1 Introduction

The main purpose of this work is to study an advanced version of the coupled Burgers’ equations involving non-singular variable-order (VO) fractional derivatives. Technically, these systems can be described as follows:

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial ^{\xi (x,t)}u(x,t)}{\partial t^{\xi (x,t)}}+\mu _{1}u(x,t)\frac{\partial u(x,t)}{\partial x}+\sigma _{1}\frac{\partial \left( u(x,t)v(x,t)\right) }{\partial x}=\kappa _{1}\frac{\partial ^{2} u(x,t)}{\partial x^{2}}+f_{1}(x,t),\\ \displaystyle \frac{\partial ^{\vartheta (x,t)}v(x,t)}{\partial t^{\vartheta (x,t)}}+\mu _{2}v(x,t)\frac{\partial v(x,t)}{\partial x}+\sigma _{2}\frac{\partial \left( u(x,t)v(x,t)\right) }{\partial x}=\kappa _{2}\frac{\partial ^{2} v(x,t)}{\partial x^{2}}+f_{2}(x,t), \end{array}\,\,\,(x,t)\in [0,{\mathcal {L}}]\times [0,{\mathcal {T}}], \end{aligned}$$
(1.1)

where \(\xi (x,t),\,\vartheta (x,t)\in (0,1)\), \(f_{1}(x,t)\) and \(f_{2}(x,t)\) are continuous functions on their domains, and \(\mu _{i}\), \(\sigma _{i}\) and \(\kappa _{i}\) for \(i=1,2\) are constants. Note that the VO fractional differentiations are considered in the concept of Atangana–Baleanu–Caputo (ABC) [1]. In this study, we focus on the model associated with the following initial conditions

$$\begin{aligned} \begin{array}{lll} u(x,0)=u_{0}(x),&\,&v_{0}(x,0)=v_{0}(x), \end{array} \end{aligned}$$
(1.2)

and the boundary conditions

$$\begin{aligned} \begin{array}{lll} \displaystyle u(0,t)=u_{1}(t),&{}&{}\displaystyle u({\mathcal {L}},t)=u_{2}(t),\\ \displaystyle v(0,t)=v_{1}(t),&{}&{}\displaystyle v({\mathcal {L}},t)=v_{2}(t). \end{array} \end{aligned}$$
(1.3)

We remind that different classes of constant-order fractional coupled Burgers’ equations have been studied by many scholars in the last years, e.g. [2,3,4,5,6,7]. It is worth noting that when \(\xi (x,t)=1\) and \(\vartheta (x,t)=1\), the above model reduces to the classical coupled Burgers’ equations. Recently, the Burgers’ and coupled Burgers’ equations are successfully used in many physical phenomena like, wave propagation, traffic flow, nuclear fusion reactor, soil water, turbulent mixing and others [6].

In recent years, the kernel singularity in the previous fractional operators (derivative and integral) has aroused the interest of researchers to define non-singular fractional operators, for example, see [1, 8, 9]. As it is known, mathematical and dynamical systems including operators with singular kernel might face inconsistency in some points. Versus, it has been expressed that fractional equations including non-singular operators can model different problems in physics and engineering more precisely than fractional equations with singular operators [10]. In fact, eliminating the singularity avoids possible inconsistency of models and guarantees the feasibility of numerical results.

During the last decades, substantial attention has been directed to the VO fractional calculus (derivative and integral operators of variable orders) and its applications, e.g. [11,12,13]. Therefore, different numerical approximation methods have been studied to solve problems which involve with VO fractional derivatives in spite of the complexity of these operators. A meshless technique is applied in [14] to solve VO fractional reaction–diffusion equation in two dimensions. The artificial neural networks are used in [15] for some categories of VO fractional differential equations. A numerical technique is used in [16] for VO fractional 1D reaction–diffusion equation. A finite difference scheme is applied in [17] for VO fractional telegraph equation. An interpolation scheme by using the Lagrange polynomials is expressed in [18] for some classes of VO fractional differential equations. A meshless method is described in [19] to solve VO fractional 2D telegraph equation. The Legendre wavelets are used in [20] for VO fractional 2D reaction–diffusion equation. Legendre cardinal functions are used in [21] for nonlinear VO fractional Schrödinger equation. In [22], Chebyshev cardinal functions are utilized to solve a class of nonlinear VO fractional optimization problems. Recently, orthonormal Bernoulli polynomials are used in [23] to solve VO fractional coupled Boussinesq–Burger’s equations.

Discrete orthogonal polynomials as a useful class of basis functions have successfully been used instead of continuous orthogonal polynomials for various problems. They have been used in birth and death processes [24], chemical engineering [25], image coding [26] and optimization problems [27]. We note that there are some advantages in applying discrete polynomials, particularly computation efficiency. In fact, in order to approximate a continuous function, the coefficients of expansion can be obtained precisely by only computing a summation. Consequently, the implementation of numerical methods based on these polynomials is less complex and costly while they often fulfill spectral accuracy [27]. Briefly, any sufficiently differentiable function can be approximated by these polynomials efficiently and accurately.

Recently, the Hahn polynomials (HPs) as a useful class of discreet polynomials have been used for solving some classes of differential equations. In [28], the HPs used for solving VO fractional mobile-immobile advection-dispersion equation. These polynomials used in [29] for solving VO fractional Rayleigh–Stokes equation.

Since no previous study has been accomplished for the non-singular VO fractional model of the coupled Burgers’ equations, we first introduce the new type of VO time fractional coupled Burgers’ equations (1.1) and then we design a numerical method by using the HPs for its solution due to the mentioned advantages of the discrete HPs. The proposed method uses the operational matrices of classical and VO fractional derivatives of the shifted HPs for converting the mentioned system into an algebraic system of equations.

This work is organized in the following sections: In Sect. 2, the VO fractional derivative in the ABC sense is described. The HPs and their properties are reviewed in Sect. 3. Approximation by the shifted HPs is explained in Sect. 4. Some operational matrices of derivative of the shifted HPs are obtained in Sect. 5. Formulation of the expressed method is explained in Sect. 6. The designed method is implemented in Sect. 7 on some numerical examples. The conclusions of study are listed in Sect. 8.

2 The VO fractional calculus what non-singular kernel

Definition 2.1

[30] The Mittag–Leffler functions are defined by

$$\begin{aligned} {\mathbf{E }}_{\zeta }(t)=\sum _{j=0}^{\infty }\frac{t^{j}}{\Gamma {\left( j\zeta +1\right) }}, \quad t\in {\mathbb {R}},\,\,\zeta \in {\mathbb {R}}^{+}, \end{aligned}$$
(2.1)

and

$$\begin{aligned} {\mathbf{E }}_{\zeta _{1},\zeta _{2}}(t)=\sum _{j=0}^{\infty }\frac{t^{j}}{\Gamma {\left( j\zeta _{1}+\zeta _{2}\right) }},\quad t\in {\mathbb {R}},\,\,\zeta _{1},\,\zeta _{2}\in {\mathbb {R}}^{+}. \end{aligned}$$
(2.2)

Definition 2.2

[1] If u(x, t) defined on \(\left[ 0,{\mathcal {L}}\right] \times [0,{\mathcal {T}}]\) is differentiable with respect to t and \(\xi :\left[ 0,{\mathcal {L}}\right] \times [0,{\mathcal {T}}]\longrightarrow (0,1)\) is continuous, the VO time fractional differentiation of order \(\xi (x,t)\) of u(x, t) in the ABC sense is defined by

$$\begin{aligned} \frac{\partial ^{\xi (x,t)} u(x,t)}{\partial t^{\xi (x,t)}}= & {} \frac{{\mathbf {C}}(\xi (x,t))}{1-\xi (x,t)}\int _{0}^{t}\frac{\partial u (x,s)}{\partial s}{\mathbf{E }}_{\xi (x,t)}\nonumber \\&\times \left( \frac{-\xi (x,t)(t-s)^{\xi (x,t)}}{1-\xi (x,t)}\right) {\mathrm{d}}s, \end{aligned}$$
(2.3)

where \({\mathbf {C}}(\xi (x,t))=1-\xi (x,t)+\frac{\xi (x,t)}{\Gamma \left( \xi (x,t)\right) }\).

Corollary 2.3

[22] If \(k\in {\mathbb {N}}\cup \{0\}\) and \(\xi :\left[ 0,{\mathcal {L}}\right] \times [0,{\mathcal {T}}]\longrightarrow (0,1)\) is continuous, we have

$$\begin{aligned} \frac{\partial ^{\xi (x,t)}}{\partial t^{\xi (x,t)}}t^{k}=\left\{ \begin{array}{lll} 0,&{}&{}k=0,\\ \displaystyle \frac{{\mathbf {C}}(\xi (x,t))\,k!\, t^{k}}{1-\xi (x,t)}\,{\mathbf {E}}_{\xi (x,t),k+1}\left( \frac{-\xi (x,t)\,t^{\xi (x,t)}}{1-\xi (x,t))}\right) ,&{}&{}k=1,2,\ldots . \end{array}\right. \end{aligned}$$
(2.4)

3 The Hahn polynomials (HPs)

Definition 3.1

[31] The Pochhammer symbol \((\varpi )_{k}\) is defined as follows:

$$\begin{aligned} \begin{array}{l} (\varpi )_{0}=1,\\ (\varpi )_{k}=\varpi (\varpi +1)(\varpi +2)\ldots (\varpi +k-1),\quad k \in {\mathbb {N}},\ \varpi \in {\mathbb {C}},\ \text {Real}(\varpi )>0. \end{array} \end{aligned}$$
(3.1)

Definition 3.2

[31] The first kind Stirling numbers are defined by

$$\begin{aligned} S_{k}^{(i)}=\sum _{r=0}^{k-i}(-1)^{r}{k+r-1\atopwithdelims ()k+r-i}{2k-i\atopwithdelims ()k-r-i}s_{k+r-i}^{(r)},k,i,r\in {\mathbb {N}}, \end{aligned}$$
(3.2)

where \(s_{k}^{(i)}\)’s are the second kind Stirling numbers as follows:

$$\begin{aligned} s_{k}^{(i)}=\frac{1}{i!}\sum _{r=0}^{i}(-1)^{r}{i\atopwithdelims ()r}(i-r)^{k}. \end{aligned}$$
(3.3)

Definition 3.3

[32] Let \(a,b>-1\) be real constants and \(M\in {\mathbb {N}}\). The Hahn polynomial \(h_{m}(x;a,b,M)\) of degree m for \(m=0,1,\ldots ,M\) is defined over [0, M] by

$$\begin{aligned} h_{m}(x;a,b,M)=\sum _{k=0}^{m}\frac{(-m)_{k}(m+a+b+1)_{k}(-x)_{k}}{(m+1)_{k}(-M)_{k}k!}, \end{aligned}$$
(3.4)

where \((\varpi )_{k}\) is the Pochhammer symbol.

Remark 1

[32] The HPs are orthogonal with the following discrete inner product over [0, M]:

$$\begin{aligned} \left<u,v\right>=\sum _{k=0}^{M}u(k)v(k)w(k), \end{aligned}$$
(3.5)

where

$$\begin{aligned} w(k)={a+k\atopwithdelims ()k}{M+b-k\atopwithdelims ()M-k}. \end{aligned}$$
(3.6)

Remark 2

Note that \((-x)_{k}\) in Eq. (3.4) can also be rewritten as follows:

$$\begin{aligned} (-x)_{k}=(-1)^{k}\sum _{n=0}^{k}S_{k}^{(n)}x^{n}, \end{aligned}$$
(3.7)

where \(S_{k}^{(n)}\)’s are the first kind Stirling numbers.

Corollary 3.4

From Definition 3.3 and Remark 2,, one has

$$\begin{aligned} h_{m}(x;a,b,M)=\sum _{k=0}^{m}\sum _{n=0}^{k}(-1)^{k}\frac{(-m)_{k}(m+a+b+1)_{k}}{(a+1)_{k}(-M)_{k}k!}S_{k}^{(n)}x^{n}. \end{aligned}$$
(3.8)

Remark 3

For the practical use of the HPs on the interval \([0,{\mathcal {L}}]\), it is necessary to shift the domain of the definition by using the substitution \(z=\frac{M}{{\mathcal {L}}}x\), \(0\le x\le {\mathcal {L}}\).

Definition 3.5

The shifted HPs \(h_{m}^{*}(x;a,b,M)\) can be defined on the interval \([0,{\mathcal {L}}]\) by

$$\begin{aligned}&h_{m}^{*}(x;a,b,M,{\mathcal {L}})=\sum _{k=0}^{m}\sum _{n=0}^{k}(-1)^{k}\frac{(-m)_{k}(m+a+b+1)_{k}}{(a+1)_{k}(-M)_{k}k!}\left( \frac{M}{{\mathcal {L}}}\right) ^{n}S_{k}^{(n)}x^{n}. \end{aligned}$$
(3.9)

4 Function approximation

A function \(u(x)\in C \left( [0,{\mathcal {L}}]\right)\) can be approximated via the shifted HPs in the form

$$\begin{aligned} u(x)\simeq \sum _{i=0}^{M}u_{i}h_{i}^{*}(x;a,b,M,{\mathcal {L}}), \end{aligned}$$
(4.1)

where

$$\begin{aligned} u_{i}=\frac{1}{\eta (i;a,b,M)}\sum _{r=0}^{M}u\left( \frac{{\mathcal {L}}}{M}r\right) h_{i}(r;a,b,M)w(r), \end{aligned}$$
(4.2)

in which

$$\begin{aligned} \eta (i;a,b,M)=\frac{(-1)^{i}\left( i+a+b+1\right) _{M+1}\left( b+1\right) _{i}i!}{(2i+a+b+1)\left( a+1\right) _{i}\left( -M\right) _{i}M!}. \end{aligned}$$

Note that Eq. (4.1) can be written in the matrix form as

$$\begin{aligned} u(x)\simeq U^{T}H^{({\mathcal {L}})}_{M}(x), \end{aligned}$$
(4.3)

where

$$\begin{aligned}&U=\left[ u_{0}\,\,u_{1}\,\,\ldots \,\,u_{M}\right] ^{T}, \nonumber \\&H^{({\mathcal {L}})}_{M}(x)=\left[ h_{0}^{*}(x;a,b,M,{\mathcal {L}})\,\,h_{1}^{*}(x;a,b,M,{\mathcal {L}})\right. \nonumber \\&\quad \left. \ldots \,\,h_{M}^{*}(x;a,b,M,{\mathcal {L}})\right] ^{T}. \end{aligned}$$
(4.4)

Similarly, a function \(u(x,t)\in C\left( [0,{\mathcal {L}}]\times [0,{\mathcal {T}}]\right)\) can be approximated via the shifted HPs in the form

$$\begin{aligned} u(x,t)\simeq & {} \sum _{i=0}^{M}\sum _{j=0}^{N}\lambda _{ij}h_{i}^{*}(x;a,b,M,{\mathcal {L}})h_{j}^{*}(t;a,b,N,{\mathcal {T}})\nonumber \\\triangleq & {} H^{({\mathcal {L}})}_{M}(x)^{T}\Lambda H^{({\mathcal {T}})}_{N}(t), \end{aligned}$$
(4.5)

where \(\Lambda =\left[ {\lambda }_{ij}\right]\) is an \((M+1)\times (N+1)\) matrix with

$$\begin{aligned} {\lambda }_{ij}&=\frac{1}{\eta (i-1;a,b,M)\,\eta (j-1;a,b,N)}\nonumber \\&\quad \sum _{r=0}^{M}\sum _{l=0}^{N}u\left( \frac{{\mathcal {L}}}{M}r,\frac{{\mathcal {T}}}{N}l\right) h_{i-1}(r;a,b,M)h_{j-1}(l;a,b,N)w(r)w(l),\nonumber \\&\quad 1\le i \le M+1,\,1\le j\le N+1. \end{aligned}$$
(4.6)

Theorem 4.1

If u(x, t) is sufficiently differentiable over \([0,{\mathcal {L}}]\times [0,{\mathcal {T}}]\) with boundedness

$$\begin{aligned}&\left\| \frac{\partial ^{M+1}u(x,t)}{\partial x^{M+1}}\right\| _{\infty }\le \omega _{1}, \quad \left\| \frac{\partial ^{N+1}u(x,t)}{\partial t^{N+1}}\right\| _{\infty }\le \omega _{2}, \nonumber \\&\quad \left\| \frac{\partial ^{M+N+2}u(x,t)}{\partial x^{M+1}\partial t^{N+1}}\right\| _{\infty }\le \omega _{3}, \end{aligned}$$
(4.7)

\(\, and \, {\tilde{u}}(x,t)=H^{({\mathcal {L}})}_{M}(x)^{T}\Lambda H^{({\mathcal {T}})}_{N}(t)\) is the approximation of u(x, t) in terms of the shifted HPs, then

$$\begin{aligned}&\left\| u(x,t)-{\tilde{u}}(x,t)\right\| _{\infty }\le \frac{\omega _{1}}{4(M+1)}\left( \frac{{\mathcal {L}}}{M}\right) ^{M+1}\nonumber \\&\quad +\frac{\omega _{2}}{4(N+1)}\left( \frac{{\mathcal {T}}}{N}\right) ^{N+1} \nonumber \\&\quad +\frac{\omega _{3}}{16(M+1)(N+1)}\left( \frac{{\mathcal {L}}}{M}\right) ^{M+1}\left( \frac{{\mathcal {T}}}{N}\right) ^{N+1}. \end{aligned}$$
(4.8)

Proof

The proof is similar to the result obtained in [28] for the error bound, so we omit it. \(\square\)

5 Operational matrices

Lemma 5.1

The shifted HPs vector in Eq. (4.4) can be expressed as follows:

$$\begin{aligned} H^{({\mathcal {L}})}_{M}(x)=A_{M}^{({\mathcal {L}})}K_{M}(x), \end{aligned}$$
(5.1)

where \(A_{M}^{({\mathcal {L}})}\) is a square matrix of order \((M+1)\) with entries

$$\begin{aligned} \displaystyle \left[ A_{M}^{({\mathcal {L}})}\right] _{ij}=\left\{ \begin{array}{ll} \displaystyle \sum _{k=j-1}^{i-1}(-1)^{k}\frac{(-i+1)_{k}(i+a+b)_{k}}{(a+1)_{k}(-M)_{k}k!}\left( \frac{M}{{\mathcal {L}}}\right) ^{j-1}S_{k}^{(j-1)}&{} \quad i\ge j,\\ 0,&{}\quad i<j, \end{array} \right. \quad i,j=1,2,\ldots ,M+1, \end{aligned}$$
(5.2)

and \(K_{M}(x)\) is a column vector of order \((M+1)\) in the form

$$\begin{aligned} K_{M}(x)=\left[ 1\,\,x\,\,x^{2}\,\,\ldots \,\,x^{M}\right] ^{T}. \end{aligned}$$
(5.3)

Proof

Using Definition 3.5, the proof is clear. \(\square\)

Remark 4

The matrix \(A_{M}^{({\mathcal {L}})}\) is an upper triangular matrix which its determinant is computed as follows:

$$\begin{aligned} \text {det}\left( A_{M}^{({\mathcal {L}})}\right)= & {} \prod _{i=1}^{M+1}(-1)^{i-1}\frac{(-i+1)_{i-1}(i+a+b)_{i-1}}{(a+1)_{i-1}(-M)_{i-1}(i-1)!}\nonumber \\&\times \left( \frac{M}{{\mathcal {L}}}\right) ^{i-1}S_{i-1}^{(i-1)}. \end{aligned}$$
(5.4)

Meanwhile, regarding to Definition 3.5 for \(i=1,2,\ldots ,M+1\) and \(a,b>-1\), we have

$$\begin{aligned}&(-i+1)_{i-1}\ne 0,\quad (i+a+b)_{i-1}\ne 0,\quad (a+1)_{i-1},\nonumber \\&\quad (-M)_{i-1}\ne 0. \end{aligned}$$
(5.5)

Moreover, we have \(S_{i-1}^{(i-1)}=1\) and consequently \(\text {det}\left( A_{M}^{({\mathcal {L}})}\right) \ne 0\).

Lemma 5.2

The derivative of the vector introduced in Eq. (5.3) can be presented as

$$\begin{aligned} \frac{{\mathrm{d}} K_{M}(x)}{{\mathrm{d}}x}={\mathbb {D}}^{(1)}_{M}K_{M}(x), \end{aligned}$$
(5.6)

where \({\mathbb {D}}^{(1)}\) is a matrix of order \((M+1)\) with

$$\begin{aligned} \left[ {\mathbb {D}}^{(1)}_{M}\right] _{ij}=\left\{ \begin{array}{ll} 0,&{} \quad i=1,\ 1\le j\le M+1,\\ i-1,&{} \quad 2\le i\le M+1, \ 1\le j\le M+1,\ i-j=1. \end{array}\right. \end{aligned}$$

Generally, we have

$$\begin{aligned} \frac{{\mathrm{d}}^{r} K_{M}(x)}{{\mathrm{d}}x^r}={\mathbb {D}}^{(r)}_{M}K_{M}(x), \end{aligned}$$
(5.7)

where \({\mathbb {D}}^{(r)}_{M}\) is the rth power of \({\mathbb {D}}^{(1)}_{M}\).

Proof

The proof is straightforward, so we omit it. \(\square\)

Lemma 5.3

If \(\xi : [0,{\mathcal {L}}]\times [0,{\mathcal {T}}]\rightarrow (0,1)\) is a two-variable continuous function, we have

$$\begin{aligned} \frac{\partial ^{\xi (x,t)}K_{N}(t)}{\partial t^{\xi (x,t)}}={\mathbb {W}}^{(\xi (x,t))}_{ N}K_{N}(t), \end{aligned}$$
(5.8)

where \({\mathbb {W}}^{(\xi (x,t))}_{N}\) is a matrix of order \((N+1)\) with

$$\begin{aligned} \left[ {\mathbb {W}}^{(\xi (x,t))}_{N}\right] _{ij}=\left\{ \begin{array}{ll} 0,&{} \quad i=1,\,1\le j\le N+1,\\ \displaystyle \frac{(i-1)!{\mathbf {C}}(\xi (x,t))}{1-\xi (x,t)}\mathbf{E }_{\xi (x,t),i}\left( \frac{-\xi (x,t)\,t^{\xi (x,t)}}{1-\xi (x,t)}\right) ,&{}\quad 2\le i\le N+1,\, 1\le j\le N+1, i=j. \end{array}\right. \end{aligned}$$

Proof

Using Corollary 2.3, the proof is straightforward. \(\square\)

Theorem 5.4

Let \(H_{M}^{({\mathcal {L}})}(x)\) be the shifted HPs vector defined in Eq. (4.4). Then, we have

$$\begin{aligned} \frac{{\mathrm{d}} H_{M}^{({\mathcal {L}})}(x)}{{\mathrm{d}}x}={\mathbf {D}}^{(1;{\mathcal {L}})}_{M}H_{M}^{({\mathcal {L}})}(x), \end{aligned}$$
(5.9)

where \({\mathbf {D}}^{\left( 1;{\mathcal {L}}\right) }_{M}\) is a matrix (called the derivative operational matrix of the shifted HPs) of order \((M+1)\) as follows:

$$\begin{aligned} {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}=A_{M}^{({\mathcal {L}})}{\mathbb {D}}^{(1)}_{M}\left( A_{M}^{{\mathcal {L}}}\right) ^{-1}. \end{aligned}$$

Generally, we have

$$\begin{aligned} \frac{{\mathrm{d}}^{r} H_{M}^{({\mathcal {L}})}(x)}{{\mathrm{d}}x^r}={\mathbf {D}}^{(r;{\mathcal {L}})}_{M}H_{M}^{({\mathcal {L}})}(x), \end{aligned}$$
(5.10)

where \({\mathbf {D}}^{(r;{\mathcal {L}})}_{M}\) is the rth power of \({\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\).

Theorem 5.5

If \(H_{N}^{({\mathcal {T}})}(t)\) is the shifted HPs vector given in Eq. (4.5) and \(\xi : [0,{\mathcal {L}}]\times [0,{\mathcal {T}}]\rightarrow (0,1)\) is continuous, then

$$\begin{aligned} \frac{\partial ^{\xi (x,t)}H_{N}^{({\mathcal {T}})}(t)}{\partial t^{\xi (x,t)}}={\mathbf {W}}^{\left( \xi (x,t);{\mathcal {T}}\right) }_{N}H_{N}^{({\mathcal {T}})}(t), \end{aligned}$$
(5.11)

where \({\mathbf {W}}^{(\xi (x,t);{\mathcal {T}})}_{N}\) is a matrix (called the VO fractional derivative operator matrix of order \(\xi (x,t)\) of the shifted HPs) of order \((N+1)\) in the form

$$\begin{aligned} {\mathbf {W}}^{(\xi (x,t),{\mathcal {T}})}_{N}=A_{N}^{({\mathcal {T}})}{\mathbb {W}}^{(\xi (x,t))}_{N}\left( A_{N}^{({\mathcal {T}})}\right) ^{-1}. \end{aligned}$$

Proof

Regarding to Lemmas 5.1 and 5.3, the proof is straightforward. \(\square\)

6 The computational method

In order to solve the system (1.1), we approximate the solution via the shifted HPs as

$$\begin{aligned} u(x,t)\simeq & {} \sum _{i=0}^{M} \sum _{j=0}^{N} \lambda _{ij}h_{i}^{*}(x;a,b,M,{\mathcal {L}})h_{j}^{*}(t;a,b,N,{\mathcal {T}}) \nonumber \\\triangleq & {} H^{({\mathcal {L}})}_{M}(x)^{T}{\Lambda } H^{({\mathcal {T}})}_{N}(t),\nonumber \\ u(x,t)\simeq & {} \sum _{i=0}^{M} \sum _{j=0}^{N} \theta _{ij}h_{i}^{*}(x;a,b,M,{\mathcal {L}})h_{j}^{*}(t;a,b,N,{\mathcal {T}})\nonumber \\\triangleq & {} H^{({\mathcal {L}})}_{M}(x)^{T}{\Theta } H^{({\mathcal {T}})}_{N}(t), \end{aligned}$$
(6.1)

where \(\Lambda =[\lambda _{ij}]\) and \(\Theta =[\theta _{ij}]\) are unknown \((M+1)\times (N+1)\) matrices, and \(H^{({\mathcal {L}})}_{M}(x)\) and \(H^{({\mathcal {T}})}_{N}(t)\) are the vectors given in Eq. (4.5). Theorem 5.4 yields

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial u(x,t)}{\partial x}\simeq H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) ^{T}{\Lambda } H^{({\mathcal {T}})}_{N}(t),\\ \displaystyle \frac{\partial v(x,t)}{\partial x}\simeq H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) ^{T}{\Theta } H^{({\mathcal {T}})}_{N}(t), \end{array} \end{aligned}$$
(6.2)

and

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial ^{2} u(x,t)}{\partial x^{2}}\simeq H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(2;{\mathcal {L}})}_{M}\right) ^{T}{\Lambda } H^{({\mathcal {T}})}_{N}(t),\\ \displaystyle \frac{\partial ^{2} v(x,t)}{\partial x^{2}}\simeq H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(2;{\mathcal {L}})}_{M}\right) ^{T}{\Theta } H^{({\mathcal {T}})}_{N}(t). \end{array} \end{aligned}$$
(6.3)

Also, using Theorem 5.5, one has

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial ^{\xi (x,t)}u(x,t)}{\partial t^{\xi (x,t)}}\simeq H^{({\mathcal {L}})}_{M}(x)^{T}{\Lambda } {\mathbf {W}}^{(\xi (x,t);{\mathcal {T}})}_{N} H^{({\mathcal {T}})}_{N}(t),\\ \displaystyle \frac{\partial ^{\vartheta (x,t)}v(x,t)}{\partial t^{\vartheta (x,t)}}\simeq H^{({\mathcal {L}})}_{M}(x)^{T}{\Theta } {\mathbf {W}}^{(\vartheta (x,t);{\mathcal {T}})}_{N} H^{({\mathcal {T}})}_{N}(t). \end{array} \end{aligned}$$
(6.4)

By inserting Eqs. (6.1)–(6.4) into Eq. (1.1), one can introduce the residual functions

$$\begin{aligned} {\mathbf {R}}_{1}(x,t)&\triangleq H^{({\mathcal {L}})}_{M}(x)^{T}{\Lambda } {\mathbf {W}}^{(\xi (x,t);{\mathcal {T}})}_{N} H^{({\mathcal {T}})}_{N}(t)\nonumber \\&\quad +\mu _{1}\left( H^{({\mathcal {L}})}_{M}(x)^{T}{\Lambda } H^{({\mathcal {T}})}_{N}(t)\right) \nonumber \\&\quad \left( H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) ^{T}{\Lambda }H^{({\mathcal {T}})}_{N}(t)\right) \nonumber \\&\quad +\sigma _{1}\left[ \left( H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) ^{T}{\Lambda }H^{({\mathcal {T}})}_{N}(t)\right) \right. \nonumber \\&\quad \left. \left( H^{({\mathcal {L}})}_{M}(x)^{T}{\Theta } H^{({\mathcal {T}})}_{N}(t)\right) +\left( H^{({\mathcal {L}})}_{M}(x)^{T}{\Lambda } H^{({\mathcal {T}})}_{N}(t)\right) \right. \nonumber \\&\quad \left. \left( H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) {\Theta } H^{({\mathcal {T}})}_{N}(t)\right) \right] \nonumber \\&\quad -\kappa _{1} H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(2;{\mathcal {L}})}_{M}\right) ^{T}{\Lambda }H^{({\mathcal {T}})}_{N}(t)-f_{1}(x,t),\nonumber \\ {\mathbf {R}}_{2}(x,t)&\triangleq H^{({\mathcal {L}})}_{M}(x)^{T}{\Theta } {\mathbf {W}}^{(\vartheta (x,t);{\mathcal {T}})}_{N} H^{({\mathcal {T}})}_{N}(t)\nonumber \\&\quad +\mu _{2}\left( H^{({\mathcal {L}})}_{M}(x)^{T}{\Theta } H^{({\mathcal {T}})}_{N}(t)\right) \nonumber \\&\quad \left( H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) ^{T}{\Theta }H^{({\mathcal {T}})}_{N}(t)\right) \nonumber \\&\quad +\sigma _{2}\left[ \left( H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) ^{T}{\Lambda }H^{({\mathcal {T}})}_{N}(t)\right) \right. \nonumber \\&\quad \left. \left( H^{({\mathcal {L}})}_{M}(x)^{T}{\Theta } H^{({\mathcal {T}})}_{N}(t)\right) +\left( H^{({\mathcal {L}})}_{M}(x)^{T}{\Lambda } H^{({\mathcal {T}})}_{N}(t)\right) \right. \nonumber \\&\quad \left. \left( H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(1;{\mathcal {L}})}_{M}\right) {\Theta } H^{({\mathcal {T}})}_{N}(t)\right) \right] \nonumber \\&\quad -\kappa _{2} H^{({\mathcal {L}})}_{M}(x)^{T}\left( {\mathbf {D}}^{(2;{\mathcal {L}})}_{M}\right) ^{T}{\Theta }H^{({\mathcal {T}})}_{N}(t)-f_{2}(x,t). \end{aligned}$$
(6.5)

Beside, one can approximate the functions given in Eqs. (1.2) and (1.3) by the shifted HPs as follows:

$$\begin{aligned} \begin{array}{lll} u_{0}(x)\simeq H^{({\mathcal {L}})}_{M}(x)^{T}F_{0},&\,&v_{0}(x)\simeq H^{({\mathcal {L}})}_{M}(x)^{T}G_{0}, \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{lll} u_{1}(t)\simeq F_{1}^{T}H^{({\mathcal {T}})}_{N}(t),&{}&{}u_{2}(t)\simeq F_{2}^{T}H^{({\mathcal {T}})}_{N}(t),\\ v_{1}(t)\simeq G_{1}^{T}H^{({\mathcal {T}})}_{N}(t),&{}&{}v_{2}(t)\simeq G_{2}^{T}H^{({\mathcal {T}})}_{N}(t), \end{array} \end{aligned}$$

where \(F_{i}\) and \(G_{i}\) for \(i=0,1,2\) are given coefficients vectors. So, from Eqs. (1.1), (1.2), (6.1) and the above two relations, we have

$$\begin{aligned} \begin{array}{lll} \Lambda H^{({\mathcal {T}})}_{N}(0)-F_{0}\triangleq U_{0}\simeq 0,&\,&\Theta H^{({\mathcal {T}})}_{N}(0)-G_{0}\triangleq V_{0}\simeq 0, \end{array} \end{aligned}$$
(6.6)

and

$$\begin{aligned} \begin{array}{lll} H^{({\mathcal {L}})}_{M}(0)^{T}{\Lambda }- F_{1}^{T}\triangleq U_{1}\simeq 0,&{}&{}H^{({\mathcal {L}})}_{M}({\mathcal {L}})^{T}{\Lambda }- F_{2}^{T}\triangleq U_{2}\simeq 0,\\ H^{({\mathcal {L}})}_{M}(0)^{T}{\Theta }- G_{1}^{T}\triangleq V_{1}\simeq 0,&{}&{}H^{({\mathcal {L}})}_{M}({\mathcal {L}})^{T}{\Theta }- G_{2}^{T}\triangleq V_{2}\simeq 0. \end{array} \end{aligned}$$
(6.7)

Eventually, the following system with \(2(M+1)\times (N+1)\) equations are generated from Eqs. (6.5), (6.6) and (6.7):

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathbf {R}}_{r}\left( x_{i},t_{j}\right) =0,&{} \quad r=1,2, \ 2\le i\le M,\ 2\le j\le N+1,\\ {[}U_{0}]_{i}=0,\, [V_{0}]_{i}=0,&{} \quad 1\le i\le M+1,\\ {[}U_{r}]_{j}=0,\, [V_{r}]_{j}=0, &{}\quad r=1,2, \ 2\le j\le N+1, \end{array} \right. \end{aligned}$$
(6.8)

where \(x_{i}=\frac{{\mathcal {L}}}{2}\left( 1-\cos \left( \frac{(2i-1)\pi }{2(M+1)}\right) \right)\) for \(i=1,2,\ldots ,M+1\) and \(t_{j}=\frac{{\mathcal {T}}}{2}\left( 1-\cos \left( \frac{(2j-1)\pi }{2(N+1)}\right) \right)\) for \(j=1,2,\ldots ,N+1\). The obtained system should be solved for computing the matrices \(\Lambda\) and \(\Theta\) in Eq. (6.1), and eventually deriving a numerical solution for the original system.

7 Numerical examples

Herein, we examine the precision of the presented algorithm by solving three test problems. The simulations are done via Maple 18 (with 45 decimal digits). The experimental convergence order (ECO) of the method is given by

$$\begin{aligned} \text {ECO}=\log _{\frac{{Q}_{1}}{{Q}_{2}}}\left( \frac{{\mathbf {e}}_{2}}{{\mathbf {e}}_{1}}\right) , \end{aligned}$$

where \({\mathbf {e}}_{1}\) and \({\mathbf {e}}_{2}\) are, respectively, the maximum absolute error (MAE) arisen in the first and second simulations. In addition, \(Q_{i}=(M_{i}+1)(N_{i}+1)\) for \(i=1, 2\) express the number of the shifted HPs applied in the first and second implementations, respectively. Meanwhile, we use only the first 35th terms of the series expressing the Mittag–Leffler functions.

Example 1

Consider Eq. (1.1) on \([0,1]\times [0,2]\) with \(\mu _{1}=\frac{1}{2}\), \(\mu _{2}=1\), \(\sigma _{1}=1\), \(\sigma _{2}=\frac{1}{2}\), \(\kappa _{1}=2\), \(\kappa _{2}=1\) and

$$\begin{aligned} f_{1}(x,t)&=\left( \frac{6{\mathbf {C}}(\xi (x,t))t^{3}}{1-\xi (x,t)}\mathbf{E }_{\xi (x,t),4}\left( \frac{-\xi (x,t)\,t^{\xi (x,t)}}{1-\xi (x,t))}\right) \right) \sin (x)\\&\quad +\,\frac{1}{2}\mu _{1}t^{6}\sin (2x)+\sigma _{1}t^{6}\left( \cos ^{2}(x)-\sin ^{2}(x)\right) \\&\quad +\,\kappa _{1}t^{3}\sin (x), \\ f_{2}(x,t)&=\left( \frac{6{\mathbf {C}}(\vartheta (x,t))t^{3}}{1-\vartheta (x,t)}\mathbf{E }_{\vartheta (x,t),4}\left( \frac{-\vartheta (x,t)\,t^{\vartheta (x,t)}}{1-\vartheta (x,t))}\right) \right) \cos (x)\\&\quad -\,\frac{1}{2}\mu _{2}t^{6}\sin (2x)+\sigma _{2}t^{6}\left( \cos ^{2}(x)-\sin ^{2}(x)\right) \\&\quad +\,\kappa _{2}t^{3}\cos (x). \end{aligned}$$

The required initial and boundary conditions (IBCs) are obtained by the exact solution

$$\begin{aligned} \left<u(x,t),v(x,t)\right>= \left<t^{3}\sin (x),t^{3}\cos (x)\right>. \end{aligned}$$

The expressed method via two values (a, b) is used for solving this system where \(\xi (x,t)=0.45+0.33\sin (xt)\) and \(\vartheta (x,t)=0.85-e^{-xt}\). The obtained solutions are expressed in Table 1 and Fig. 1. The illustrated results emphasize the applicability and the high accuracy of the expressed method. Meanwhile, from Table 1, it can be seen that the obtained results are similar for two selections (a, b).

Table 1 The results extracted for Example 1 with two values (a, b)
Full size table
Fig. 1
figure 1

The results elicited for Example 1 with \(\left( a=-\frac{1}{2},b=-\frac{1}{2}\right)\) and \((M=N=8)\)

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Example 2

Consider Eq. (1.1) on \([0,\pi ]\times [0,1]\) with \(\mu _{1}=1\), \(\mu _{2}=2\), \(\sigma _{1}=\frac{1}{2}\), \(\sigma _{2}=\frac{1}{3}\), \(\kappa _{1}=2\), \(\kappa _{2}=3\) and

$$\begin{aligned} f_{1}(x,t)&=\left( \frac{{\mathbf {C}}(\xi (x,t))}{1-\xi (x,t)}\sum _{k=0}^{\infty }(-t)^{k+1}\mathbf{E }_{\xi (x,t),k+2} \right. \\&\quad \left. \left( \frac{-\xi (x,t)\,t^{\xi (x,t)}}{1-\xi (x,t))}\right) \right) \sin (x)+\frac{1}{2}\mu _{1}e^{-2t}\sin (2x)\\&\quad +\sigma _{1}e^{-3t}\left( \cos ^{2}(x)-\sin ^{2}(x)\right) +\kappa _{1}e^{-t}\sin (x),\\ f_{2}(x,t)&=\left( \frac{{\mathbf {C}}(\vartheta (x,t))}{1-\vartheta (x,t)}\sum _{k=0}^{\infty }(-2t)^{k+1}\mathbf{E }_{\vartheta (x,t),k+2} \right. \\&\quad \left. \left( \frac{-\vartheta (x,t)\,t^{\vartheta (x,t)}}{1-\vartheta (x,t))}\right) \right) \cos (x)-\frac{1}{2}\mu _{2}e^{-4t}\sin (2x)\\&\quad +\sigma _{2}e^{-3t}\left( \cos ^{2}(x)-\sin ^{2}(x)\right) +\kappa _{2}e^{-2t}\cos (x). \end{aligned}$$

The associated IBCs are identified by the analytic solution

$$\begin{aligned} \left<u(x,t),v(x,t)\right>= \left<e^{-t}\sin (x),e^{-2t}\cos (x)\right>. \end{aligned}$$

The established method with \((a=2,b=1)\) is implemented to solve this equation for two selections \((\xi (x,t),\vartheta (x,t))\). The obtained solutions are portrayed in Table 2 and Fig. 2.

Table 2 The results extracted for Example 2 with two selections \((\xi (x,t),\vartheta (x,t))\)
Full size table
Fig. 2
figure 2

The results elicited for Example 2 with \(\xi (x,t)=0.35+0.25\sin (2\pi xt)\), \(\vartheta (x,t)=0.45+0.35\cos (2\pi xt)\) and \((M=N=9)\)

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Example 3

Consider Eq. (1.1) on \(\left[ 0,\frac{\pi }{2}\right] \times \left[ 0,\frac{3}{2}\right]\) with \(\mu _{1}=\frac{1}{3}\), \(\mu _{2}=\frac{1}{2}\), \(\sigma _{1}=\frac{3}{2}\), \(\sigma _{2}=\frac{4}{3}\), \(\kappa _{1}=\frac{1}{2}\), \(\kappa _{2}=2\) and

$$\begin{aligned} f_{1}(x,t)&=\left( \frac{{\mathbf {C}}(\xi (x,t))}{1-\xi (x,t)}\sum _{k=0}^{\infty }(-1)^{k}t^{2k+1}\mathbf{E }_{\xi (x,t),2k+2} \right. \\&\quad \left. \left( \frac{-\xi (x,t)\,t^{\xi (x,t)}}{1-\xi (x,t))}\right) \right) \cosh (x)+\frac{1}{2}\mu _{1}\sin ^{2}(t)\sinh (2x)\\&\quad +\frac{1}{2}\sigma _{1}\sin (2t)\left( \sinh ^{2}(x)+\cosh ^{2}(x)\right) -\kappa _{1}\sin (t)\cosh (x),\\ f_{2}(x,t)&=\left( \frac{{\mathbf {C}}(\vartheta (x,t))}{1-\vartheta (x,t)}\sum _{k=0}^{\infty }(-1)^{k+1}t^{2k+2}\mathbf{E }_{\vartheta (x,t),2k+3} \right. \\&\quad \left. \left( \frac{-\vartheta (x,t)\,t^{\vartheta (x,t)}}{1-\vartheta (x,t))}\right) \right) \sinh (x)+\frac{1}{2}\mu _{2}\cos ^{2}(t)\sinh (2x)\\&\quad +\frac{1}{2}\sigma _{2}\sin (2t)\left( \sinh ^{2}(x)+\cosh ^{2}(x)\right) -\kappa _{2}\cos (t)\sinh (x). \end{aligned}$$

The associated IBCs for this system can be extracted from the analytic solution

$$\begin{aligned} \left<u(x,t),v(x,t)\right>= \left<\sin (t)\cosh (x),\cos (t)\sinh (x)\right>. \end{aligned}$$

The expressed method with \((a=-\frac{1}{2},b=\frac{1}{2})\) is implemented to solve this equation for two selections \((\xi (x,t),\vartheta (x,t))\). The numerical solutions are shown in Table 3 and Fig. 3.

Table 3 The results extracted for Example 3 with two selections \((\xi (x,t),\vartheta (x,t))\)
Full size table
Fig. 3
figure 3

The results elicited for Example 3 with \(\xi (x,t)=0.35-0.33\sin (2xt)\), \(\vartheta (x,t)=0.55-0.25e^{-xt}\) and \((M=8,N=9)\)

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8 Conclusion

In this study, a practicable spectral method based on the shifted Hahn polynomials is employed to obtain the approximate solution for a category of non-singular variable-order time fractional coupled Burgers’ equations. The fractional differentiation is considered in the Atangana–Balaenu–Caputo sense. Some new operational matrices are extracted for the shifted Hahn polynomials. Three examples are examined in order to show and validate the efficiency of the proposed algorithm. The illustrated results manifest an exponential rate of convergence. It is worth noting that the Mittag–Leffler functions which are conventionally defined as Definition 2.1 may be represented by the elementary functions introduced in [33] as well. As the future directions, the numerical approach devised in this study may be a substantial help in many other researches such as computing the multi-fractal dimension D(t) of network traffic investigated in [34].