We seek the solution *u* of Eq. (10) of the form

$$\begin{array}{}{\displaystyle u(x,y)\approx \sum _{j=0}^{N}\sum _{i=0}^{M}u({x}_{j},{y}_{i}){L}_{ji}(x,y)}\end{array}$$(11)

where the functions *L*_{ji}(*x*,*y*) = *L*_{j}(*x*)*L*_{i}(*y*) are bivariate Lagrange polynomials defined as

$$\begin{array}{}{\displaystyle {L}_{j}(x)=\prod _{\begin{array}{c}k=0\\ k\ne j\end{array}}^{N}\frac{x-{x}_{k}}{{x}_{j}-{x}_{k}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{i}(y)=\prod _{\begin{array}{c}k=0\\ k\ne i\end{array}}^{M}\frac{y-{y}_{k}}{{y}_{i}-{x}_{k}}}\end{array}$$(12)

The functions *L*_{j}(*x*) and *L*_{i}(*y*) both obey the Kronecker delta equation, that is,

$$\begin{array}{}{\displaystyle {L}_{ji}({x}_{n},{y}_{m})=\left\{\begin{array}{l}1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{if}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j=n,\phantom{\rule{thinmathspace}{0ex}}i=m\\ 0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{otherwise}\end{array}\right.}\end{array}$$(13)

Before applying the spectral method, it is convenient to transform the physical domain [*a*,*b*] × [*c*,*d*] in the x-y axis to the computational domain [–1, 1] × [–1, 1] in the *ξ*-*η* axis using linear transformations *x*(*ξ*) = $\begin{array}{}{\displaystyle \frac{a+b}{2}+\frac{b-a}{2}}\end{array}$ *ξ* and *y*(*η*) = $\begin{array}{}{\displaystyle \frac{c+d}{2}+\frac{d-c}{2}\eta .}\end{array}$ Approximating the partial derivatives of *u* at Chebyshev-Gauss-Lobatto collocation points

$$\begin{array}{}{\displaystyle \{{x}_{j}{\}}_{j=0}^{N}=\mathrm{cos}(\frac{\pi j}{N}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\{{y}_{i}{\}}_{i=0}^{M}=\mathrm{cos}(\frac{\pi i}{M}),}\end{array}$$(14)

we have

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}u}{\mathrm{\partial}x}{|}_{\begin{array}{c}x={x}_{j}\\ y={y}_{i}\end{array}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\approx \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{p=0}^{N}\sum _{q=0}^{M}u({x}_{p},{y}_{q})\frac{d{L}_{p}({x}_{j})}{dx}{L}_{q}({y}_{i})}\end{array}$$(15)

$$\begin{array}{}{\displaystyle =\sum _{p=0}^{N}{D}_{jp}u({x}_{p},{y}_{i})=\mathbf{D}{\mathbf{U}}_{i}.}\end{array}$$(16)

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}u}{\mathrm{\partial}y}{|}_{\begin{array}{c}x={x}_{j}\\ y={y}_{i}\end{array}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\approx \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{q=0}^{M}{d}_{iq}u({x}_{j},{y}_{q})=\sum _{q=0}^{M}{d}_{iq}{\mathbf{U}}_{q},}\end{array}$$(17)

and

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}x\mathrm{\partial}y}{|}_{\begin{array}{c}x={x}_{j}\\ y={y}_{i}\end{array}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\approx \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{p=0}^{N}\sum _{q=0}^{M}u({x}_{p},{y}_{q})\frac{d{L}_{p}({x}_{j})}{dx}\frac{d{L}_{q}({y}_{i})}{dy}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{q=0}^{M}{d}_{iq}(\mathbf{D}{\mathbf{U}}_{q}),}\end{array}$$(18)

where **D̂** = $\begin{array}{}{\displaystyle (\frac{b-a}{2})}\end{array}$ and **d̂**_{iq}= $\begin{array}{}{\displaystyle (\frac{d-c}{2})}\end{array}$*d*_{iq} are the standard Chebyshev differentiation matrices [30] of orders (*N* + 1) × (*N* + 1) and (*M* + 1) × (*M* + 1), respectively, and **U**_{i} = (*u*(*x*_{0},*y*_{i}),*u*(*x*_{1},*y*_{i}), ⋯, *u*(*x*_{N},*y*_{i}))^{T}, for *i* = 0, 1, 2, ⋯, *N*. The superscript *T* denotes matrix transposition. Higher order derivatives of *u* are defined as follows:

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{n}u}{\mathrm{\partial}{x}^{n}}{|}_{\begin{array}{c}x={x}_{j}\\ y={y}_{i}\end{array}}\approx {\mathbf{D}}^{n}{\mathbf{U}}_{i},\phantom{\rule{thinmathspace}{0ex}}\frac{{\mathrm{\partial}}^{n}u}{\mathrm{\partial}{y}^{n}}{|}_{\begin{array}{c}x={x}_{j}\\ y={y}_{i}\end{array}}\approx \sum _{q=0}^{M}{d}_{iq}^{n}{\mathbf{U}}_{q},\phantom{\rule{thinmathspace}{0ex}}n=2,3,\cdots .}\end{array}$$

and

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{n+m}u}{\mathrm{\partial}{x}^{n}\mathrm{\partial}{y}^{m}}=\sum _{q=0}^{M}{d}_{iq}^{m}({\mathbf{D}}^{n}{\mathbf{U}}_{q}),\phantom{\rule{thinmathspace}{0ex}}n,m=1,2,\cdots}\end{array}$$

Expanding Eq. (10) we get

$$\begin{array}{}{\displaystyle {\alpha}_{5,r}\frac{{\mathrm{\partial}}^{2}{u}_{r+1}}{\mathrm{\partial}{x}^{2}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{4,r}\frac{{\mathrm{\partial}}^{2}{u}_{r+1}}{\mathrm{\partial}{y}^{2}}+{\alpha}_{3,r}\frac{{\mathrm{\partial}}^{2}{u}_{r+1}}{\mathrm{\partial}x\mathrm{\partial}y}+{\alpha}_{2,r}\frac{\mathrm{\partial}{u}_{r+1}}{\mathrm{\partial}x}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{1,r}\frac{\mathrm{\partial}{u}_{r+1}}{\mathrm{\partial}y}+{\alpha}_{0,r}{u}_{r+1}={K}_{r},}\end{array}$$(19)

where

$$\begin{array}{}{\displaystyle {\alpha}_{5,r}=\frac{\mathrm{\partial}F({u}_{r})}{\mathrm{\partial}{u}_{xx}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{4,r}=\frac{\mathrm{\partial}F({u}_{r})}{\mathrm{\partial}{u}_{yy}},{\alpha}_{3,r}=\frac{\mathrm{\partial}F({u}_{r})}{\mathrm{\partial}{u}_{xy}},}\\ {\displaystyle {\alpha}_{2,r}=\frac{\mathrm{\partial}F({u}_{r})}{\mathrm{\partial}{u}_{x}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{1,r}=\frac{\mathrm{\partial}F({u}_{r})}{\mathrm{\partial}{u}_{y}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{0,r}=\frac{\mathrm{\partial}F({u}_{r})}{\mathrm{\partial}u}}\end{array}$$

and

$$\begin{array}{}{\displaystyle {K}_{r}\phantom{\rule{thinmathspace}{0ex}}={\alpha}_{5,r}\frac{{\mathrm{\partial}}^{2}{u}_{r}}{\mathrm{\partial}{x}^{2}}+{\alpha}_{4,r}\frac{{\mathrm{\partial}}^{2}{u}_{r}}{\mathrm{\partial}{y}^{2}}+{\alpha}_{3,r}\frac{{\mathrm{\partial}}^{2}{u}_{r}}{\mathrm{\partial}x\mathrm{\partial}y}+{\alpha}_{2,r}\frac{\mathrm{\partial}{u}_{r}}{\mathrm{\partial}x}+{\alpha}_{1,r}\frac{\mathrm{\partial}{u}_{r}}{\mathrm{\partial}y}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+{\alpha}_{0,r}{u}_{r}-F({\mathbf{u}}_{r})+R(\mathbf{x}\mathbf{)}\mathbf{.}}\end{array}$$

Approximating the *u* and its derivatives in Eq. (19) at collocation points (*x*_{j},*y*_{i}) we get

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{5,r}(\mathbf{x},{y}_{i}){\hat{\mathbf{D}}}^{2}{\mathbf{U}}_{r+1,i}+{\alpha}_{4,r}(\mathbf{x},{y}_{i})\sum _{q=0}^{M}{\hat{d}}_{jq}^{2}{\mathbf{U}}_{r+1,q}}\\ {\displaystyle +\phantom{\rule{1em}{0ex}}{\alpha}_{3,r}(\mathbf{x},{y}_{i})\sum _{q=0}^{M}{\hat{d}}_{iq}^{2}({\hat{\mathbf{D}}}^{2}{\mathbf{U}}_{r+1,q})+{\alpha}_{2,r}(\mathbf{x},{y}_{i})\hat{\mathbf{D}}{\mathbf{U}}_{r+1,i}}\\ {\displaystyle +\phantom{\rule{1em}{0ex}}{\alpha}_{1,r}(\mathbf{x},{y}_{i})\sum _{q=0}^{M}{\hat{d}}_{iq}(\hat{\mathbf{D}}{\mathbf{U}}_{r+1,q})+{\alpha}_{0,i}(\mathbf{x},{y}_{i}){\mathbf{U}}_{r+1,i}}\\ {\displaystyle =\phantom{\rule{1em}{0ex}}{\mathbf{K}}_{r,i}}\end{array}$$(20)

where

$$\begin{array}{}{\alpha}_{a,r(\mathbf{X},{y}_{i})}\\ =\\ \phantom{\rule{1em}{0ex}}{\displaystyle \left[\begin{array}{c}{\alpha}_{a,r}({x}_{0},{y}_{i})\\ & {\alpha}_{a,r}({x}_{1},{y}_{i})\\ & & \ddots \\ & & & & {\alpha}_{a,r}({x}_{N},{y}_{i})\end{array}\right]}\end{array}$$

and

$$\begin{array}{}{\displaystyle {K}_{r,i}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{5,r}(\mathbf{x},{y}_{i}){\hat{\mathbf{D}}}^{2}{\mathbf{U}}_{r,i}+{\alpha}_{4,r}(\mathbf{x},{y}_{i})\sum _{q=0}^{M}{\hat{d}}_{jq}^{2}{\mathbf{U}}_{r,q}}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{2em}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{3,r}(\mathbf{x},{y}_{i})\sum _{q=0}^{M}{\hat{d}}_{iq}^{2}({\hat{\mathbf{D}}}^{2}{\mathbf{U}}_{r,q})+{\alpha}_{2,r}(\mathbf{x},{y}_{i})\hat{\mathbf{D}}{\mathbf{U}}_{r,i}}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{2em}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{1,r}(\mathbf{x},{y}_{i})\sum _{q=0}^{M}{\hat{d}}_{iq}(\hat{\mathbf{D}}{\mathbf{U}}_{r,q})+{\alpha}_{0,i}(\mathbf{x},{y}_{i}){\mathbf{U}}_{r,i}}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{2em}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}F({u}_{r}(\mathbf{x},{y}_{i}))+R(\mathbf{x},{y}_{i}).}\end{array}$$

In compact form, Eq. (20) can be written as:

$$\begin{array}{}{\displaystyle \left[\begin{array}{cccc}{A}_{0,0}& {A}_{0,1}& \cdots & {A}_{0,M}\\ {A}_{1,0}& {A}_{1,1}& \cdots & {A}_{1,M}\\ \vdots & \vdots & \ddots & \vdots \\ {A}_{N-1,0}& {A}_{N-1,1}& \cdots & {A}_{N-1,M}\\ {A}_{N,0}& {A}_{N,1}& \cdots & {A}_{N,M}\end{array}\right]\left[\begin{array}{c}{\mathbf{U}}_{r+1,0}\\ {\mathbf{U}}_{r+1,1}\\ \vdots \\ {\mathbf{U}}_{r+1,N-1}\\ {\mathbf{U}}_{r+1,N}\end{array}\right]=\left[\begin{array}{c}{\mathbf{K}}_{0}\\ {\mathbf{K}}_{1}\\ \vdots \\ {\mathbf{K}}_{M-1}\\ {\mathbf{K}}_{M}\end{array}\right]}\end{array}$$

where

$$\begin{array}{}{\displaystyle {A}_{ii}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{5,r}(\mathbf{x},{y}_{i}){\hat{\mathbf{D}}}^{2}+{\alpha}_{4,r}(\mathbf{x},{y}_{i}){\hat{d}}_{ii}\mathbf{I}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{3,r}(\mathbf{x},{y}_{i}){\hat{d}}_{ii}^{2}{\hat{\mathbf{D}}}^{2}+{\alpha}_{2,r}\mathbf{x},{y}_{i}\hat{\mathbf{D}}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\alpha}_{1,r}(\mathbf{x},{y}_{i}){\hat{d}}_{ii}\hat{\mathbf{D}}+{\alpha}_{0,r}(\mathbf{x},{y}_{i})}\\ {\displaystyle {A}_{ij}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\hat{d}}_{ij}\mathbf{I},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{when}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i\ne j}\end{array}$$

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