Time accurate solution to Benjamin–Bona–Mahony–Burgers equation via Taylor–Boubaker series scheme

Abstract

The object of this paper is to develop an accurate combined spectral collocation approach to numerically solve the generalized nonlinear Benjamin–Bona–Mahony–Burgers equation. The first stage is devoted to discretization in time, which is carried out with the aid of the well-known Taylor series expansions. Then the spectral collocation procedure based on the Boubaker polynomials is applied for the resulting discretized spatial operator in each time step. A detailed error analysis of the presented technique is carried out with regard to the space variable. The advantages of the hybrid technique are shown via performing several simulations through four test examples. Comparisons between our numerical results and the outcomes of some existing schemes indicate that the proposed technique is not only simple and easy-to-implement, but also sufficiently accurate using a moderate number of bases and a large time step.

1 Introduction

Diverse important physical phenomena in nature are mathematically described by means of (nonlinear) partial differential equations (PDEs) with appropriate initial and boundary conditions. Except for some particular cases, the closed-form solutions to the most PDEs either do not exist or are not intractable in practice. Thus developing an accurate approximate or numerical solution for nonlinear PDEs becomes practically important in the filed. In the past decades the subject has attracted many authors, and difference equations have been appeared as a promising research field on both applied and theoretical levels (see [1–17]).

The generalized Benjamin–Bona–Mahony–Burgers (BBMB) model (1) was first considered in [1]. Due to the importance and vast applications, many researchers have been considered the BBMB-type equations over the past decades. In this respect, diverse computational and approximation techniques were proposed in the literature. The Galerkin approach was applied to the Benjamin–Bona–Mahony (BBM) equations in [18–20]. The finite difference schemes with different combined methods in space and time are developed in [6, 7]. The B-spline collocation approaches are studied in [21–23]. The meshless-based methods were investigated in the previously published papers [24, 25]. The spectral collocation procedures based upon (orthogonal) basis functions were discussed in [26–29]. Additionally, a discussion of some other methods for solving PDEs can be found in [30–41]. In most of the proposed numerical solution algorithms for (13), the implemented time marching procedure has the first-order accuracy. In addition, to achieve a reasonable accuracy, the time step cannot be taken rather large. Alternatively, in this research work, we construct a second-order time advancement technique for the numerical treatment of the BBMB equation. Our efficient and accurate approach based upon a hybrid of Taylor method for the time discretization and exponential spectral Boubaker collocation scheme for the spatial operator. The presented hybrid technique is straightforward in implementation compared to other existing numerical models; see also [42–44].

Recently, the authors in [45] obtained new exact solitary solutions for a version of $(3+ 1)$-dimensional Wazwaz–Benjamin–Bona–Mahony equation formulated in the sense of conformable derivative with the help of two novel techniques: the generalized Kudryashov method and $\exp (-\phi (\aleph ))$ method. The general fractional formulation of the Wazwaz–Benjamin–Bona–Mahony equation can be expressed as follows:

$$\begin{aligned} \mathcal{D}_{t}^{\zeta }\Psi + \mathcal{D}_{x}^{\zeta } \Psi + \mathcal{D}_{y}^{\zeta }\Psi - \mathcal{D}_{xzt}^{\zeta } \Psi =0, \end{aligned}$$

where $\mathcal{D}^{\zeta }$ is the fractional operator of order $0 < \zeta \leq 1$, and the function $\Psi: [0, \infty ) \to \mathbb{R}$ is ζ-differentiable at a point t. Matar et al. [46], studied the following fractional differential equation:

$$\begin{aligned} \textstyle\begin{cases} \frac{{\mathrm{d}}}{{\mathrm{d}} t} ( \varphi _{p} ( {}^{C }\mathcal{D}^{ \alpha, \rho } x(t) ) ) = q (t, x(t), {}^{C }\mathcal{D}^{ \gamma, \rho } x(t) ),\quad 0 \leq t \leq 1, \\ x(0) + \mu x(1) = \theta _{1} (x(0), x(1)), \\ x'(1) = \theta _{2} (x(0), x(1)), \end{cases}\displaystyle \end{aligned}$$

where ${}^{C }\mathcal{D}^{ \alpha, \rho }$ and ${}^{C }\mathcal{D}^{ \gamma, \rho }$ are the generalized Caputo fractional derivatives of orders $0 < \alpha <1$ and $0 < \gamma < 1$, respectively, $\rho >1$, and s $q: [0,1] \times \mathbb{R}^{2} \to \mathbb{R}$ and $\theta _{i}: \mathbb{R}^{2} \to \mathbb{R}$, $i=1,2$, are given nonlinear continuous functions. They investigated possible solutions to the following fractional boundary value problem for an implicit nonlinear an implicit nonlinear fractional differential equation:

$$\begin{aligned} \textstyle\begin{cases} \mathcal{D}_{ 0^{+}}^{ q(t)} x(t) = \theta (t, x(t), \mathcal{D}_{ 0^{+}}^{ q(t)} x(t) ), \quad 0 \leq t \leq \Omega, \\ x(0) =0, \qquad x(\Omega ) = 0, \end{cases}\displaystyle \end{aligned}$$

where $\Omega >0$, $q: [0, \Omega ] \to (1. 2]$ and $\theta: [0, \Omega ]\times \mathbb{R}^{2} \to \mathbb{R}$ are continuous functions, and $\mathcal{D}_{ 0^{+}}^{ q(t)}$ is the Riemann–Liouville fractional derivative in the context of variable order $q(t)$ [47]. Matar et al. [48] consider the FDE

$$\begin{aligned} \textstyle\begin{cases} \frac{{\mathrm{d}}}{{\mathrm{d}} \mathfrak{t}} ( \varphi _{p} ( {{}^{C} \mathcal{D}}^{ \varsigma _{1}, \rho } \wp ( t) ) ) = \mathfrak{w} (t, \wp (t), {{}^{C} \mathcal{D}}^{\varsigma _{2}, \rho } \wp ( t) ),\quad t \in [0,1], \\ \wp (0) +\mu \wp (1) =\mathfrak{v}_{1} (\wp (0), \wp (1)), \quad \mu \neq 1, \\ \wp '(1) =\mathfrak{v}_{2} (\wp (0), \wp (1)), \end{cases}\displaystyle \end{aligned}$$

where ${{}^{C} \mathcal{D}}^{\varsigma _{1}, \rho }$ and ${{}^{C} \mathcal{D}}^{\varsigma _{2}, \rho } \wp (t )$ are generalized Caputo fractional derivatives of orders $1 < \sigma _{1}<2$ and $0 < \sigma _{2}<1$, respectively, $\rho >1$, $\varphi _{p} (p >1)$ is a p-Laplacian operator, and $\mathfrak{w}: [0,1]\times \mathbb{R}^{2} \to \mathbb{R}$ and $\mathfrak{v}_{1}, \mathfrak{v}_{2}: \mathbb{R}^{2} \to \mathbb{R}$ are given nonlinear continuous functions. Also, the authors in [49] found a theoretical method to investigate the existence of solutions for the strongly singular fractional model of thermostat control given as

$$\begin{aligned} \textstyle\begin{cases} {}^{C }\mathcal{D}^{ \omega } x(t) + q(t) \theta (x(t)) =0, \quad \omega >0, w \in (n-1, n], \\ x^{(j)}=0, \quad j=1,2,\dots, n-1, \\ ( p(t) x(t) )^{\prime }|_{t=1} + a x(\eta ) =0, \quad a>0, 0 < \eta < 1, \end{cases}\displaystyle \end{aligned}$$

where $(\omega - k - 1) p(1) > a \eta ^{k}$, with $j\neq k$ coupled, $q: [0,1 ] \to \mathbb{R}$ is singular or strongly singular at some points of $[0,1]$, $p: [0,1] \to [0, \infty )$ is differentiable at $t=1$, $\theta \in C(\mathbb{R}, \mathbb{R})$, and ${}^{C }\mathcal{D}^{ \omega }$ is the Caputo derivative of order ω. For more studied related application, see [50, 51]. Rezapour et al. investigated the multisingular integro-differential pointwise equation

$$\begin{aligned} \textstyle\begin{cases} { \mathcal{D}}_{\mathrm{q}}^{\varsigma } \wp (t) = \mathfrak{w} (t, \wp (t), \wp '(t), { \mathcal{D}}^{\gamma _{1} } \wp (t), { \mathcal{I}}^{\gamma _{1} } \wp (t) ), \quad t \in [0,1], \\ \wp '(0) = \wp (a), \\ \wp (1) = \int _{0}^{b} \wp (\xi ) {\,\mathrm{d}} \xi, \end{cases}\displaystyle \end{aligned}$$

$\wp ^{(j)} (0) = 0, j = 2, \dots, [\sigma ]-1$, where $\wp \in C^{1}([0,1])$, $\gamma _{1}, \gamma _{2} \in [2,\infty ), a, b \in (0,1)$, ${ \mathcal{D}}^{\varsigma }$ is the Caputo fractional q-derivative of order ς, and $\mathfrak{w}:[0,1] \times \mathbb{R}^{4} \to \mathbb{R}$ is a function such that $\mathfrak{w}(t,.,.,.,.)$ is singular at some points $0 \leq t\leq 1$ [52]. In 2021, Izadi [42] presented an effective approximation algorithm to solve the nonlinear Hunter–Saxton equation

$$\begin{aligned} \textstyle\begin{cases} w_{xt} + ww_{xx} + 0.5 w_{x}^{2} =0, \\ w|_{t=0} = w_{0}(x), \end{cases}\displaystyle \end{aligned}$$

subjected to the boundary condition $\lim_{x\to \infty } w(x, t) =0$. Abdelwahed and Chorf [53] consider the nonlinear heat equation: Find a solution φ of

$$\begin{aligned} \textstyle\begin{cases} \frac{\partial \varphi }{\partial t} - \operatorname{div} ( \lambda (\varphi ) \nabla \varphi )= f & \text{in } \Omega \times ]0, T[, \\ \varphi (x, t) = 0 & \text{on } \partial \Omega \times ]0, T[, \\ \varphi (x, 0) = \varphi _{0} & \text{in } \partial \Omega, \end{cases}\displaystyle \end{aligned}$$

where Ω is a bounded simply connected domain of $\mathbb{R}^{d}$ $(d= 1,2,3)$, ∂Ω is its connected Lipschitz continuous boundary, and T is a positive constant. They considered families of large eddy simulation models, which are variants of the classical Smagorinsky model, and similarly to the model of Cottet, Jiroveanu, and Michaux, proposed a selective model based on the local behavior of the angle of the vorticity direction [54].

The chief goal of this research paper is developing an effective time-accurate computational procedure for finding the solutions of a PDE model problem arising in broad branches of science and engineering. We consider a hybrid approximation technique to numerically treat the following BBMB equation:

$$\begin{aligned} \begin{aligned}& \frac{\partial w(x,t)}{\partial t}- \mu \frac{\partial ^{3} w(x,t) }{\partial t\partial x^{2}} - \gamma \frac{\partial ^{2} w(x,t)}{\partial x^{2}} + \sigma \frac{\partial w(x,t)}{\partial x} \\ &\quad{} + \eta w(x,t) \frac{\partial w(x,t)}{\partial x}=h(x,t), \end{aligned} \end{aligned}$$

(1)

where $x \in [x_{L},x_{R}]$ and $t\in [0, T_{f}]$. Also, two coefficient parameters $\mu,\gamma,\sigma$ are positive constants, $\eta \in \mathbb{R}$, and $h(x,t)$ is a familiar real-valued function. Along with this equation, an initial condition is given as

$$\begin{aligned} w(x,t=0) = w_{0}(x),\quad x\in [x_{L},x_{R}]. \end{aligned}$$

(2)

Moreover, with the initial-value problem (1)–(2), the following boundary conditions are supplemented:

$$\begin{aligned} \textstyle\begin{cases} w(x=x_{L},t) = w_{L}(t), \\ w(x=x_{R},t)=w_{R}(t), \end{cases}\displaystyle \end{aligned}$$

(3)

for $t\in [0,T_{f}]$, where $w_{L}(t)$ and $w_{R}(t)$ are two prescribed functions. If $\gamma =0$, then equation (1) is called the BBM equation, which first studied as a model for the propagation of long waves in nonlinear dispersive systems [2]. Some facts on physical significance of this model were given in [3, 5]. For $\mu =0$ and $\eta =-1$, the model (1) reduces to (generalized) Burgers equation [4].

Our plan in the rest of the paper is as follows. In the next Sect. 2, we illustrate the time advancement procedure for BBMB equation (1), which is relied on the Taylor series expansion. Afterward, in Sect. 3, we give an overview of the Boubaker functions together with some important their properties. In this section, we also discuss the convergence for this class of polynomials. The hybrid strategy of Taylor and Boubaker functions is illustrated in Sect. 4 in detail. The results of numerical simulations presented through tables and figures are given in Sect. 5. We end the paper with the conclusion Sect. 6.

2 Time-marching procedure

To gain an accurate time solution of the nonlinear BBMB equation (1), we first consider the Taylor expansion series to discretize it in time. For this purpose, we uniformly partition the time interval $[0,T_{f}]$ into J uniform subdivisions. The corresponding grid points and time-step Δt are given by

$$\begin{aligned} t_{0} = 0< t_{1} = \Delta t< \cdots < t_{J}=J \Delta t=T_{f}, \end{aligned}$$

where $\Delta t =t_{j}-t_{j-1}$, and $j=1,2,\ldots, J$. In what follows, by $w^{j}$ we denote the approximate solution at time level $t_{j}$, that is,

$$\begin{aligned} w^{j}(x)\approx w(x,t_{j}). \end{aligned}$$

We then evaluate the original BBMB equation (1) at time level $t_{j}$ to arrive at

$$\begin{aligned} \frac{\partial }{\partial t} \biggl(w^{j} - \mu \frac{\partial ^{2} w^{j}}{\partial x^{2}} \biggr) = \gamma \frac{\partial ^{2} w^{j}}{\partial x^{2}} - \sigma \frac{\partial w^{j}}{\partial x} - \eta w^{j} \frac{\partial w^{j}}{\partial x} + h(x, t_{j}). \end{aligned}$$

(4)

Let us for convenience define

$$\begin{aligned} u^{j}:= w^{j} - \mu \frac{\partial ^{2} w^{j}}{\partial x^{2}}. \end{aligned}$$

Application of the Taylor formula to $u^{j}$ yields

$$\begin{aligned} \frac{\partial u^{j}}{\partial t} = \frac{u^{j+1}-u^{j}}{\Delta t}- \frac{1}{2}\Delta t \frac{\partial ^{2} u^{j}}{\partial t^{2}}+ \mathcal{O}\bigl(\Delta t^{2}\bigr). \end{aligned}$$

(5)

Differentiating (4) with regard to t reveals

$$\begin{aligned} \frac{\partial ^{2} u^{j}}{\partial t^{2}} = \gamma \frac{\partial ^{2} w^{j}}{\partial t \partial x^{2}} - \sigma \frac{\partial ^{2} w^{j}}{ \partial t \partial x} - \eta \frac{\partial w^{j}}{\partial t} \frac{\partial w^{j}}{\partial x} - \eta w^{j} \frac{\partial ^{2} w^{j}}{\partial t \partial x} + \frac{ h(x,t_{j})}{\partial t}. \end{aligned}$$

Now we replace all first-order derivatives $\frac{\partial w^{j}}{\partial t}$ by the forward difference relation

$$\begin{aligned} \frac{w^{j+1}-w^{j}}{\Delta t}. \end{aligned}$$

After multiplying both sides by Δt and some manipulations, the resultant equation becomes

$$\begin{aligned} \Delta t \frac{\partial ^{2} u^{j}}{\partial t^{2}} = {}&\gamma \biggl( \frac{\partial ^{2} w^{j+1}}{\partial x^{2}} - \frac{\partial ^{2} w^{j}}{\partial x^{2}} \biggr) - \bigl( \sigma + \eta w^{j} \bigr) \biggl( \frac{\partial w^{j+1}}{\partial x} - \frac{ \partial w^{j}}{\partial x} \biggr) \\ & {}-\eta \frac{\partial w^{j}}{\partial x} \bigl({w^{j+1}-w^{j}} \bigr) +h^{j+1}-h^{j}, \end{aligned}$$

(6)

where $h^{j}:=h(x,t_{j})$. Our next task is to place (6) into the right-hand side of (5) followed by equating to (4). After multiplying by $2\Delta t$, we get the time-discretized equation for the nonlinear model (1) with second-order accuracy in time. Again for simplicity of notations, we introduce

$$\begin{aligned} \chi _{j+1}(x):=w^{j+1}(x). \end{aligned}$$

Similarly, we define

$$\begin{aligned} &a_{j}(x):=- (\gamma \Delta t+2\mu ), \\ &b_{j}(x):= \Delta t \bigl(\eta w^{j}(x)+ \sigma \bigr), \\ &c_{j}(x):=2+\Delta t \eta \frac{\partial w^{j}}{\partial x}(x), \\ &f_{j}(x):=2w^{j}(x)- [2\mu -\Delta t \gamma ] \frac{ \partial ^{2} w^{j}}{\partial x^{2}} - \Delta t \sigma \frac{\partial w^{j}}{\partial x} + \Delta t \bigl(h^{j+1}+h^{j}\bigr). \end{aligned}$$

Therefore the resulting second-order equation with regard to space variable can be written as

$$\begin{aligned} a_{j}(x) \chi ''_{j+1}(x)+b_{j}(x) \chi '_{j+1}(x)+c_{j}(x) \chi _{j+1}(x)=f_{j}(x),\quad x\in [x_{L},x_{R}], \end{aligned}$$

(7)

for $j=0,1,\ldots, J-1$. In each time level, we must solve the linear equation (7). So we need the given initial condition $w^{0}=w_{0}(x)$ and its first- and second-order derivatives as they appear in the coefficients of (7). In other words, for $j=0$, we have $\chi _{0}(x)=w_{0}(x)$ for $x\in [x_{L}, x_{R}]$. At each time level $t_{j+1}$, we exploit the boundary conditions obtained from (3) at two end-points $x=x_{L},x_{R}$ as follows:

$$\begin{aligned} \textstyle\begin{cases} \chi _{j+1}(x_{L}):=\chi _{L}^{j+1}=w_{L}(t_{j+1}), \\ \chi _{j+1}(x_{R}):=\chi _{R}^{j+1}=w_{R}(t_{j+1}), \end{cases}\displaystyle \end{aligned}$$

(8)

for $j=0,1,\ldots,J-1$.

3 Boubaker functions: an overview

The family of nonorthogonal Boubaker polynomials ${\mathcal{B}}_{\ell }(x)$ naturally appeared in the study of heat transfer equation when its solution can be expanded in terms of Bessel functions of the first kind [55]. These polynomials are very closely related to the Chebyshev polynomials ${\mathrm{U}}_{\ell }(x)$ of the second kind denoted by the relation

$$\begin{aligned} {\mathcal{B}}_{\ell }(x) = {\mathrm{U}}_{\ell } \biggl( \frac{x}{2} \biggr) + 3 { \mathrm{U}}_{ \ell -2} \biggl( \frac{x}{2} \biggr). \end{aligned}$$

These polynomials are obtained via the following recursive formula:

$$\begin{aligned} {\mathcal{B}}_{\ell }(x)=x{\mathcal{B}}_{\ell -1}(x)-{ \mathcal{B}}_{ \ell -2}(x),\quad \ell =3,4,\ldots, \end{aligned}$$

(9)

where the first three Boubaker functions are ${\mathcal{B}}_{0}(x)=1$, ${\mathcal{B}}_{1}(x)=x$, and

$$\begin{aligned} {\mathcal{B}}_{2}(x)=x^{2}+2. \end{aligned}$$

For $\ell >0$, these functions are the unique solutions of the second-order differential equation

$$\begin{aligned} & \bigl(x^{2}-1\bigr) \bigl(3\ell x^{2}+\ell -2 \bigr)y'' +3x\bigl(\ell x^{2}+3\ell -2 \bigr)y' \\ &\quad{} -\ell \bigl(3\ell ^{2} x^{2}+\ell ^{2}-6\ell +8 \bigr)y=0 \quad (\ell \in \mathbb{N}), \end{aligned}$$

(10)

where $y(x)= {\mathcal{B}}_{\ell }(x)$. Furthermore, the explicit series solution of Boubaker polynomial ${\mathcal{B}}_{\ell }(x)$ of degree ℓ is given by

$$\begin{aligned} {\mathcal{B}}_{\ell }(x)=\sum _{r=0}^{\lfloor \frac{\ell }{2}\rfloor }(-1)^{r} \binom{\ell -r}{r} \frac{\ell -4r}{\ell -r} x^{\ell -2r}\quad (\ell >1). \end{aligned}$$

(11)

The next theorem establishes the distribution of the zeros of these polynomials on $[-2,2]$; see [56, Theorem 2.4].

Theorem 3.1

The function ${\mathcal{B}}_{\ell }(x)$ of degree ℓ has has $\ell -2$ real zeros and two nonreal, purely imaginary complex zeros. All zeros of ${\mathcal{B}}_{\ell }(x)$ are located in $[-2,2]$, and the two purely imaginary zeros lie outside the unit circle.

Remark 3.1

Based on the observation established in Theorem 3.1, the shifted version of these polynomials were previously considered on an arbitrary domain $[x_{L},x_{R}]$ in [57]. This can be accomplished trough the change of variable

$$\begin{aligned} \eta =\frac{4x-2(x_{L}+x_{R})}{x_{R}-x_{L}}, \end{aligned}$$

where $x\in [x_{L},x_{R}]$ and $\eta \in [ -2, 2 ]$. Thus we get the shifted version of these polynomial by using ${\mathcal{B}}_{\ell }^{*}(\eta )={\mathcal{B}}_{\ell }(x)$. In the numerical examples, we may appropriately use the shifted Boubaker polynomials.

3.1 Convergence results

To proceed, we define $\Omega =[x_{L},x_{R}]$ and the related weighted $L_{2}$ space as [57]

$$\begin{aligned} L_{2,\omega }(\Omega ) = \bigl\{ s: \Omega \to \mathbb{R} | s \text{ is measurable, and } \Vert s \Vert _{\omega }< \infty \bigr\} , \end{aligned}$$

where

$$\begin{aligned} \Vert s \Vert ^{2}_{\omega } = \int _{x_{L}}^{x_{R}} \bigl\lvert s(x)\bigr\rvert ^{2} \omega (x) {\,\mathrm{d}}x \end{aligned}$$

with $\omega (x)=\frac{1}{x_{R}-x_{L}}$ denotes the induced norm resulting from the following inner product of the space $L_{2,\omega }(\Omega )$:

$$\begin{aligned} \bigl\langle s(x),p(x)\bigr\rangle _{\omega }= \int _{x_{L}}^{x_{R}} s(x) p(x) \omega (x) {\, \mathrm{d}}x. \end{aligned}$$

In practice, we consider a finite-dimensional subspace of $L_{2,\omega }(\Omega )$ of the form

$$\begin{aligned} \mathbb{L}_{M} = \operatorname{span} \bigl\langle {\mathcal{B}}_{0}^{*}(x),{ \mathcal{B}}_{1}^{*}(x),\ldots,{\mathcal{B}}_{M}^{*}(x) \bigr\rangle . \end{aligned}$$

Clearly, $\dim (\mathbb{L}_{M})=M+1$, and $\mathbb{L}_{M}$ is a closed and thus complete subspace of $L_{2,\omega }(\Omega )$. Thus any function $u\in L_{2,\omega }(\Omega )$ has a unique best approximation $u^{*}\in \mathbb{L}_{M}$ in the sense that

$$\begin{aligned} \bigl\Vert u(x)-u^{*}(x) \bigr\Vert _{\omega }\leq \bigl\Vert u(x)-v(x) \bigr\Vert _{\omega } \quad\forall v \in \mathbb{L}_{M}. \end{aligned}$$

(12)

We further invoke the following result from the approximation theory [58].

Theorem 3.2

Let u be an M times continuously differentiable function on Ω. Also, let $P_{M}$ denote the interpolating function of u at M Chebyshev nodes in the interval Ω. Then, for every $x\in \Omega$, we have

$$\begin{aligned} \bigl\lvert u(x)-P_{M}(x)\bigr\rvert \leq \frac{(x_{R}-x_{L})^{M} \Vert u \Vert _{\infty }}{2^{2M-1}M!}, \end{aligned}$$

where $\|u\|_{\infty }: = \max_{ x \in \Omega } \vert u^{(M)}(x) \vert$.

Now assume that an arbitrary function $u\in L_{2,\omega }(\Omega )$ can be written in terms of shifted Boubaker functions as

$$\begin{aligned} u(x)=\sum_{\ell =0}^{\infty } {\lambda }_{\ell } {\mathcal{B}}_{\ell }^{*}(x). \end{aligned}$$

By restricting our attention to the finite subsets $\mathbb{L}_{M}$ of $L_{2,\omega }(\Omega )$ we may write a truncated series for u as

$$\begin{aligned} u(x)\approx u_{M}(x)=\sum_{\ell =0}^{M} {\lambda }_{\ell } { \mathcal{B}}_{\ell }^{*}(x). \end{aligned}$$

The next theorem provides an error bound for $E_{M}(x)=u(x)-u_{M}(x)$.

Theorem 3.3

Let

$$\begin{aligned} u\in C^{(M+1)}(\Omega )\cap L_{2,\omega }(\Omega ). \end{aligned}$$

Let $u_{M}$ be the best approximation to u in the space $\mathbb{L}_{M}$ in the sense of (12). Then we have

$$\begin{aligned} \bigl\Vert E_{M}(x) \bigr\Vert _{\omega } \to 0 \end{aligned}$$

as $M \to \infty$.

Proof

Since $u_{M}$ is the best approximation belonging to $\mathbb{L}_{M}$, we obtain

$$\begin{aligned} \bigl\Vert E_{M}(x) \bigr\Vert ^{2}_{\omega } = \bigl\Vert u(x)-u_{M}(x) \bigr\Vert ^{2}_{\omega } \leq \bigl\Vert u(x) - v(x) \bigr\Vert ^{2}_{\omega } \quad\forall v \in \mathbb{L}_{M}. \end{aligned}$$

The foregoing inequality is valid in particular for $v=P\in \mathbb{L}_{M}$. We therefore conclude that

$$\begin{aligned} \bigl\Vert E_{M}(x) \bigr\Vert ^{2}_{\omega } & \leq \bigl\Vert u(x)-P_{M+1}(x) \bigr\Vert ^{2}_{\omega } = \int _{\Omega } \bigl\lvert u(x)-P_{M+1}(x) \bigr\rvert ^{2} \omega (x) { \,\mathrm{d}}x. \end{aligned}$$

According to Theorem 3.2 with $(M+1)$ nodes, we get

$$\begin{aligned} \bigl\Vert E_{M}(x) \bigr\Vert ^{2}_{\omega } & \leq \int _{x_{L}}^{x_{R}} \biggl\lvert \frac{ \Vert u \Vert _{\infty } (x_{R}-x_{L})^{M+1}}{(M+1)! 2^{2M+1}} \biggr\rvert ^{2} \omega (x) {\,\mathrm{d}}x \\ & \leq \biggl[ \frac{ \Vert u \Vert _{\infty } (x_{R}-x_{L})^{M+1}}{(M+1)! 2^{2M+1}} \biggr]^{2}. \end{aligned}$$

The proof is finished by tending M to infinity. □

4 The hybrid procedure

The discretization of the BBMB equation (1) with regard to time is already carried out by relation (7). Also, the given boundary conditions (3) are converted to the boundary conditions (8) for the second-order differential equation (7). So the main objective is to treat the initial-boundary value problem (7)–(8) numerically with respect to the space variable x. Suppose that the solutions $\chi _{j+1}(x)$ of (7) at each time level j can be written as combinations of the (shifted) Boubaker functions defined by (11). Let assume that ${\mathcal{X}}_{j,M}(x)$ are the computed Boubaker approximations to $\chi _{j}(x)$ at the time steps $t_{j}$. To start, we need $\chi _{0}(x)$, which can be derived from the given initial condition $w_{0}(x)$. In the next time step $t_{j+1}$, we look for the approximate solution ${\mathcal{X}}_{j+1,M}(x)$ for $j=0,1,\ldots,J-1$ as follows:

$$\begin{aligned} {\mathcal{X}}_{j+1,M}(x)=\sum _{\ell =0}^{M} {\lambda }^{j+1}_{\ell } { \mathcal{B}}_{\ell }(x),\quad x\in \Omega. \end{aligned}$$

(13)

Here our ultimate goal is to find the unknowns ${\lambda }^{j+1}_{\ell }$ for $\ell =0,1, \ldots, M$ at time level $j\geq 0$. For convenience, we set

$$\begin{aligned} &\pmb{O}_{M}(x) = \bigl[ {\mathcal{B}}_{0}(x)\quad { \mathcal{B}}_{1}(x) \quad\ldots \quad{\mathcal{B}}_{M}(x) \bigr], \\ &\pmb{{\Lambda }}^{j+1}_{M} = \bigl[ \lambda _{0}^{k+1} \quad\lambda _{1}^{k+1}\quad \ldots\quad \lambda _{M}^{k+1} \bigr]^{T}. \end{aligned}$$

With the aid of these vectors, the $(M+1)$-term finite expansion series (13) can be expressed compactly in the matrix form as

$$\begin{aligned} {\mathcal{X}}_{j+1,M}(x)=\pmb{O}_{M}(x) \pmb{{\Lambda }}^{j+1}_{M}. \end{aligned}$$

(14)

After defining the vector of monomials

$$\begin{aligned} \pmb{V}_{M}(x)= \bigl[1 \quad x\quad x^{2} \quad\ldots\quad x^{M} \bigr], \end{aligned}$$

we further decompose the vector of Boubaker bases as

$$\begin{aligned} \pmb{O}_{M}(x)=\pmb{V}_{M}(x) \pmb{N}_{M}, \end{aligned}$$

(15)

where the upper-triangular matrix

$$\begin{aligned} \pmb{N}_{M}=(n_{i,j})_{i=j-2 \lfloor \frac{j}{2} \rfloor,j=2}^{M}\quad (i\leq j) \end{aligned}$$

has the following elements:

$$\begin{aligned} n_{j,i}:= \textstyle\begin{cases} (-1)^{\frac{i-j}{2}} (\frac{4j-2i}{i+j} ) \frac{\Gamma ( \frac{i+j}{2}+1 )}{\Gamma (j+1) \Gamma ( \frac{i-j}{2}+1 ) } & \text{if } (i-j) \text{ is even}, \\ 0 & \text{if } (i-j) \text{ is odd or } i>j, \\ 1 & \text{if } i=j, \end{cases}\displaystyle \end{aligned}$$

where the only nonzero elements on the first and second columns are $n_{0,0},n_{1,1}=1$. For example, for $M=6$, we get

$$\begin{aligned} \pmb{N}_{6}= \begin{pmatrix} 1 & 0 & 2 & 0 & -2 & 0&2 \\ 0 & 1 & 0 & 1 & 0 &-3&0 \\ 0 & 0 & 1 & 0 & 0 &0 &-3 \\ 0 & 0 & 0 & 1 & 0 &-1&0 \\ 0 & 0 & 0 & 0 & 1 &0 &-2 \\ 0 & 0 & 0 & 0 & 0 &1 &0 \\ 0 & 0 & 0 & 0 & 0 &0 &1 \end{pmatrix} . \end{aligned}$$

We note that determinant of $\pmb{N}_{M}$ is equal to unity. Ultimately, we requires a set of collocation points on $[x_{L},x_{R}]$ to acquire an approximate solution of the discretized model problem (7) in the form (9). In this respect, we use the zeros of the shifted Chebyshev polynomials on $[x_{L},x_{R}]$ given by

$$\begin{aligned} x_{\tau }=\frac{x_{L}+x_{R}}{2} + \frac{x_{R}-x_{L}}{2} s_{\tau },\quad \tau =0,1,\ldots,M, \end{aligned}$$

(16)

where $s_{\tau }$ are roots of Chebyshev functions of degree $(M+1)$ on $(-1,1)$. Further, we combine relationships (14) and (15) to express the approximate solution ${\mathcal{X}}_{j+1,M}(x)$ in (13) in a compact form as

$$\begin{aligned} {\mathcal{X}}_{j+1,M}(x)=\pmb{O}_{M}(x) \pmb{{\Lambda }}^{j+1}_{M}= \pmb{V}_{M}(x) \pmb{N}_{M} \pmb{{\Lambda }}^{j+1}_{M}. \end{aligned}$$

(17)

Using points of collocation (16) and putting them into (17), we obtain

$$\begin{aligned} {\pmb{X}}_{j+1}=\pmb{V} \pmb{N}_{M} \pmb{{\Lambda }}^{j+1}_{M}, \qquad{\pmb{X}}_{j+1}= \begin{pmatrix} {\mathcal{X}}_{j+1,M}(x_{0}) \\ {\mathcal{X}}_{j+1,M}(x_{1}) \\ \vdots \\ {\mathcal{X}}_{j+1,M}(x_{M}) \end{pmatrix} , \qquad\pmb{V}= \begin{pmatrix} \pmb{V}_{M}(x_{0}) \\ \pmb{V}_{M}(x_{1}) \\ \vdots \\ \pmb{V}_{M}(x_{M}) \end{pmatrix} . \end{aligned}$$

(18)

The next objective would be finding a connection between $\pmb{V}_{M}(x)$ and its derivatives. We can show that the derivatives of $\pmb{V}_{M}(x)$ can be stated in terms of differentiation matrix $\pmb{D}$ as

$$\begin{aligned} \frac{d^{k}}{dx^{k}}\pmb{V}_{M}(x)=\pmb{V}_{M}(x) \pmb{E},\qquad \pmb{E}= \begin{pmatrix} 0 & 1 & 0 &\ldots & 0 \\ 0 & 0 & 2 &\ldots & 0 \\ \vdots & \vdots & \ddots &\vdots & \vdots \\ 0 & 0 & 0 &\ddots & M \\ 0 & 0 & 0 & \ldots & 0 \end{pmatrix} _{(M+1)\times (M+1)}, \end{aligned}$$

(19)

for $k=1,2$. Twice differentiation of relation (17) with respect to x and using the former relation (19) reveal that

$$\begin{aligned} \textstyle\begin{cases} \chi '_{j+1}(x)\approx {\mathcal{X}}^{(1)}_{j+1,M}(x)=\pmb{V}_{M}(x) \pmb{E} \pmb{N}_{M} \pmb{{\Lambda }}^{j+1}_{M}, \\ \chi ''_{j+1}(x)\approx {\mathcal{X}}^{(2)}_{j+1,M}(x)=\pmb{V}_{M}(x) \pmb{E}^{2} \pmb{N}_{M} \pmb{{\Lambda }}^{j+1}_{M}. \end{cases}\displaystyle \end{aligned}$$

(20)

Now it is sufficient to insert the points of collocations (16) into the last formulas to obtain the following matrix expressions for the first and second derivatives in (7):

$$\begin{aligned} & {{{\pmb{X}}}^{(1)}_{j+1}}=\pmb{V} \pmb{E} \pmb{ N}_{M} \pmb{{\Lambda }}^{j+1}_{M},\qquad {{ \pmb{X}}}^{(1)}_{j+1}= \begin{pmatrix} {\mathcal{X}}^{(1)}_{j+1,M}(x_{0}) \\ {\mathcal{X}}^{(1)}_{j+1,M}(x_{1}) \\ \vdots \\ {\mathcal{X}}^{(1)}_{j+1,M}(x_{M}) \end{pmatrix} , \end{aligned}$$

(21)

$$\begin{aligned} & {{{\pmb{X}}}^{(2)}_{j+1}}=\pmb{V} \pmb{E}^{2} \pmb{N}_{M} \pmb{{\Lambda }}^{j+1}_{M},\qquad {{\pmb{X}}}^{(2)}_{j+1}= \begin{pmatrix} {\mathcal{X}}^{(2)}_{j+1,M}(x_{0}) \\ {\mathcal{X}}^{(2)}_{j+1,M}(x_{1}) \\ \vdots \\ {\mathcal{X}}^{(2)}_{j+1,M}(x_{M}) \end{pmatrix} . \end{aligned}$$

(22)

If we substitute the approximate solution ${\mathcal{X}}_{j+1,M}(x)$ and its two derivatives

$$\begin{aligned} {\mathcal{X}}^{(1)}_{j+1,M}(x),\qquad {\mathcal{X}}^{(2)}_{j+1,M}(x) \end{aligned}$$

into (7), then the resulting equation for $j=0,1,\ldots,J-1$ is

$$\begin{aligned} a_{j}(x) {\mathcal{X}}^{(2)}_{j+1,M}(x)+b_{j}(x) {\mathcal{X}}^{(1)}_{j+1,M}(x)+c_{j}(x) { \mathcal{X}}_{j+1,M}(x)=f_{j}(x) \end{aligned}$$

(23)

for $x\in [x_{L},x_{R}]$. Inserting the collocation points inserted into the foregoing equation followed by writing it in a matrix representation yields

$$\begin{aligned} \pmb{A} {{\pmb{X}}}^{(2)}_{j+1}+\pmb{B} {{ \pmb{X}}}^{(1)}_{j+1}+ \pmb{C} {\pmb{X}}_{j+1}= \pmb{F}_{j}, \quad j=0,1,\ldots,J-1, \end{aligned}$$

(24)

where we have used the coefficient matrices $\pmb{A}_{j}\ (\pmb{B}_{j} \text{ or } \pmb{C}_{j})$ of size $(M+1)\times (M+1)$ and the vectors $\pmb{F}_{j}$ of size $(M+1)\times 1$ defined as

$$\begin{aligned} &\pmb{A}_{j}(\pmb{B}_{j} \text{ or } \pmb{C}_{j}) = \begin{pmatrix} a_{j}(b_{j} \text{ or } c_{j})(x_{0}) & 0 &\ldots & 0 \\ 0 & a_{j}(b_{j} \text{ or } c_{j})&\ldots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 & \ldots & a_{j}(b_{j} \text{ or } c_{j})(x_{M}) \end{pmatrix}, \\ &\pmb{F}_{j} = \begin{pmatrix} f_{j}(x_{0}) \\ f_{j}(x_{1}) \\ \vdots \\ f_{j}(x_{M}) \end{pmatrix}. \end{aligned}$$

Based on the matrix forms derived in relations (18)–(22), we get the fundamental matrix equation

$$\begin{aligned} \pmb{Y}_{j} \pmb{{\Lambda }}^{j+1}_{M}= \pmb{F}_{j},\qquad [\pmb{Y}_{j}; \pmb{F}_{j}], \end{aligned}$$

(25)

where

$$\begin{aligned} \pmb{Y}_{j}:= \bigl\{ \pmb{A}_{j} \pmb{V} \pmb{E}^{2}+\pmb{B}_{j} \pmb{V} \pmb{E}+\pmb{C}_{j} \pmb{V} \bigr\} \pmb{N}_{M}. \end{aligned}$$

To obtain the unknowns $\lambda _{0}^{j+1}, \lambda _{1}^{j+1}, \ldots, \lambda _{M}^{j+1}$, we may use any linear solver to solve the matrix equation (25) consisting of $M+1$ linear equations and $M+1$ unknowns. The incorporation of the boundary conditions (8) into the fundamental matrix equation (25) is still demanding to get a unique solution. By means of representation (15) the boundary conditions ${\mathcal{X}}_{j+1,M}(x_{L})=\chi _{L}^{j+1}$ and ${\mathcal{X}}_{j+1,M}(x_{R})=\chi _{R}^{j+1}$ can be written in the matrix notation:

After the replacements of two rows (first and last rows) of the augmented matrix $[\pmb{Y}_{j}; \pmb{F}_{j}]$ by two row matrices and , we get the linear algebraic system of equations

(26)

Ultimately, once the above linear system is solved, we get the unknown Boubaker coefficients in (13) or (17).

5 Graphical and experimental results

We performed some numerical experiments to testify the exactness of theoretical findings as well as the practicability and utility of the presented hybrid algorithm for the model problem (1) with initial and boundary conditions (2)–(3). Our calculations were performed utilizing Matlab 2021a on a laptop computer with a 2.2 GHz Intel Core i7 CPU and 16 GB memory.

Additionally, the numerical errors are assessed by defining the (pointwise) absolute errors at time step $t=t_{j}$ via

$$\begin{aligned} E_{j,M}(x): = \bigl\lvert w(x,t_{j}) - { \mathcal{X}}_{j, M}(x) \bigr\rvert ,\quad x\in [x_{L},x_{R}]\ (j=1,\ldots,J). \end{aligned}$$

The calculation of the $L_{2}$ and the $L_{\infty }$ error norms at $t=T_{f}$ are accomplished through defining

$$\begin{aligned} &\mathcal{E}_{\infty } :=\max_{x_{L}\leq x\leq x_{R}} E_{J,M}(x), \\ &\mathcal{E}_{2} := \biggl( \int _{x_{L}}^{x_{R}} \frac{[w(x,T_{f})-{\mathcal{X}}_{J,M}(x)]^{2}}{M+1}\,\mathrm{d}x \biggr)^{1/2}. \end{aligned}$$

Furthermore, the estimated rates of convergence of $\mathcal{E}_{\infty }$ and $\mathcal{E}_{2}$ are calculated as the the number bases and grid size are halved successively. Suppose that $\mathcal{E}_{\ell }\equiv \mathcal{E}_{\ell }(J,M)$ for $\ell =2,\infty$. Then the rates of convergence (ROCs) with regard to x and t are estimated as

$$\begin{aligned} &\operatorname{ROCx}_{\ell }:=\log _{2} \frac{\mathcal{E}_{\ell }(J,2M)}{\mathcal{E}_{\ell }(J,M)}, \\ &\operatorname{ROCt}_{\ell }:=\log _{2} \frac{\mathcal{E}_{\ell }(J,M)}{\mathcal{E}_{\ell }(2J,M)},\quad \ell =2, \infty. \end{aligned}$$

Test Problem 5.1

We first consider (1) using the following coefficients and right-hand side [7]: $\mu =1, \gamma =1, \sigma =1, \eta =1$,

$$\begin{aligned} h(x,t)=\pi e^{-2t} \sin (2\pi x)+2\pi e^{-t} \cos (2\pi x)-e^{-t} \sin (2\pi x). \end{aligned}$$

The given periodic initial and boundary conditions are

$$\begin{aligned} \textstyle\begin{cases} w(x,0)=\sin (2\pi x), \\ w(x_{L},t)=0, \\ w(x_{L},t)=0,\quad [x_{L},x_{R}]=[0,1]. \end{cases}\displaystyle \end{aligned}$$

The exact analytical solution takes the form

$$\begin{aligned} w(x,t)=e^{-t} \sin (2\pi x),\quad x\in [0,1], t\in [0,T_{f}]. \end{aligned}$$

We first consider $\Delta t=0.1$ and $T_{f}=1$. We use (13) with $M=8$ to obtain the following approximate solutions at the first and last time steps $t=\Delta t$ and $t=T_{f}$ for $0\leq x\leq 1$:

$$\begin{aligned} {\mathcal{X}}_{1,8}(x) = {}&{-}0.0253645 x^{8} +51.0847 x^{7} - 178.607 x^{6} \\ & {}+ 197.218 x^{5}-46.6584 x^{4} - 27.7309 x^{3} \\ & {}-1.00867 x^{2} + 5.72667 x - 3.48812\times 10^{-105} \end{aligned}$$

and

$$\begin{aligned} {\mathcal{X}}_{10,8}(x)={}&{-}0.210542 x^{8} +21.5219 x^{7} - 73.7469 x^{6} \\ &{} + 81.07 x^{5}-19.3787 x^{4} -11.1573 x^{3} \\ &{} -0.426105 x^{2} + 2.3278 x. \end{aligned}$$

The profile of the approximate solution using these parameters are presented in Fig. 1, left. In Fig. 1, right, we further present the achieved absolute errors $E_{j,M}(x)$ for $x\in [0,1]$, $\Delta t=0.1$, and $T_{f}=1$, evaluated at various time levels

$$t_{j} = j\Delta t, \quad j=1,2,\ldots,10. $$

In Tables 1 and 2, some comparisons are done to validate our computational results.

Figure 1

Graphs of approximate solution (a) and absolute errors (b) in Test Problem 5.1 for $\Delta t=0.1, M=8$, and $T_{f}=1$

Table 1 The results of $L_{\infty }$ error norms, CPU times, and the related spatial rate of convergence in Test Problem 5.1 with $\Delta t=0.01$ and diverse M

Table 2 The results of $L_{\infty }$ error norms, CPU times, and the related temporal rate of convergence in Test Problem 5.1 with $M=10$ and diverse Δt

For this purpose, the error norms in the $L_{\infty }$ evaluated at the final time $t=T_{f}$ are calculated. In Table 1 the spatial rate of convergence of the proposed hybrid technique for a moderate time step $\Delta t=0.01$ and some

$$M=2^{i},\quad i=1,2,3,4, $$

are reported. In addition, the elapsed CPU time (in seconds) are further presented in Table 1. It should be emphasized that in each case the required times for solving the modified linear system of equations (26) in all time steps are added together. Here we have 100 time steps. Moreover, analogue results obtained by the fourth-order difference method (FODM) [7] using the parameters $k=10^{-4}$ and different

$$\begin{aligned} h = \frac{0.1}{2^{i}},\quad i=0,1,2,3, \end{aligned}$$

are tabulated in Table 1. We can clearly see that the performance of the proposed hybrid technique is better than the FODM with less computational efforts and the number of bases.

The second-order accuracy of the present technique with respect to time variable is investigated in Table 2. In this respect, we use a fixed $M=10$ and various

$$\begin{aligned} \Delta t=\frac{1}{2^{i}},\quad i=1,2,3,4. \end{aligned}$$

The related outcomes of the existing FODM [7] are also presented in Table 2 for comparison. Obviously, our method with considerably larger time steps Δt shows its order of accuracy as $\mathcal{O}(\Delta t^{2})$ in comparison with FODM.

Finally, for this example, we go beyond the unit time interval and consider $T_{f}=3\pi$. In this case the approximate solutions at time $t=T_{f}$ is given by

$$\begin{aligned} {\mathcal{X}}_{94,10}(x)={}&0.000214278 x^{10} - 0.00364302 x^{9}+0.0137945 x^{8} \\ &{} -0.0196859 x^{7} +0.00787041 x^{6} + 0.00367164 x^{5} \\ &{} +0.000685197 x^{4}-0.00340879 x^{3} -0.00000189864 x^{2} \\ & {}+ 0.000503632 x- 5.32245\times 10^{-110}. \end{aligned}$$

Here we have used $\Delta t=0.1$ and $M=10$. The achieved absolute errors $E_{j,10}(x)$ at all time steps are depicted in Fig. 2.

Figure 2

Plot of numerical solutions in Test Problem 5.1 at diverse time instants $t = \ell \Delta t, \ell =1,2, \ldots, \lfloor 30\pi \rfloor$, for $\Delta t=0.1, M=10$, and $T_{f}=3\pi$

Test Problem 5.2

As the second test problem, let us consider [6, 27, 28] $\mu =1, \gamma =1, \sigma =1, \eta =1$,

$$\begin{aligned} h(x,t)=e^{-t} \cos (x)-e^{-t} \sin (x)+\frac{e^{-2t}}{2} \sin (2x). \end{aligned}$$

The initial and boundary conditions are taken for $x\in [x_{L},x_{R}]=[0,\pi ]$ and $t\in [0,T_{f}]$ as

$$\begin{aligned} \textstyle\begin{cases} w(x,0)=\sin (x), \\ w(x_{L},t)=0, \\ w(x_{R},t)=0.\end{cases}\displaystyle \end{aligned}$$

The exact solution for this model problem is

$$\begin{aligned} w(x,t)=e^{-t} \sin (x). \end{aligned}$$

Let us first consider $\Delta t=0.1$, $T_{f}=1,2,4$, and $T_{f}=10$. The approximated solutions using $M=8$ obtained at these $t=T_{f}$ are given as follows:

$$\begin{aligned} {\mathcal{X}}_{10,8}(x) ={}&9.3804\times 10^{-6} x^{8}-1.1566\times 10^{-4} x^{7}+1.0527\times 10^{-4} x^{6} \\ & {}+0.00293115 x^{5} + 1.3398\times 10^{-4} x^{4} - 0.0614951 x^{3} \\ & {}- 2.3178\times 10^{-5} x^{2} + 0.367838 x + 1.70\times 10^{-8}, \\ {\mathcal{X}}_{20,8}(x) ={}&3.8074\times 10^{-6} x^{8}-4.7055\times 10^{-5} x^{7}+5.8875\times 10^{-5} x^{6} \\ &{} +1.0406\times 10^{-3} x^{5} + 8.5359\times 10^{-5} x^{4} - 0.02265 x^{3} \\ &{} -1.0890\times 10^{-6} x^{2} + 0.135264 x, \\ {\mathcal{X}}_{40,8}(x) ={}&5.6989\times 10^{-7} x^{8} - 7.2805 \times 10^{-6} x^{7} + 1.3162 \times 10^{-5} x^{6} \\ &{} +1.2898\times 10^{-4} x^{5} + 2.4261\times 10^{-5} x^{4} - 0.00307087 x^{3} \\ &{} -1.4795\times 10^{-5} x^{2}+ 0.0182886 x \end{aligned}$$

and

$$\begin{aligned} {\mathcal{X}}_{100,8}(x) ={}&3.3011\times 10^{-10} x^{8} - 7.4345 \times 10^{-9} x^{7} + 1.2907 \times 10^{-8} x^{6} \\ &{} +3.3227\times 10^{-7} x^{5} + 5.6666\times 10^{-8} x^{4} - 7.7921 \times 10^{-6} x^{3} \\ &{} -1.5728\times 10^{-7} x^{2} + 4.5477\times 10^{-6} x. \end{aligned}$$

Graphics of the whole approximate solutions on $(x,t)\in [0,\pi ]\times [0,T_{f}]$ using various $T_{f}=1,2,4,10$ are presented in Figs. 3and 4. The maximum absolute errors in all four cases is are than $\mathcal{E}_{\infty }=5\times 10^{-5}$. However, to get more accurate results, we may use a smaller time step Δt or increase the number of bases M. To show this fact, we utilize $M=8$ and set $\Delta t=0.01$. The snapshots of absolute errors at all time steps

$$t=s\Delta t, \quad s=1,\ldots,1000, $$

are visualized in Fig. 5.

Figure 3

Plots of approximate solution for $T_{f}=1$ (a) $T_{f}=2$ (b) in Test Problem 5.2 for $\Delta t=0.1, M=8$

Figure 4

Plots of approximate solution for $T_{f}=4$ (left) $T_{f}=10$ (right) in Test Problem 5.2 for $\Delta t=0.1, M=8$

Figure 5

Plot of absolute errors in Test Problem 5.2 at different time instants $t=\ell \Delta t,\ell =1,2,\ldots,10^{3}$, for $\Delta t=0.01, M=8$, and $T_{f}=10$

A comparison of the outcomes obtained by the present Taylor–Boubaker method (TBM) with the outcomes of some existing approaches are carried out in the next experiments. For this purpose, we exploit the quartic B-spline collocation method (QSCM) [21], the nonpolynomial spline method (NPSM) [22], and the improvised cubic B-spline collocation method (ICSCM) [23].

The results using fixed $M=15$ and $\Delta t=0.01$, and diverse values of $T_{f}=1,2,4,10$ are listed in Table 3. Clearly, our approach with less computational efforts is more accurate than the QSCM/ICSCM and is comparable with the NPSM.

Table 3 The comparison results of the $L_{\infty }$ and $L_{2}$ error norms for Test Problem 5.2 using $M=15$, $\Delta t=0.01$, and various $T_{f}$

We next fix $M=12$ and vary Δt as $1/2,1/4,1/8,1/16$. The estimated order of accuracy in time ($\operatorname{ROCt}_{\infty }$) is investigated at time $T_{f}=10$ for this Test Problem as well. The results of achieved $\mathcal{E}_{\infty }$ error norms, together with related $\operatorname{ROCt}_{\infty }$, are shown in Table 4, where a further comparison with some other available computational methods is done.

Table 4 The results of $L_{\infty }$ error norms and the related temporal rates of convergence in Test Problem 5.2 with $M=12$, $T_{f}=10$, and diverse Δt

We have used the combined method based on a finite difference procedure and a new class of polynomials (FDCP) [28], the FDM [6], and the method based on Lucas polynomials [27]. Obviously, the presented results confirm that the order of accuracy of two is obtainable for the used time-marching algorithm and with relatively large time steps in comparison with other approaches.

Test Problem 5.3

The third test problem is devoted to the following nonhomogeneous model with nonunity coefficients [27, 28], $\mu =0.1, \gamma =1, \sigma =0.01, \eta =1$:

$$\begin{aligned} h(x,t) = {}&\biggl( \frac{9}{5}+\frac{189}{100}\bigl(x^{2}-1 \bigr)+\frac{179}{50}x \biggr)e^{x-t} \\ &{} + \bigl(x^{2}-1 \bigr) \bigl(x^{2}+2x-1 \bigr)e^{2x-2t}. \end{aligned}$$

The computational domain is taken as

$$\begin{aligned} x\in [x_{L},x_{R}]=[-1,1],\quad t\in [0,T_{f}], \end{aligned}$$

and the initial and boundary conditions are

$$\begin{aligned} \textstyle\begin{cases} w(x,0)=e^{x} (1-x^{2}), \\ w(x_{L},t)=0, \\ w(x_{R},t)=0. \end{cases}\displaystyle \end{aligned}$$

The exact solution of this example is

$$\begin{aligned} w(x,t)=e^{x-t} \bigl(1-x^{2}\bigr). \end{aligned}$$

We first set $M=8$ and $T_{f}=1$ and use $\Delta t=0.1$ in the computations. The approximate solution at $t=T_{f}$ on $[x_{L},x_{R}]$ is obtained as

$$\begin{aligned} {\mathcal{X}}_{10,8}(x)={}&{-}6.1430\times 10^{-4} x^{8} - 0.00321614 x^{7} - 0.0147067 x^{6} \\ & {}- 0.0578943 x^{5} - 0.168302 x^{4}- 0.306555 x^{3} \\ &{} - 0.18433 x^{2} + 0.367665 x + 0.367954. \end{aligned}$$

The profiles of the approximate solutions along with related exact solutions at diverse time steps

$$\begin{aligned} t=\ell \Delta t,\quad \ell =1,2,\ldots,10, \end{aligned}$$

are shown in Fig. 6, left. The related absolute errors are further shown in Fig. 6, right. In the next experiments, we indicate the second-order convergence of our approach with respect to time discretization. In Table 5, using $M=10$ and various

$$\Delta t=1/2^{i},\quad i=1,2,3,4, $$

we present the achieved $\mathcal{E}_{\infty }$ and calculate the corresponding $\operatorname{ROCt}_{\infty }$. Similar results reported by other the existing schemes but with larger number of resources are also presented in Table 5. These approaches are the FDCP [28] with $N=18$, the Chebyshev–Legendre method (CLM) [26] with $N=36$, and the method based on Lucas polynomials [27] with $N=18$. Clearly, the TBM using a few bases, together with relatively large time steps, confirms the order of convergence two.

Figure 6

Plot of numerical solutions (a) and related absolute errors (b) in Test Problem 5.3 at $t=\ell \Delta t$, $\ell =1,2,\ldots,10$, for $\Delta t=0.1, M=8$, and $T_{f}=1$

Table 5 The results of $L_{\infty }$ error norms and the related temporal rate of convergence in Test Problem 5.3 with $M=10$, $T_{f}=1$ and diverse Δt

The spectral accuracy with regard to the achieved $\mathcal{E}_{2}/\mathcal{E}_{\infty }$ error norms for the space variable is also investigated in Fig. 7. In these results, we have used $\Delta t=10^{-3}$ and diverse

$$\begin{aligned} M=2^{i},\quad i=1,2,3,4. \end{aligned}$$

Figure 7

The results of $\mathcal{E}_{2}/\mathcal{E}_{\infty }$ errors (a) and related rate of convergence (b) in Test Problem 5.3 at $t=T_{f}$ for $\Delta t=0.001$, $T_{f}=1$, and various M

The corresponding rates of convergence ROCx₂ and $\operatorname{ROCx}_{\infty }$ are also visualized in Fig. 7. Finally, for this test problem, let us see the behavior of numerical solution and error norms when $T_{f}=5$. These results using $M=12$ are depicted in Fig. 8. Note that in the left plot, we used $\Delta t=0.05$ whereas for the right plotted absolute errors, we used $\Delta t=0.005$. Additionally, the approximate solution for $\Delta t=0.05$ on $x\in [-1,1]$ at $t=T_{f}$ is given by

$$\begin{aligned} {\mathcal{X}}_{100,12}(x) ={}&{ -}1.920\times 10^{-9} x^{12} - 1.897 \times 10^{-8} x^{11} - 1.659\times 10^{-7} x^{10} \\ & {}- 1.320\times 10^{-6} x^{9} - 9.196\times 10^{-6} x^{8} - 5.482 \times 10^{-5} x^{7} \\ &{} - 2.714\times 10^{-4} x^{6} - 0.0010669 x^{5}- 0.0030883 x^{4} \\ &{} - 0.00561447 x^{3} - 0.00336744 x^{2} + 0.00673753 x \\ &{} + 0.00673652. \end{aligned}$$

Figure 8

Plot of numerical solutions with $\Delta t=0.05$ (a) and graphs of absolute errors with $\Delta t=0.005$ (b) in Test Problem 5.3 using $M=12$ and $T_{f}=5$

Test Problem 5.4

In the last test problem, we pay attention to [24, 25] with $\mu =1, \gamma =1, \sigma =1$, and $\eta =\pm 1$. For $\eta =\pm 1$, we have the right-hand side functions $h(x,t)$, respectively,

$$\begin{aligned} \cosh ^{4}(t - x) h(x,t) ={}&{-}5\sinh (t - x) + 2\cosh (t - x) \pm \sinh (2t - 2x)/2 \\ & {}- \cosh ^{3}(t - x) + \sinh ^{3}(t - x). \end{aligned}$$

The model problem (1) is solved on

$$\begin{aligned} x\in [x_{L},x_{R}]=[0,1], [-1,1],\quad t\in [0,T_{f}]. \end{aligned}$$

The corresponding exact solution is

$$\begin{aligned} w(x,t)=\sec h(x-t). \end{aligned}$$

The initial and boundary conditions are obtained from the exact solution for this example. We first consider $M=7$ and $T_{f}=1$ and use $\Delta t=0.1$ in the computations. The approximate solutions for $\eta =\pm 1$ at $t=T_{f}$ on $[-1,1]$ are obtained, respectively, as

$$\begin{aligned} \mathcal{X}_{10,7}^{+}(x)={}&0.00842799 x^{7} +0.0184003 x^{6} - 0.00867362 x^{5} \\ &{} - 0.0846306 x^{4} - 0.126626 x^{3}+ 0.0502792 x^{2} \\ &{} + 0.49397 x + 0.648852 \end{aligned}$$

and

$$\begin{aligned} \mathcal{X}_{10,7}^{-}(x)={}&0.00869142 x^{7} +0.018303 x^{6} - 0.00922897 x^{5} \\ &{} - 0.0843227 x^{4}- 0.126124 x^{3}+ 0.0499579 x^{2} \\ &{} + 0.493761 x + 0.648963. \end{aligned}$$

As we can see, both approximate solutions are very close to each other. We thus only plot the the graphs of $\mathcal{X}_{10, 7}^{-}(x)$ and the related absolute errors for $\eta =-1$ in Fig. 9.

Figure 9

Plots of approximate solution (a) and absolute errors (b) in Test Problem 5.4 for $\Delta t=0.1, M=7$, $\eta =-1$, and $T_{f}=1$

Next, we examine the behavior of $\mathcal{E}_{\infty }$ error norms when $M=8$ is fixed and Δt varies as $1/2,1/4,1/8,1/16$. In addition, the final time is taken as $T_{f}=1$, and $[x_{L},x_{R}]=[0,1]$. These results for $\eta =-1$ are reported in Table 6. For comparison, the outcomes of the previously existing computational procedures are also displayed in Table 5. To this end, we use the Lie-group approach based on radial basis functions (LG-RBFs) [25], the meshless method [24], and the Legendre spectral element method (LSEM) [29]. We can observe from the results shown in Table 6that our numerical results provide second-order accuracy in time while employing a smaller number of bases and time steps compared to other well-established numerical procedures in the literature.

Table 6 The results of $L_{\infty }$ error norms and the related temporal rate of convergence in Test Problem 5.4 with $M=8$, $T_{f}=1$, $\eta =-1$, and diverse Δt on $[0,1]$

Similar results for $\eta =+1$ are further reported in Table 7. However, here we have used $T_{f}=0.1$, and the values of Δt are $1/10,1/20,1/40$ as used in the results presented in the ICSCM [23] and meshless methods [24] with parameter $h=1/10$. Clearly, the results obtained by the TBM are more accurate in comparison with two other methods. The next experiments show further the accuracy of our approach in the framework of results evaluated at time $t=T_{f}$, where $T_{f}=5$. We use $\eta =-1$ and $M=8$ and compare with the methods used in Table 6. The aforementioned results are shown in Table 8, which indicate the superiority of our procedure compared to the LG-RBFs, LSEM, and meshless approaches. Besides, the second-order accuracy in time is visible from the given results if Table 8. For completeness, the graphical representations of approximate solutions together with their absolute errors at diverse time instants

$$t=\ell \Delta t,\quad s=1,2,\ldots,50, $$

are visualized in Fig. 10. Finally, for the last test example, we also investigate the spectral accuracy of the numerical solutions for $\Delta t=0.001$ and moderate values of

$$M=2^{i}, \quad i=1,2,3,4. $$

For the simulations, we employ both $\eta =\pm 1$ and compute the results at $T_{f}=1$ and the spatial domain $[-1,1]$. The results are shown in Table 9.

Figure 10

Plot of approximate solutions (a) and absolute errors (b) in Test Problem 5.4 for $\Delta t=0.1, M=8$, $\eta =-1$, and $T_{f}=5$

Table 7 The results of $L_{\infty }$ error norms and the related temporal rate of convergence in Test Problem 5.4 with $M=8$, $T_{f}=0.1$, $\eta =+1$, and diverse Δt on $[0,1]$

Table 8 The results of $L_{\infty }$ error norms and the related temporal rate of convergence in Test Problem 5.4 with $M=8$, $T_{f}=5$, $\eta =-1$, and diverse Δt on $[0,1]$

Table 9 The results of $\mathcal{E}_{2}/\mathcal{E}_{\infty }$ error norms and the related spatial rate of convergence in Test Problem 5.4 with $\Delta t=0.001$, $T_{f}=1$, $\eta =\pm 1$, and diverse M on $[-1,1]$

6 Conclusions

We have developed an accurate time numerical solution algorithm (using a large time step) for the generalized BBMB-type equations (1) arising in diverse disciplines of engineering science. The proposed technique is constructed based on a combination of the Boubaker collocation procedure for the spatial variable and the Taylor series formula for the temporal discretization. The main characteristic of the presented work is that we need to solve an algebraic system of equations at each time step rather than solving a global system obtained in the spectral collocation methods developed in the past. The convergence analysis of the hybrid technique is discussed. The numerical results shown in tables and figures justify the second-order accuracy in time and the high-order accuracy in the space of the presented technique in comparison with some existing well-established computational schemes.