)+\cdots \cdots \cdots \cdots \cdots \cdots$$(5.20) (5.21) $$\mathrm{e}(x+\mathrm{\gamma }\mathrm{h})=\frac{{\mathrm{h}}^9\mathrm{\gamma }[3{\mathrm{H}}^9-2{\mathrm{H}}^9{\mathrm{\gamma }}^2(10-21{\mathrm{\gamma }}^2+30{\mathrm{\gamma }}^4\left)\right]}{3628800}-\frac{{\mathrm{h}}^{10}{\mathrm{H}}^{10}{\mathrm{\gamma }}^2\left(21-50{\mathrm{\gamma }}^2+42{\mathrm{\gamma }}^4\right)}{7257600}+\mathrm{ }\frac{{\mathrm{h}}^{11}{\mathrm{H}}^{11}\mathrm{\gamma }(-75+231{\mathrm{\gamma }}^2-220{\mathrm{\gamma }}^4+66{\mathrm{\gamma }}^6)}{79833600}+\frac{{\mathrm{h}}^{12}{\mathrm{H}}^{12}{\mathrm{\gamma }}^2(150-231{\mathrm{\gamma }}^2+88{\mathrm{\gamma }}^4)}{191600640}+\mathrm{O}\left({\mathrm{h}}^{13}\right)\mathrm{ }$$(5.21)

Where $a\le \mathrm{\gamma }\le b.$

The following theorem condenses the preceding results:

Theorem 5.3.

For sufficiently small h, the truncation error is O(h^9), where H(x) is the analytical solution and w(x) is the approximate solution of (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) with boundary conditions. This reveals that the method has O(h^9) accuracy.

Theorem 5.4.

Refer (Kadalbajoo & Awasthi, 2008)

If $\frac{d^{r}w}{d{t}^r}\le \mathrm{ }Q,r=0,1\mathrm{ }(x,t)\mathrm{\in }[a,b],$ the estimation of local error $E_{n+1}$ is given by $E_{n+1}\le Q{(\Delta t)}^3.$

Theorem 5.5.

The global error estimate $E_{n+1}$ at ${(n+1)}^{th}$ time level with global truncation error $E_n$ at the time of discretization can be expressed as: (5.22) $$E_{n+1}=\sum _{m=1}^n{E}_m$$(5.22)

Proof.

Applying norm for equation (5.22)(5.22) $$E_{n+1}=\sum _{m=1}^n{E}_m$$(5.22) $${\Vert E_{n+1}\Vert }_{\mathrm{\infty }}={\Vert \sum _{m=1}^{n}E\Vert }_{\mathrm{\infty }}\le \sum _{m=1}^n{\Vert E\Vert }_{\mathrm{\infty }}\le \sum _{m=1}^{n}Q{\left(\Delta t\right)}^3=Q_n{\left(\Delta t\right)}^3=Q_n\left(\frac{T}{n}\right){(\Delta t)}^2$$ (5.23) $${\Vert E_{n+1}\Vert }_{\mathrm{\infty }}\le S{\left(\Delta t\right)}^2=O{(\Delta t)}^2$$(5.23) where $S=QT.$

Equations (5.21) and (5.23)(5.23) $${\Vert E_{n+1}\Vert }_{\mathrm{\infty }}\le S{\left(\Delta t\right)}^2=O{(\Delta t)}^2$$(5.23) generate $${\mathrm{ }\Vert w(x,t)-W(x,t)\Vert }_{\mathrm{\infty }}=O(h^9+(\Delta {t)}^2)$$

6. Numerical computation

This section bears with the formulae and numerical computation of $L_2$ and $L_{\mathrm{\infty }}\mathrm{ }$ error norms and invariants $I_1,$ $I_2$ and $I_3$ invariants via four tests. $$L_2=\sqrt{h\sum _{m=0}^N{\left|W_m-w_m\right|}^2}$$ $$L_{\mathrm{\infty }}=\max \left|W_m-w_m\right|$$ $$I_1={\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}W\mathrm{ }dx\cong h\sum _{m=0}^{N}w(x_m,t)$$ $$I_2={\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}\left[W^2+{{(W}_x)}^2\right]dx\cong h\sum _{m=0}^N\left[{w(x_m,t)}^2+{{(w}_x(x_m,t))}^2\right]$$ $$I_3={\int }_{\text{-}\mathrm{\infty }}^{\mathrm{\infty }}\left[W^3+3W^2\right]dx\cong h\sum _{m=0}^N\left[{w(x_m,t)}^3+3{\left(w\right(x_m,t\left)\right)}^2\right]$$

Example 1.

For β = 0 and λ = 1 in the governing BBMB equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) with the initial condition $w\left(x,0\right)=\sec {\mathrm{h}}^2\left(\frac{x}{4}\right)\mathrm{ }$ in the space domain [−10, 30] and the time domain [1, 4], the exact solution is $w(x,t)=\sec {\mathrm{h}}^2(\frac{x}{4}-\frac{t}{3}).$ (See Zarebna & Parvaz, 2020). Here $w(x,t)$ stands for the soliton to single solitary wave of equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) . Table 2 reports the $\mathrm{ }{L}_2,L_{\mathrm{\infty }}$ error norms at different values N and temporal status. Table 2 also presents a comparison of the $\mathrm{ }{L}_2$ error norms, with many BBMB equations (Zarebnia & Parvaz, 2016, 2020). Tables 3 and 5 report the invariants $I_1,I_2\mathrm{ }\mathit{and}\mathrm{ }{I}_3$ collocations with cubic Hermitian splines (Arora et al., 2020), for different values of N. Table 4 represents the comparison $I_1,I_2\mathrm{ }\mathit{and}\mathrm{ }{I}_3$ invariants at different time levels by present method with existing results (Arora et al., 2020; Zarebnia & Parvaz, 2016). The result is found to be more accurate in present scheme. Figure 1(a) and (b) surf approximate and analytical solutions for time level t = 4, N = 100, Δt = 0.01, respectively. Figure 2 depicts a comparison of the approximate and analytical solutions at various time points for Δt = 0.01, N = 100. For clarity, we compare the approximate and exact solutions for Δt = 0.1 and N = 100 at time t = 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 Δt = 0.01 and N = 900.

Figure 1. Solution of Example 1.

Figure 2. Comparison of numerical solution with analytic solution Example 1.

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

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

Figure 5. Numerical simulation of Example 2.

Table 2. $L_2,L_{\mathrm{\infty }}$ are compared for $\Delta t=0.01$ of Example 1.

t	(Zarebna & Parvaz, 2020) ${\mathit{L}}_2\times {10}^3$	${\mathrm{ }\mathit{L}}_2$ (Arora et al., 2020)		${\mathit{L}}_2$ (Zarebnia & Parvaz, 2016)		Present method
	N = 100	N = 100	N = 400	N = 100	N = 400	N = 100		N = 400
						${\mathrm{ }\mathit{L}}_2$	${\mathrm{ }\mathit{L}}_{\mathrm{\infty }}$	${\mathrm{ }\mathit{L}}_2$	${\mathrm{ }\mathit{L}}_{\mathrm{\infty }}$
1	1.071410	2.49e-05	6.80e-06	1.51e-02	1.40e-02	8.3929e-06	1.9803e-06	3.3414e-06	2.5981e-06
2	1.103980	3.64e-05	1.08e-05	1.25e-02	1.21e-02	9.1453e-06	4.1552e-06	6.5727e-06	5.2773e-06
3	No data	4.44e-05	1.41e-05	1.21e-02	1.18e-02	1.2735e-05	1.0941e-05	1.1152e-05	1.0242e-05
4	No data	5.10e-05	1.74e-05	1.24e-02	1.20e-02	1.3632e-05	1.2756e-05	1.9217e-05	1.9762e-05

Table 3. Computation of $I_1,I_2 \mathit{and}\mathrm{ }{I}_3$ invariants of Example 1 for Δt = 0.01.

$t\mathrm{ }$	N = 100			N = 900
	$I_1$	$I_2$	$I_3$	$I_1$	$I_2$	$I_3$
1	7.9751	5.5998	3.3905	7.9727	5.5998	3.3905
1.5	7.9821	5.5999	3.905	7.9804	5.5999	3.3905
2	7.9872	5.5999	3.3905	7.9860	5.5999	3.3905
2.5	7.9908	5.6000	3.3905	7.9899	5.6000	3.3905
3	7.9934	5.6000	3.3905	7.9928	5.6000	3.3905
3.5	7.9953	5.6000	3.3905	7.9948	5.6000	3.3905
4	7.9966	5.6000	3.3905	7.9963	5.6000	3.3905

Table 4. Comparison of $I_1,I_2 \mathit{and}\mathrm{ }{I}_3$ invariants for Example 1 at N = 400 and $\Delta t=0.01.$

$t\mathrm{ }$	Zarebnia and Parvaz (2016)			Arora et al. (2020)			Present method
	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$
2	7.96026	5.59845	20.26020	7.98574	5.37594	20.26650	7.9862	5.5999	20.2665
6	7.96211	5.59863	20.26100	7.99871	5.37600	20.26665	7.9990	5.5999	20.2667
8	7.95947	5.59872	20.26140	7.99903	5.37600	20.26665	7.9995	5.6000	20.2667
10	7.95735	5.59881	20.26170	7.99778	5.37600	20.26665	7.9990	5.6000	20.2667
12	7.95140	5.59891	20.26200	7.99246	5.37598	20.26661	7.9965	5.6003	20.2668
14	7.92685	5.59908	20.26130	7.97219	5.37580	20.26608	7.9965	5.6003	20.2668

Table 5. Computation for $L_2,L_{\mathrm{\infty }}$ error norms of Example-1$\mathrm{ }$ for $N=$500 and 1000 and Δt = 0.05.

Partition	Error	$t$ =1	$t$ =1.5	$t=$ 2	$t$ =2.5
$N=$500	$L_2$	5.3427e-05	8.0059e-05	1.0661e-04	1.3305e-04
	$L_{\mathrm{\infty }}$	1.9588e-06	2.7549e-06	3.9375e-06	5.6035e-06
$N=1000$	$L_2$	5.3428e-05	8.0062e-05	1.0663e-04	1.3306e-04
	$L_{\mathrm{\infty }}$	1.9927e-06	2.9010e-06	4.1471e-06	5.8647e-06

Example 2.

Employ $\lambda =1$ and $\beta =0\mathrm{ }$ in the governing equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) in the space domain [−10] and time domain [1, 5] with $\mathrm{w}\left(\mathrm{x},0\right)={\mathrm{e}}^{{\text{-}\mathrm{x}}^2}$ (Arora et al., 2020; Zarebna & Parvaz, 2020). The numerical solution is reported in Table 6 with N = 100 and $\Delta t=0.0$1 at spatial domain $x=-6,-4,$ −2, 0,2,4,6. at time levels $t=1,2,3,$ 4 and 5. For $x=-6,-4,$ −2 and 0 all the numerical values are with negative sign and for $\mathrm{ }x=$0,2,4 and 6 they are with positive sign as shown in Table 6. Table 7 reports the approximate solution of present method with (Zarebna & Parvaz, 2020) at time levels $t=1,4,7$ and 10 at spatial domain $x=-5,0$ and 5. Table 8 reports the approximate result for $t=1,2,3$ and 4 at spatial domain $x=-8.-4,0,$ 4 and 8 with N = 500 and $\Delta \mathrm{t}=0.01.$ For node $x=-8\mathrm{ }\mathit{and}-4\mathrm{ }$ all the approximate results are with negative sign and for node $\mathrm{ }x=$0,4 and 8 they are with positive sign as displayed in Table 8. Figure 5 depicts the approximate results for N = 500 with $\Delta t=0.05\mathrm{ }$ at time levels $t=1,2,3,4,5,6,7,8,9\mathrm{ }\mathit{and}\mathrm{ }1.$ For more visibility, the numerical solutions with $\mathrm{ }\Delta t=0.01$ and N = 400 at $t=2,4,6,8$ are surfed in Figures 6(a), 6(b), 6(c) and 6(d) respectively.

Figure 6. 3-D solution profile of individual numerical solution of Example 2.

Table 6. Numerical computation of soliton of Example 2 with N = 300 and $\Delta t=0.0$1.

	$x=-6$	$-4$	$-2$	$0$	$2$	$4$	$6$
	All the values are with $(-)$ sign
$t=1$	0.000401407199036673	0.00296926071540682	0.0137853553459947	0.572751292927330	0.274509933438081	0.0777159540147300	0.0197903504300199
2	0.000384159167862869	0.00284746387335175	0.0173436311409643	0.286989033752241	0.358758642371988	0.175392656056146	0.0656775408289371
3	0.000284790057354907	0.00211993103547228	0.0139653235369269	0.132001264271730	0.321767815829540	0.243249700247327	0.125000097703736
4	0.000192022267456392	0.00143967421158459	0.00985540035771939	0.0555421709038957	0.240064176271104	0.261377306737874	0.177844713102755
5	0.000123404468588533	0.000935106688780686	0.00655408687641935	0.0201696314501513	0.160588717493677	0.238522463907760	0.208661367257681

Table 7. Comparison of approximate soliton for Example 2 with N = 300 and $\mathrm{\Delta }t=0.01$ with (Zarebna & Parvaz, 2020).

Method	time	$x=-5$	$x=0$	$x=0$
Current	1	−0.00109238399373050	0.572751314989190	0.0396681211805501
	4	−0.000528271685838057	0.0555423977601240	0.225218897847576
	7	−0.000133794912337084	−0.000907991236768601	0.179543488485821
	10	−2.98537503520385e-05	−0.00218832743049637	0.0736309183741807
Zarebna and Parvaz (2020)	1	−0.0010923500	0.5727480000	0.0396689000
	4	−0.0005281950	0.0555430000	0.2252390000
	7	−0.0001337230	−0.0009074750	0.1795830000
	10	−0.0000298137	−0.0021879900	0.0736685000

Table 8. Computational approach of Example 2.

t x	−8	−4	0	4	8
	All solutions are with $(-)$ sign
1	5.21333916471900e-05	0.00296888713460812	0.572761388045681	0.0776979215841071	0.00455903219193619
2	4.82689873458118e-05	0.00284715449485855	0.286981241066854	0.175376917930275	0.0205492425444438
3	3.42680974029302e-05	0.00211970208249819	0.131986158128780	0.243258423667514	0.0498848222167189
4	2.19172938772908e-05	0.00143950495578777	0.0555280899061477	0.261414870926295	0.0881030718545412

Example 3.

Consider the problem in the in the spatial domain [−40, 60] and $t\le 20$ with $\beta =0\text{ and λ}=1$ in the governing equation (1.1)(1.1) $$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=0,\mathrm{ }x\in [a,b],\mathrm{ }t\in [0,T]\mathrm{ }$$(1.1) . The exact solution (Arora et al., 2020) is given by $w(x,t)=3\mathrm{c}{\mathit{sech}}^2[\phi (x-\delta t)].$ Where $\phi =\frac{1}{2}\sqrt{c/\delta },$ and wave velocity $\delta =1+c$. Table 9 reports the $L_2\mathrm{ }\mathit{and}\mathrm{ }{L}_{\mathrm{\infty }}$ error norms for N = 100, 500 and 1000 respectively at time levels t = 2, 4, 6, 8 and 10 with $c=0.1.$ Table 10 studies the $I_1,I_2\text{ and }{I}_3$ invariants at t = 2, 4, 6, 8 for shake of Zarebnia and Parvaz (2016), Gardner et al. (1995] and Al-Khaled et al. (2005) with $c=0.1\mathrm{ }.\mathrm{Table}\mathrm{ }11$ executes the comparative study of $L_2,L_{\mathrm{\infty }}$ by Zarebnia and Parvaz (2016), Arora et al. (2014), Gardner et al. (1995), Kutluay and Esen (2006), and Arora et al. (2020) with present method for N = 100 and $\Delta \mathrm{t}=0.01$ and $c={10}^{-1}$. The results are found to be more accurate in present scheme. Figures 7(a) and 7(b) surf the numerical and exact solution with $\mathrm{ }\Delta \mathrm{t}$ = 0. 01 and N = 100 at t = 6 respectively. Figure 8 depicts comparative study of approximate and exact results at time levels t = 2, 4,6,8 and 10 for N = 100 and $\Delta \mathrm{t}=0.01$.

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

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

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

$t$	$N=100,\Delta t=0.01$		N = 500 $\Delta t=0.05$		N = 1000 $\Delta t=0.08$
	${\mathrm{ }L}_2$	${\mathrm{ }L}_{\mathrm{\infty }}$	$L_2$	${\mathrm{ }L}_{\mathrm{\infty }}$	$L_2$	$L_{\mathrm{\infty }}$
$2$	2.1705e-07	2.4734e-08	5.4006e-06	2.7363e-08	1.3824e-05	2.6749e-08
$4$	4.2940e-07	4.2500e-08	1.0784e-05	5.4352e-08	2.7603e-05	5.4595e-08
6	6.4795e-07	7.3901e-08	1.6132e-05	1.0406e-07	4.1294e-05	1.0633e-07
8	8.5275e-07	1.5091e-07	2.1432e-05	2.0016e-07	5.4859e-05	2.0574e-07
10	1.1179e-06	2.8770e-07	2.6672e-05	3.8694e-07	6.8268e-05	3.9840e-07

Table 10. Comparison of $I_1\mathrm{ }{I}_2$ and $I_3$ invariants of Example 3 $\mathrm{for}\mathrm{ }N=100,\Delta t=0.01$ at c = 0.1.

$t$	$\mathrm{ }N=100,\Delta t=0.01$			Gardner et al. (1995)			Zarebnia and Parvaz (2016)			Arora et al. (2020)
	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$	${\mathit{I}}_1$	${\mathit{I}}_2$	${\mathit{I}}_3$
2	3.979943	0.810462	2.57901	3.97993	0.810465	2.57901	3.97993	0.810461	2.57900	3.97994	0.810462	2.57901
4	3.979943	0.810462	2.57901	3.97993	0.810464	2.57901	3.97993	0.810462	2.57900	3.97994	0.810462	2.57901
6	3.97995	0.810462	2.57901	3.97992	0.8105	2.57901	3.97993	0.810462	2.57901	3.97995	0.810462	2.57901
8	3.97995	0.810462	2.57901	3.97993	0.8105	2.57901	3.97993	0.810462	2.57901	3.97995	0.810462	2.57901
10	3.97995	0.810462	2.57901	3.97993	0.8105	2.57901	3.97993	0.810462	2.57901	3.97995	0.810462	2.57901

Table 11. Comparison of $L_2,\mathrm{ }{L}_{\mathrm{\infty }}\mathrm{ }\mathit{error}\mathrm{ }\mathit{norms}\mathrm{ }of\mathrm{ }$ Example 3 with N = 100 and $\Delta t=0.01$ with $c=.$ ${10}^{-1}$

$t$	Present		Zarebnia and Parvaz (2016)		Arora et al. (2014)		Gardner et al. (1995)		Kutluay and Esen (2006)		Arora et al. (2020)
	$L_2$	$L_{\mathrm{\infty }}$	$L_2$	$L_{\mathrm{\infty }}$	$L_2$	$L_{\mathrm{\infty }}$	$L_2$	$L_{\mathrm{\infty }}$	$L_2$	$L_{\mathrm{\infty }}$	$L_2$	$L_{\mathrm{\infty }}$
4	4.29e-07	1.59e-07	1.49e-05	5.96e-06	4.20e-05	1.50e-05	3.98e-02	1.37e-02	1.20e-04	5.00e-05	4.31e-07	1.62e-07
8	8.52e-07	3.24e-07	2.71e-05	1.05e-05	3.30e-05	3.30e-05	7.95e-02	2.77e-02	2.30e-04	9.00e-05	8.51e-07	3.29e-07
12	1.25e-06	4.82e-06	3.64e-05	1.32e-05	1.30e-04	4.90e-05	1.19e-01	4.14e-02	3.40e-04	1.40e-04	1.27e-06	4.90e-07
16	1.65e-06	6.32e-07	4.29e-05	1.45e-05	1.60e-04	6.40e-05	1.58e-01	5.46e-02	4.50e-04	1.80e-04	1.68e-06	6.40e-07
20	2.03e-06	7.75e-07	4.76e-05	1.50e-05	2.00e-04	7.80e-05	1.96e-01	6.74e-02	5.50e-04	2.10e-04	2.07e-06	7.81e-07

Example 4.

Consider the non-homogeneous BBMB equation

$w_t-w_{\mathit{xxt}}-\beta w_{xx}+\lambda w_x+w{w}_x=e^{-t}\left(\mathit{cosx}-\mathit{sinx}+\frac{1}{2}{e}^{-t}\hspace{0.17em}\sin \hspace{0.17em}2x\right).$ For $\beta =\lambda =1\mathrm{ }\mathit{the}$ exact solution is $w\left(x,t\right)=e^{-t}s\mathit{inx}$ in the interval [0, $\pi$] and N = 121 (Arora et al., 2020; Zarebna & Parvaz, 2020). Table 12 reports the comparison of $L_2,L_{\mathrm{\infty }}$ error norms by Arora et al. (2014) and Arora et al. (2020) with present method for N = 121 and N = 140 respectively with $\Delta t=0.01$ in [0, $\pi$] at different times t = 1, 2,4 and 10. The results are found to be more accurate in present scheme. The distinguish on numerical solution with analytical at different times t = 0, 1,2,3 and 4 in [0, $\pi$] for N = 121 with $\Delta t=0.01$ is plotted in Figure 9.

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

Table 12. Computational error norms of present method with others of Example 3 at $\Delta t=0.01$ in [0, $\pi$].

$t$	(N = 121) (Arora et al., 2014)		(N = 400) (Arora et al., 2020)		Present method
	${\mathrm{ }\mathit{L}}_2$	${\mathrm{ }\mathit{L}}_{\mathrm{\infty }}$	${\mathrm{ }\mathit{L}}_2$	${\mathrm{ }\mathit{L}}_{\mathrm{\infty }}$	${\mathrm{ }\mathit{L}}_2$	${\mathrm{ }\mathit{L}}_{\mathrm{\infty }}$
1	5.13e-03	7.67e-03	1.51e-03	1.46e-03	1.4912e-03	1.4367e-03
2	1.73e-03	2.84e-03	1.21e-03	1.18e-03	1.1991e-03	1.1576e-03
4	2.12e-04	3.84e-04	3.95e-04	3.96e-04	2.015e-04	3.8125e-04
10	4.08e-06	4.06e-06	2.75e-06	3.22e-06	2.7056e-06	3.1604e-06

7. 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 $L_{\mathrm{\infty }}$ 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 $O(h^9+(\Delta {t)}^2)$ 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.

Acknowledgements

The authors would like to thank the anonymous referees, Managing Editor and Editor in Chief for their valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.