We propose a pseudospectral algorithm based on Chebyshev polynomials to integrate the spatial variable for the \((1+1)\) parabolic PDEs with discrete time delay. The problem is then transformed into a system of ODEs with time delay. The algorithm is extended to treat the coupled parabolic PDEs and \((2+1)\) parabolic PDEs with time delay. The spatial variables are integrated based on the Chebyshev pseudospectral scheme, in which we use the Gauss-Lobatto interpolation nodes, to convert the problems to those of ODEs with time delay. The continuous RK scheme is employed to solve the resulting systems of ODEs.

###
\((1+1)\) Parabolic PDEs with time delay

Consider the \((1+1)\) parabolic PDEs with time delay of the form

$$ \frac{\partial u(x,t)}{\partial{t}}= \eta \frac{\partial^{2} u(x,t)}{\partial{x^{2}}}+ \lambda_{1} u(x,t-\tau)+\lambda_{2} u^{2}(x,t-\tau) +f(x,t),\quad(x, t)\in\mathrm{I} \times\mathrm{T}, $$

(8)

with the boundary conditions

$$ u(-1,t)=c_{1}(t),\qquad u(1,t)=c_{2}(t),\quad t \in\mathrm{T}, $$

(9)

and the initial state

$$ u(x,t)=c_{3}(x,t),\quad (x, t)\in \mathrm{I} \times[-\tau,0], $$

(10)

where \(\mathrm{I}\equiv[-1,1]\), \(\mathrm{T}\equiv(0,T]\), and *η* and \(\tau>0\) are the diffusion coefficient and delay parameter.

The polynomial approximation of degree *N* to \(u(x,t)\) may be expressed in terms of the orthogonal family \(\{T_{i}(x)\} \) in the form

$$ u(x,t)=\sum_{i=0}^{N}a_{i}(t) T_{i}(x). $$

(11)

It follows from (4) and (5) that

$$ a_{i}(t)=\frac{1}{h_{i}} \int_{-1}^{1}u(x,t) w(x) T_{i}(x)\,dx. $$

(12)

It can be seen directly from (4) that \(a_{j}(t)\) may be expanded in the form

$$a_{i}(t)=\frac{1}{h_{i}} \sum _{j=0}^{N}T_{i}( \zeta_{N,j}) \varpi_{N,j} u(\zeta_{N,j},t), $$

(13)

where \(\zeta_{N,j}\) (\(0\leq j\leq N\)) are the zeros and \(\varpi_{N,j}\) (\(0\leq j\leq N\)) are the corresponding quadrature weights.

We can further rewrite (11) as

$$ u(x,t)=\sum_{j=0}^{N} \Biggl(\sum_{i=0}^{N}\frac{1}{h_{i}}T_{i} (\zeta_{N,j})T_{i}(x)\varpi_{N,j} \Biggr) u( \zeta_{N,j},t). $$

(14)

The first-order spatial partial derivative at a specific collocation node \(\zeta_{N,n}\) can be obtained from (14) as

$$ u_{x}(\zeta_{N,n},t)= \sum _{i=0}^{N}\mu_{ni}u( \zeta_{N,i},t),\quad 0\leq n \leq N, $$

(15)

where

$$ \mu_{ni}=\sum_{j=0}^{N} \frac{1}{h_{j}}T_{j} ( \zeta_{N,i})\partial_{x} \bigl(T_{j}( \zeta_{N,n}) \bigr) \varpi_{N,i}. $$

(16)

This result can be extended to compute the second-order spatial partial derivative at a specific collocation node \(\zeta_{N,n}\) as

$$ u_{xx}(\zeta_{N,n},t)= \sum _{i=0}^{N}\gamma_{ni}u( \zeta_{N,i},t),\quad 0\leq n \leq N, $$

(17)

where

$$ \gamma_{ni}=\sum _{j=0}^{N}\frac{1}{h_{j}}T_{j} ( \zeta_{N,i})\partial_{xx} \bigl(T_{j}( \zeta_{N,n}) \bigr) \varpi_{N,i}. $$

(18)

In the context of Chebyshev pseudospectral approximation, putting (14) and (17) in (8) gives

$$\begin{aligned} &\dot{u}_{n}(t)=\eta \sum _{i=0}^{N}\gamma_{ni}u_{i}(t)+ \lambda_{1} u_{n}(t-\tau)+\lambda_{2} u^{2}_{n}(t-\tau)+f_{n}(t), \\ &\quad 1\leq n \leq N-1, \end{aligned}$$

(19)

where

$$\begin{aligned}& u_{k}(t)=u(\zeta_{N,k},t),\qquad u_{k}(t- \tau)=u(\zeta _{N,k},-\tau), \\& f_{k}(t)=f(\zeta_{N,k},t),\quad 1\leq k \leq N-1. \end{aligned}$$

The boundary conditions (9) are satisfied exactly at the two collocation points \(\zeta_{N,0}=-1\) and \(\zeta_{N,N}=1\). Furthermore, the preceding equation provides a system of \((N-1)\) ODEs with discrete time delay *τ*, namely

$$ \dot{u}_{n}(t)=\eta \sum _{i=1}^{N-1}\gamma_{ni}u_{i}(t)+ \rho_{n}(t)+\lambda_{1} u_{n}(t-\tau)+ \lambda_{2} u^{2}_{n}(t-\tau)+f_{n}(t), $$

(20)

subject to

$$ u_{n}(t)=c_{3}( \zeta_{N,n},t),\quad n=1,\ldots, N-1, t\in[-\tau,0], $$

(21)

where

$$\rho_{n}(t)=\gamma_{n0} c_{1}(t)+ \gamma_{nN} c_{2}(t). $$

Finally, the system (20)-(21) can be written in a matrix form as

$$ \begin{aligned} &\dot{\mathbf{U}}(t)=\mathbf{G} \bigl(t,u(t), u(t-\tau) \bigr), \\ &\mathbf{U}(0)=\mathbf{c},\quad t\in[-\tau,0], \end{aligned} $$

(22)

where

$$\begin{aligned}& \dot{\mathbf{U}}(t)= \bigl[\dot{u}_{1}(t),\dot{u}_{2}(t), \ldots,\dot{u}_{N-1}(t) \bigr]^{T}, \\& \mathbf{c}= \bigl[c_{3}(\zeta_{N,1}),c_{3}( \zeta_{N,2}),\ldots,c_{3}(\zeta _{N,N-1}) \bigr]^{T}, \\& \mathbf{G} \bigl(t,u(t),u(t-\tau) \bigr)= \bigl[g_{1} \bigl(t,u(t), u(t- \tau) \bigr),\ldots ,g_{N-1} \bigl(t,u(t), u(t-\tau) \bigr) \bigr]^{T}, \end{aligned}$$

and

$$g_{i} \bigl(t,u(t), u(t-\tau) \bigr)=\eta \sum _{j=1}^{N-1} \gamma _{ij}u_{j}(t)+ \rho_{i}(t)+ \lambda_{1} u_{i}(t-\tau)+ \lambda_{2} u^{2}_{i}(t-\tau)+f_{i}(t). $$

The previous system of first-order ODEs with time delay (22) is integrated by the continuous RK scheme [35, 36].

### Coupled \((1+1)\) parabolic PDEs with time delay

The main objective of this section is to develop the Chebyshev pseudospectral method to numerically solve the coupled parabolic PDEs with discrete time delay. This converts the following coupled parabolic PDEs with time delay into system of ODEs with time delay:

$$ \left . \textstyle\begin{array}{l@{}} \frac{\partial u(x,t)}{\partial{t}}= \eta_{1} \frac{\partial^{2} u(x,t)}{\partial{x^{2}}} +u(x,t) (1+\lambda_{1} u(x,t- \tau)+\kappa_{1} v(x,t-\tau) )+r(x,t),\\ \frac{\partial v(x,t)}{\partial{t}}=\eta_{2} \frac{\partial^{2} v(x,t)}{\partial{x^{2}}} +v(x,t) (1+ \lambda_{2} u(x,t-\tau)+\kappa_{2} v(x,t-\tau) )+s(x,t),\\ \quad (x,t)\in \mathrm{I} \times\mathrm{T}, \end{array}\displaystyle \right \} $$

(23)

subject to

$$ \begin{aligned} &u(-1,t)=c_{1}(t),\qquad u(1,t)=c_{2}(t), \\ &v(-1,t)=c_{3}(t),\qquad v(1,t)=c_{4}(t),\quad t\in \mathrm{T}, \end{aligned} $$

(24)

and initial data

$$ u(x,t)=c_{5}(x,t),\qquad v(x,t)=c_{6}(x,t), \quad (x,t) \in\mathrm{I} \times[-\tau,0]. $$

(25)

Now, we outline the main step of the Chebyshev pseudospectral algorithm for the coupled parabolic PDEs with time delay. Let us expand the approximate solutions as

$$ \left . \textstyle\begin{array}{l@{}} u(x,t)=\sum_{i=0}^{N}a_{i}(t) T_{i}(x),\\ v(x,t)=\sum_{i=0}^{N}b_{i}(t) T_{i}(x). \end{array}\displaystyle \right \} $$

(26)

The expansion coefficients \(\{a_{j}(t)\mbox{ and }b_{j}(t)\}\), can be described in terms of the solution at the Chebyshev Gauss-Lobatto quadrature points \(\{\zeta_{N,j}\}\), \(j=0,1,\ldots,N\), as

$$ \left . \textstyle\begin{array}{l@{}} a_{i}(t)= \frac{1}{h_{i}}\sum_{j=0}^{N}T_{i}( \zeta_{N,j}) \varpi_{N,j} u(\zeta_{N,j},t),\\ b_{i}(t)=\frac{1}{h_{i}}\sum_{j=0}^{N}T_{i} (\zeta_{N,j})\varpi_{N,j} v(\zeta_{N,j},t). \end{array}\displaystyle \right \} $$

(27)

Therefore, the approximate solutions (26) can be expressed in terms of Chebyshev polynomials, at the Chebyshev Gauss-Lobatto quadrature points, in the form

$$ \left . \textstyle\begin{array}{l@{}} u(x,t)=\sum_{j=0}^{N} (\sum_{i=0}^{N}\frac{1}{h_{i}}T_{i} (\zeta_{N,j})T_{i}(x)\varpi_{N,j} ) u( \zeta_{N,j},t),\\ v(x,t)=\sum_{j=0}^{N} (\sum_{i=0}^{N}\frac {1}{h_{i}}T_{i} ( \zeta_{N,j})T_{i}(x)\varpi_{N,j} ) v( \zeta_{N,j},t). \end{array}\displaystyle \right \} $$

(28)

The first-order partial derivative with respect to *x* for the solutions (28), at a specific collocation node \(\zeta_{N,n}\) can be written as

$$ \left . \textstyle\begin{array}{l@{}} u_{x}( \zeta_{N,i},t)=\sum_{j=0}^{N} \mu_{ij}u(\zeta _{N,j},t),\\ v_{x}(\zeta_{N,i},t)=\sum_{j=0}^{N} \mu_{ij}v(\zeta_{N,j},t), \quad i=0,1,\ldots,N . \end{array}\displaystyle \right \} $$

(29)

According to the previous step, the second-order spatial derivatives of the approximate solutions (28) are

$$ \left . \textstyle\begin{array}{l@{}} u_{xx}( \zeta_{N,i},t)=\sum_{j=0}^{N} \gamma_{ij}u(\zeta_{N,j},t),\\ v_{xx}(\zeta_{N,i},t)=\sum_{j=0}^{N} \gamma_{ij}v(\zeta_{N,j},t),\quad i=0,1,\ldots,N. \end{array}\displaystyle \right \} $$

(30)

Applying the Chebyshev pseudospectral approximation [37, 38] for the coupled parabolic PDEs with discrete time delay (23), at the nodes of Chebyshev-Gauss-Lobatto quadrature. Thanks to (28)-(30), the coupled equations (23) may be written in the form

$$ \left . \textstyle\begin{array}{l@{}} \dot{u}_{n}(t)= \eta_{1} \sum_{i=0}^{N} \gamma_{ni}u_{i}(t)+u_{n}(t) (1+ \lambda_{1} u_{n}(t-\tau)+\kappa_{1} v_{n}(t-\tau) )+r_{n}(t),\\ \dot{v}_{n}(t)=\eta_{2} \sum_{i=0}^{N} \gamma _{ni}v_{i}(t)+v_{n}(t) (1+ \lambda_{2} u_{n}(t-\tau)+\kappa_{2} v_{n}(t-\tau) )+s_{n}(t), \end{array}\displaystyle \right \} $$

(31)

where

$$\begin{aligned}& u_{k}(t)=u(\zeta_{N,k},t),\qquad v_{k}(t)=v( \zeta_{N,k},t), \\& r_{k}(t)=r(\zeta_{N,k},t), \qquad s_{k}(t)=s( \zeta_{N,k},t), \quad k=1,\ldots,N-1, n=1,\ldots,N-1. \end{aligned}$$

The boundary conditions (24) have been satisfied exactly at the two collocation points \(\zeta_{N,0}=-1\) and \(\zeta_{N,N}=1\).

Let us denote

$$\rho_{n}(t)=\gamma_{n0} c_{1}(t)+ \gamma_{nN} c_{2}(t),\qquad \sigma _{n}(t)= \gamma_{n0} c_{3}(t)+\gamma_{nN} c_{4}(t), $$

then the coupled parabolic PDEs (23) with the set of boundary conditions (24) are transformed into the following ODEs with time delay:

$$ \left . \textstyle\begin{array}{l@{}} \dot{u}_{n}(t)= \eta_{1} \sum_{i=1}^{N-1} \gamma_{ni}u_{i}(t)+u_{n}(t) (1+ \lambda_{1} u_{n}(t-\tau)+\kappa_{1} v_{n}(t-\tau) )+r_{n}(t)+\eta_{1} \rho_{n}(t), \\ \dot{v}_{n}(t)=\eta_{2} \sum_{i=1}^{N-1} \gamma _{ni}v_{i}(t)+v_{n}(t) (1+ \lambda_{2} u_{n}(t-\tau)+\kappa_{2} v_{n}(t-\tau ) )+s_{n}(t)+\eta_{2} \sigma_{n}(t), \end{array}\displaystyle \right \} $$

(32)

and from (25) we get the initial values

$$ \left . \textstyle\begin{array}{l@{}} u_{n}(t)=c_{5}( \zeta_{N,n},t),\\ v_{n}(t)=c_{6}(\zeta_{N,n},t), \quad n=1,\ldots, N-1, t \in[-\tau,0]. \end{array}\displaystyle \right \} $$

(33)

The matrix formulation of the previous systems is

$$\begin{aligned} &\begin{pmatrix} \dot{u}_{1}(t)\\ \dot{u}_{2}(t)\\ \vdots\\ \dot{u}_{N-1}(t)\\ \dot{v}_{1}(t)\\ \dot{v}_{2}(t)\\ \vdots\\ \dot{v}_{N-1}(t) \end{pmatrix} \\ &\quad= \begin{pmatrix} \eta_{1} \sum_{i=0}^{N}\gamma_{1i}u_{i}(t)+u_{1}(t)(1+\lambda_{1} u_{1}(t-\tau)+\kappa_{1} v_{1}(t-\tau))+r_{1}(t)+\eta_{1} \rho_{1}(t)\\ \eta_{1} \sum_{i=0}^{N}\gamma_{2i}u_{i}(t)+u_{2}(t)(1+\lambda_{1} u_{2}(t-\tau)+\kappa_{1} v_{2}(t-\tau))+r_{2}(t)+\eta_{1} \rho_{2}(t)\\ \cdots\\ \cdots\\ \cdots\\ \eta_{1} \sum_{i=0}^{N}\gamma _{N-1i}u_{i}(t)+u_{N-1}(t)(1+\lambda_{1} u_{N-1}(t-\tau)+\kappa_{1} v_{N-1}(t-\tau))+r_{N-1}(t)+\eta_{1} \rho_{N-1}(t)\\ \eta_{2} \sum_{i=0}^{N}\gamma_{1i}v_{i}(t)+v_{1}(t)(1+\lambda_{2} u_{1}(t-\tau)+\kappa_{2} v_{1}(t-\tau))+s_{1}(t)+\eta_{2}\sigma_{1}(t) \\\eta_{2} \sum_{i=0}^{N}\gamma_{2i}v_{i}(t)+v_{2}(t)(1+\lambda_{2} u_{2}(t-\tau)+\kappa_{2} v_{2}(t-\tau))+s_{2}(t)\eta_{2}\sigma_{2}(t)\\ \cdots\\ \cdots\\ \cdots\\ \eta_{2} \sum_{i=0}^{N}\gamma_{N-1i}v_{i}(t)+v_{N-1}(t)(1+\lambda_{2} u_{N-1}(t-\tau)+\kappa_{2} v_{N-1}(t-\tau))+s_{N-1}(t)\eta_{2}\sigma_{N-1}(t) \end{pmatrix}, \end{aligned}$$

(34)

subject to the vector of initials

$$ \begin{pmatrix} u_{1}(t)\\ u_{2}(t)\\ \vdots\\ u_{N-1}(t)\\ v_{1}(t)\\ v_{2}(t)\\ \vdots\\ v_{N-1}(t) \end{pmatrix}= \begin{pmatrix} c_{5}(\zeta_{N,1},t)\\ c_{5}(\zeta_{N,2},t)\\ \vdots\\ c_{5}(\zeta_{N,N-1},t)\\ c_{6}(\zeta_{N,1},t)\\ c_{6}(\zeta_{N,2},t)\\ \vdots\\ c_{6}(\zeta_{N,N-1},t) \end{pmatrix}, \quad t\in[-\tau,0]. $$

(35)

###
\((2+1)\) Parabolic PDEs with time delay

In the present subsection, we extend the application of the Chebyshev pseudospectral approximation for a numerical treatment of the \((2+1)\) parabolic PDEs with time delay, namely

$$\begin{aligned} &\frac{\partial v(p,q,t)}{\partial t} =\mu_{1} \frac{\partial^{2} v(p,q,t)}{\partial p^{2}}+\mu_{2} \frac{\partial^{2} v(p,q,t)}{\partial q^{2}}+\mu_{3} v(p,q,t- \tau)+f_{1}(p,q,t), \\ &\quad (p,q, t)\in \Omega_{1}\times\Omega_{2}\times \Omega_{3} , \end{aligned}$$

(36)

where

$$\Omega_{1}\equiv [l_{0},l_{1}],\qquad \Omega_{2}\equiv[l_{2},l_{3}],\quad\mbox{and}\quad \Omega_{3}\equiv (0,T], $$

subject to the initial data

$$ v(p,q,t)=c_{1}(p,q,t),\quad (p,q,t) \in \Omega_{1} \times \Omega_{2}\times[-\tau,0], $$

(37)

and the boundary conditions

$$ \begin{aligned} &v(l_{0},q,t)=c_{2}(q,t), \qquad v(l_{1},q,t)=c_{3}(q,t),\quad(q,t) \in \Omega_{2}\times\Omega_{3}, \\ & v(p,l_{2},t)=c_{4}(p,t),\qquad v(p,l_{3},t)=c_{5}(p,t), \quad (p,t)\in\Omega_{1}\times\Omega_{3}. \end{aligned} $$

(38)

Firstly, we suppose we have the change of spatial variables

$$x=\frac{2}{l_{1}-l_{0}}p+\frac{l_{0}+l_{1}}{l_{0}-l_{1}},\qquad y=\frac {2}{l_{3}-l_{2}}q+ \frac{l_{2}+l_{3}}{l_{2}-l_{3}}, $$

and also \(v(p,q,t)=u(x,y,t)\). For the space variables, the above-mentioned problem transforms into another two-dimensional problem in \([-1,1]\times[-1,1]\),

$$\begin{aligned} \frac{\partial u(x,y,t)}{\partial t} ={}&\mu_{1} \biggl( \frac{2}{l_{1}-l_{0}} \biggr)^{2} \frac{\partial^{2} u(x,y,t)}{\partial x^{2}} \\ &{}+\mu_{2} \biggl(\frac{2}{l_{3}-l_{2}} \biggr)^{2} \frac{\partial^{2} u(x,y,t)}{\partial y^{2}}+\mu_{3} u(x,y,t-\tau)+f_{2}(x,y,t), \\ &{} (x,y, t)\in\mathrm{I}^{2}\times(0,T], \end{aligned}$$

(39)

subject to the initial data

$$ u(x,y,t)=c_{6}(x,y,t),\quad (x,y,t) \in \mathrm{I}^{2}\times[-\tau,0], $$

(40)

and the boundary conditions

$$ \begin{aligned} &u(-1,y,t)=c_{7}(y,t),\qquad u(1,y,t)=c_{8}(y,t),\quad (y,t)\in \mathrm{I}\times\mathrm{T}, \\ &u(x,-1,t)=c_{9}(x,t),\qquad u(x,1,t)=c_{10}(x,t),\quad (x,t)\in \mathrm{I}\times \mathrm{T}. \end{aligned} $$

(41)

Secondly, we construct an algorithm based on the Chebyshev pseudospectral approximation to convert the previous two-dimensional PPDE into a system of ODEs with time delay. We expand the approximate solution in the form

$$ u(x,y,t)=\sum_{i=0}^{N} \sum_{j=0}^{M}a_{i,j}(t) T_{i}(x) T_{j}(y). $$

(42)

Let \(\{\zeta_{N,i}; 0\leq i\leq N\}\) and \(\{\lambda _{M,j}; 0\leq j\leq M\}\) be the zeros of the Chebyshev polynomials \(T_{N}(x)\) and \(T_{M}(y)\), respectively. Making use of the Chebyshev Gauss-Lobatto quadrature and due to the orthogonality relation (4), we have the inversion formulas for \(a_{i,j}(t)\), thus

$$ a_{i,j}(t)=\frac{1}{h_{i} h_{j}} \sum _{l=0}^{N}\sum_{k=0}^{M} \bigl(T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} \bigr) u( \zeta_{N,l},\lambda _{M,k},t). $$

(43)

The approximate solution (42) may be expressed in the form

$$ u(x,y,t)=\sum_{i=0}^{N} \sum_{j=0}^{M}\sum _{l=0}^{N}\sum_{k=0}^{M} \frac{ (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i}h_{j}} T_{i}(x)T_{j}(y) u_{l,k}(t), $$

(44)

where

$$u(\zeta_{N,n},\lambda _{M,m},t)=u_{n,m}(t). $$

In what follows, the first-order partial derivative with respect to *x* for the solutions (44), at the specific collocation nodes \(\zeta_{N,n}\) and \(\lambda _{M,m}\) can be written as

$$\begin{aligned} &\partial_{x} u_{n,m}(t)=\sum _{l=0}^{N}\sum_{k=0}^{M} \sum_{i=0}^{N}\sum _{j=0}^{M}\frac{ (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i} h_{j}} \partial_{x} \bigl(T_{i}(\zeta_{N,n}) \bigr)T_{j}( \lambda _{M,m}) u_{l,k}(t), \\ &\quad n=0,1,\ldots, N,m=0,1,\ldots, M, \end{aligned}$$

(45)

or

$$ \partial_{x} u_{n,m}(t)=\sum _{l=0}^{N}\sum_{k=0}^{M} \gamma^{n,m}_{l,k} u_{l,k}(t), $$

(46)

where \(\gamma^{n,m}_{i,j}\) are the expansion coefficients of the derivative

$$ \gamma^{n,m}_{l,k}=\sum _{i=0}^{N}\sum_{j=0}^{M} \frac{ (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i} h_{j}} \partial_{x} \bigl(T_{i}( \zeta_{N,n}) \bigr)T_{j}(\lambda _{M,m}). $$

(47)

Accordingly, the first-order derivative with respect to *y* for \(u(x,y,t)\), at the nodes \(\zeta_{N,n}\) and \(\lambda _{M,m}\), is

$$\begin{aligned} &\partial_{y} u_{n,m}(t)=\sum _{l=0}^{N}\sum_{k=0}^{M} \sum_{i=0}^{N}\sum _{j=0}^{M}\frac{ (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i} h_{j}} T_{i}( \zeta_{N,n})\partial_{y} \bigl(T_{j}( \lambda _{M,m}) \bigr) u_{l,k}(t), \\ &\quad n=0,1,\ldots, N,m=0,1,\ldots, M, \end{aligned}$$

(48)

or

$$ \partial_{y} u_{n,m}(t)=\sum _{l=0}^{N}\sum_{k=0}^{M} \delta^{n,m}_{l,k} u_{l,k}(t), $$

(49)

where

$$ \delta^{n,m}_{l,k}=\sum _{i=0}^{N}\sum_{j=0}^{M} \frac{ (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i} h_{j}} T_{i}(\zeta_{N,n})\partial_{y} \bigl(T_{j}(\lambda _{M,m}) \bigr). $$

(50)

Also, the second-order partial derivatives with respect to *x* and *y* are, respectively, given by

$$ \partial^{2}_{x} u_{n,m}(t)=\sum_{l=0}^{N}\sum _{k=0}^{M} \xi^{n,m}_{l,k} u_{l,k}(t), $$

(51)

where

$$ \xi^{n,m}_{l,k}=\sum _{i=0}^{N}\sum_{j=0}^{M} \frac { (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i} h_{j}} \partial^{2}_{x} \bigl(T_{i}( \zeta_{N,n}) \bigr) T_{j}(\lambda _{M,m}), $$

(52)

and

$$ \partial^{2}_{y} u_{n,m}(t)=\sum_{l=0}^{N}\sum _{k=0}^{M}\eta^{n,m}_{l,k} u_{l,k}(t), $$

(53)

where

$$ \eta^{n,m}_{l,k}=\sum _{i=0}^{N}\sum_{j=0}^{M} \frac { (T_{j}(\lambda _{M,k})\varpi_{M,k} T_{i}(\zeta_{N,l})\varpi_{N,l} )}{h_{i} h_{j}} T_{i}(\zeta_{N,n})\partial^{2}_{y} \bigl(T_{j}(\lambda _{M,m}) \bigr). $$

(54)

Let us denote

$$\begin{aligned}& \epsilon_{1}=\mu_{1} \biggl(\frac{2}{l_{1}-l_{0}} \biggr)^{2},\qquad \epsilon_{2}=\mu _{2} \biggl( \frac{2}{l_{3}-l_{2}} \biggr)^{2}, \\& f_{2}(\zeta_{N,n},\lambda _{M,m},t)=f_{n,m}(t), \qquad c_{6}( \zeta_{N,n},\lambda _{M,m},t)=c_{n,m}(t). \end{aligned}$$

Moreover, the values of \(u_{0,k}(t)\), \(u_{N,k}(t)\), \(u_{l,0}(t)\), and \(u_{l,N}(t)\) can be given by

$$ \begin{aligned} &u_{0,k}(t)=c_{7}( \lambda _{M,k},t),\qquad u_{N,k}(t)=c_{8}( \lambda _{M,k},t),\quad k=0,\ldots,M, \\ & u_{l,0}(t)=c_{9}(\zeta_{N,l},t),\qquad u_{l,N}(t)=c_{10}( \zeta _{N,l},t),\quad l=0, \ldots,N. \end{aligned} $$

(55)

In the Chebyshev pseudospectral approximation for the two-dimensional version of parabolic PDEs with time delay, the residual of (39) is set to zero at \((N-1)\times(M-1)\) of the nodes of Chebyshev Gauss-Lobatto quadrature. Therefore, adopting (44)-(55) enables one to write (39)-(41) as a \((N-1)\times(M-1)\) system of ODEs with time delay,

$$\begin{aligned} &\dot{u}_{n,m}(t)= \epsilon_{1} \sum_{l=0}^{N}\sum _{k=0}^{M} \xi ^{n,m}_{l,k} u_{l,k}(t)+\epsilon_{2} \sum_{l=0}^{N} \sum_{k=0}^{M}\eta^{n,m}_{l,k} u_{l,k}(t)+\mu_{3} u_{n,m}(t-\tau)+f_{n,m}(t), \\ &\quad n=1,\ldots,N-1, m=1,\ldots,M-1, \end{aligned}$$

(56)

subject to the initial data

$$ u_{n,m}(t)=c_{n,m}(t),\quad n=1,\ldots,N-1,m=1, \ldots,M-1. $$

(57)

Finally, the system (56)-(57) may be arranged in the matrix form, which can be solved by the continuous RK scheme,

$$\begin{aligned} & \begin{pmatrix} \dot{u}_{1,1}(t)&\cdots&\dot{u}_{1,M-1}(t)\\ \dot{u}_{2,1}(t)&\vdots&\dot{u}_{2,M-1}(t)\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ \dot{u}_{N-2,1}(t)&\vdots&\dot{u}_{N-2,M-1}(t)\\ \dot{u}_{N-1,1}(t)&\cdots&\dot{u}_{N-1,M-1}(t) \end{pmatrix} \\ &\quad= \begin{pmatrix} {\L}_{1,1}(t, u_{1},\ldots,u_{N-1})&\cdots&{\L}_{1,M-1}(t, u_{1},\ldots ,u_{N-1})\\ {\L}_{2,1}(t, u_{1},\ldots,u_{N-1})&\vdots&{\L}_{2,M-1}(t, u_{1},\ldots ,u_{N-1})\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ {\L}_{N-2,1}(t,u_{1},\ldots,u_{N-1})&\vdots&{\L }_{N-2,M-1}(t,u_{1},\ldots,u_{N-1})\\ {\L}_{N-1,1}(t,u_{1},\ldots,u_{N-1})&\cdots&{\L }_{N-1,M-1}(t,u_{1},\ldots,u_{N-1}) \end{pmatrix}, \end{aligned}$$

(58)

$$\begin{aligned} & \begin{pmatrix} u_{1,1}(t)&\cdots&u_{1,M-1}(t)\\ u_{2,1}(t)&\vdots&u_{2,M-1}(t)\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ u_{N-2,1}(t)&\vdots&u_{N-2,M-1}(t)\\ u_{N-1,1}(t)&\cdots&u_{N-1,M-1}(t) \end{pmatrix}= \begin{pmatrix} c_{1,1}(t)&\cdots&c_{1,M-1}(t)\\ c_{2,1}(t)&\vdots&c_{2,M-1}(t)\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ \cdots& \ddots& \cdots\\ c_{N-2,1}(t)&\vdots&c_{N-2,M-1}(t)\\ c_{N-1,1}(t)&\cdots&c_{N-1,M-1}(t) \end{pmatrix}, \quad t\in[- \tau,0], \end{aligned}$$

(59)

where

$$\begin{aligned} {\L}_{n,m}(t,u_{1}, \ldots,u_{N-1})={}&\epsilon_{1} \sum _{l=0}^{N} \sum_{k=0}^{M} \xi^{n,m}_{l,k} u_{l,k}(t) \\ &{}+\epsilon_{2} \sum_{l=0}^{N} \sum_{k=0}^{M}\eta ^{n,m}_{l,k} u_{l,k}(t)+\mu_{3} u_{n,m}(t-\tau)+f_{n,m}(t), \\ &{}n=1,\ldots,N-1,m=1,\ldots,M-1. \end{aligned}$$

(60)

This system of first-order ODEs can be solved by the continuous RK scheme.