In the process of obtaining the numerical solutions of partial differential equations with the double EC method, the main idea or major step is to evaluate the necessary Chebyshev coefficients of the unknown function. So, in Section 2, we give the explicit relations between the polynomials ${E}_{r,s}(x,y)$ of an unknown function and those of its derivatives ${E}_{r,s}^{(i,j)}(x,y)$ for different nonnegative integer values of *i* and *j*.

In this section, we consider the higher-order linear PDE with variable coefficients of a general form

$\sum _{i=0}^{p}\sum _{j=0}^{r}{q}_{i,j}(x,y){u}^{(i,j)}(x,y)=f(x,y),\phantom{\rule{1em}{0ex}}-\mathrm{\infty}<x,y<\mathrm{\infty}$

(3.1)

with the conditions mentioned in [

23] as three possible cases:

$\sum _{t=1}^{\rho}\sum _{i=0}^{p}\sum _{j=0}^{r}{b}_{{t}_{i,j}}{u}^{(i,j)}({\omega}_{t},{\eta}_{t})=\lambda $

(3.2)

and/or

$\sum _{t=1}^{\upsilon}\sum _{i=0}^{p}\sum _{j=0}^{r}{c}_{{t}_{i,j}}(x){u}^{(i,j)}(x,{\gamma}_{t})=g(x)$

(3.3)

and/or

$\sum _{t=1}^{\vartheta}\sum _{i=0}^{p}\sum _{j=0}^{r}{d}_{{t}_{i,j}}(y){u}^{(i,j)}({\epsilon}_{t},y)=h(y).$

(3.4)

Here, ${u}^{(0,0)}(x,y)=u(x,y)$, ${u}^{(i,j)}(x,y)=\frac{{\partial}^{i+j}}{\partial {x}^{i}\partial {y}^{j}}u(x,y)$ and ${q}_{i,j}(x,y)$, $f(x,y)$, ${c}_{{t}_{i,j}}(x)$, $g(x)$, ${d}_{{t}_{i,j}}(y)$, $h(y)$ are known functions on the square $S(-\mathrm{\infty}<x,y<\mathrm{\infty})$. We now describe an approximate solution of this problem by means of double EC series as defined in (2.10). Our aim is to find the EC coefficients in the vector **A**. For this reason, we can represent the given problem and its conditions by a system of linear algebraic equations by using collocation points.

Now, the collocation points can be determined in the inner domain as

$\begin{array}{r}{x}_{k}=ln\left(\frac{1+cos(k\pi /m)}{1-cos(k\pi /m)}\right),\\ {y}_{l}=ln\left(\frac{1+cos(l\pi /n)}{1-cos(l\pi /n)}\right)\phantom{\rule{1em}{0ex}}(k=1,2,\dots ,m-1;l=1,2,\dots ,n-1)\end{array}$

(3.5)

and at the boundaries

- (i)
${x}_{m}\to \mathrm{\infty}$ and ${y}_{n}\to \mathrm{\infty}$,

- (ii)
${x}_{0}\to -\mathrm{\infty}$ and ${y}_{n}\to -\mathrm{\infty}$.

Since EC polynomials are convergent at both boundaries, namely their values are either 1 or −1, the appearance of infinity in the collocation points does not cause a loss in the method.

Therefore, when we substitute the collocation points into the problem (3.1), we get

$\sum _{i=0}^{p}\sum _{j=0}^{r}{q}_{i,j}({x}_{k},{y}_{l}){u}^{(i,j)}({x}_{k},{y}_{l})=f({x}_{k},{y}_{l})\phantom{\rule{1em}{0ex}}(k=0,1,\dots ,m,l=0,1,\dots ,n).$

(3.6)

The system (3.6) can be written in the matrix form as follows:

$\sum _{i=0}^{p}\sum _{j=0}^{r}{\mathbf{Q}}_{i,j}{\mathbf{U}}^{(i,j)}=\mathbf{F},\phantom{\rule{1em}{0ex}}p\le m,r\le n,$

(3.7)

where ${\mathbf{Q}}_{i,j}$ denotes the diagonal matrix with the elements ${q}_{i,j}({x}_{k},{y}_{l})$ ($k=0,1,\dots ,m$; $l=0,1,\dots ,n$) and **F** denotes the column matrix with the elements $f({x}_{k},{y}_{l})$ ($k=0,1,\dots ,m$; $l=0,1,\dots ,n$).

Putting the collocation points into derivatives of the unknown function as in Eq. (

2.28) yields

$\begin{array}{r}[{u}^{(i,j)}({x}_{k},{y}_{l})]=\mathbf{E}({x}_{k},{y}_{l}){({\mathbf{D}}_{x})}^{i}{({\mathbf{D}}_{y})}^{j}\mathbf{A},\\ {\mathbf{U}}^{(i,j)}=\left[\begin{array}{c}{u}^{(i,j)}({x}_{0},{y}_{0})\\ \vdots \\ {u}^{(i,j)}({x}_{0},{y}_{n})\\ {u}^{(i,j)}({x}_{1},{y}_{0})\\ \vdots \\ {u}^{(i,j)}({x}_{1},{y}_{n})\\ \vdots \\ {u}^{(i,j)}({x}_{m},{y}_{n})\end{array}\right]={\mathbf{E}}^{(i,j)}\mathbf{A}=\mathbf{E}{({\mathbf{D}}_{x})}^{i}{({\mathbf{D}}_{y})}^{j}\mathbf{A},\end{array}$

(3.8)

where

**E** is the block matrix given by

$\begin{array}{rcl}\mathbf{E}& =& [\mathbf{E}({x}_{0},{y}_{0})\phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{0},{y}_{1})\phantom{\rule{0.5em}{0ex}}\cdots \phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{0},{y}_{n})\phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{1},{y}_{0})\phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{1},{y}_{1})\phantom{\rule{0.5em}{0ex}}\cdots \phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{1},{y}_{n})\phantom{\rule{0.5em}{0ex}}\cdots \\ {\mathbf{E}({x}_{m},{y}_{0})\phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{m},{y}_{1})\phantom{\rule{0.5em}{0ex}}\cdots \phantom{\rule{0.5em}{0ex}}\mathbf{E}({x}_{m},{y}_{n})]}^{T}\end{array}$

and for

$i=j=0$, we see

$\mathbf{U}=\mathbf{E}\mathbf{A}.$

(3.9)

Therefore, from Eq. (

3.7), we get a system of the matrix equation for the PDE

$\left(\sum _{i=0}^{p}\sum _{j=0}^{r}{\mathbf{Q}}_{i,j}\mathbf{E}{({\mathbf{D}}_{x})}^{i}{({\mathbf{D}}_{y})}^{j}\right)\mathbf{A}=\mathbf{F},$

(3.10)

which corresponds to a system of $(m+1)(n+1)$ linear algebraic equations with unknown double EC coefficients ${a}_{r,s}$.

It is also noted that the structures of matrices ${\mathbf{Q}}_{i,j}$ and **F** vary according to the number of collocation points and the structure of the problem. However, **E**, ${\mathbf{D}}_{x}$ and ${\mathbf{D}}_{y}$ do not change their nature for fixed values of *m* and *n* which are truncation limits of the EC series. In other words, the changes in **E**, ${\mathbf{D}}_{x}$ and ${\mathbf{D}}_{y}$ are just dependent on the number of collocation points.

Briefly, we can denote the expression in the parenthesis of (3.10) by

**W** and write

$\mathbf{W}\mathbf{A}=\mathbf{F}.$

(3.11)

Then the augmented matrix of Eq. (

3.11) becomes

$[\mathbf{W}:\mathbf{F}].$

(3.12)

Applying the same procedure for the given conditions (3.2)-(3.4), we have

Then these can be written in a compact form

$\mathbf{V}\mathbf{A}=\mathbf{R},$

(3.16)

where

**V** is an

$h\times (m+1)(n+1)$ matrix and

**R** is an

$h\times 1$ matrix, so that

*h* is the rank of all row matrices belonging to the given condition. The augmented matrices of the conditions become

$[\mathbf{V}:\mathbf{R}].$

(3.17)

Consequently, (3.12) together with (3.17) can be written in a new augmented matrix form

$[{\mathbf{W}}^{\ast}:{\mathbf{F}}^{\ast}].$

(3.18)

This form can be achieved by replacing some rows of (3.12) by the rows of (3.17) accordingly, or adding those rows to the matrix (3.12) provided that

$(det{\mathbf{W}}^{\ast})\ne 0$. Then it can be written in the following compact form:

${\mathbf{W}}^{\ast}\mathbf{A}={\mathbf{F}}^{\ast}.$

(3.19)

Finally, the vector **A** (thereby the coefficients ${a}_{r,s}$) is determined by applying some numerical methods (*e.g.*, Gauss elimination) designed especially to solve the system of linear equations. Therefore, the approximate solution can be obtained. In other words, it gives the double EC series solution of the problem (3.1) with given conditions.