@KIRAN SAJJAN In the paper that you have cited, the author expands the solution into infinite series and thus reduces the PDEs to a set of ODEs (11) and boundary conditions (12).  The graphs shown are those of the solutions of that system of ODEs.

You wrote that you know how to solve ODEs, so you should be able to continue from there.

1. The equations indicate that the unknowns u, v, w, p, theta are functions of r and z, but the plots show these as functions of z only.  That's not consistent.

2. The PDEs model the motion of a fluid in a cylinder.  The radius of the cylinder is not given.  Also you need the parameter values of Pr, lambda, and epsilon.

3. Boundary conditions are given at the top and bottom (the flat parts) of the cylinder, but boundary conditions on the lateral surface of the cylinder (the curved part) is missing.

@Dkunb Most of this worksheet's functions execute in a fraction of a second on my (very old) computer.  There is not much to measure there.  Do you have any specific function in mind?

The slowest operations are in the calls to display() which take a few seconds.  To time each display() call, do

st := time():
display(... whatever ...);
time() - st;

That will print the number of seconds (and fractions of seconds) elapsed in executing the display().

As to your question regarding arranging the plots into 2 rows and three columns, do

display(< p1, p2, p3; p4, p5, p6>);

where the p1, p2, etc., are plotting commands, such as odeplot(...).  The semicolon after p3 separates the rows.

PS: Don't forget to adjust my worksheet according to the note titled Correction posted on May 10.

@Scot Gould Save mod for polynomial algebra.  It's safer using irem with numbers, as in

plot(irem(round(x),2), x=0..5,  thickness=4);

The coefficnet i (the imaginary unit) is missing from the set of equations for C in the worksheet that I have provided.  To correct, change the line

de_C := seq(diff(Cvec,tau) =~ lambda*exp(-y(tau)) * B . Cvec);

to

de_C := seq(I*diff(Cvec,tau) =~ lambda*exp(-y(tau)) * B . Cvec);

Then change the plotting range by setting t_final = 2.

@Dkunb You have specified y(0) = 20 and py(0) = -5.  That's a good start.  But we have N+2 first order differential equations, so need N+2 conditions to solve.  What are the rest?  And what is  the range of tau?

It will help if you supply a new worksheet where you specify the numerical values of all the parameters, such as m_x, lambda, N, and so on, a full set of initial conditions, and the vector E.  We don't need the differential equations themselves since we can read them in the image that you have posted.

@Dkunb The set of ODEs that you have shown at the end of the image that you have posted involves m_x, m_y, E_m, E_n, lambda.  Are these variables or constants?  What do we know about them?

The independent variable is tau.  I assume that it is a real (not complex) variable.  What is tau's range? Is it the entire real line?

What is the value of N?

Your system of ODEs involves N+2 unknowns.  To obtain a numerical solution, you need to supply N+2 initial or boundary conditions.  What are they?

I suppose one may extract the answers to these questions by sifting through your worksheet but that would be a Herculean task.  It will help if you could remove your attempt to a solution, and just present the parameter values as asked above.   We already know the system of equations from the image that you have posted.

@Paras31 For whatever it's worth, here is that diagram in tikz.  It takes fewer lines to code and the result looks more professional.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}

\tikzset{Arrow/.style={->, very thick, red}}
\begin{tikzpicture}[>=stealth]
    \pgfmathsetmacro{\R}{3}
    \pgfmathsetmacro{\r}{0.13*\R}
    \pgfmathsetmacro{\H}{0.5*\R}
    \pgfmathsetmacro{\a}{70}
    \pgfmathsetmacro{\d}{\r*cos(\a)}
    \draw[->] (-1.2*\R,0) -- (1.2*\R,0) node[right] {$\Re$};
    \draw[->] (0, -1.2*\R) -- (0,1.2*\R) node[right] {$\Im$};
    \draw[red, semithick] (0,0) circle (\R);
    \draw[red, semithick]
        (0, \H) +(-\a:\r) arc(-\a:180+\a:\r)
        -- ($(0,-\H) + (180-\a:\r)$) arc (180-\a:360+\a:\r) -- cycle;
    \draw[Arrow] (0,\R)    -- +(-0.01,0);
    \draw[Arrow] (0,\H+\r) -- +(-0.01,0);
    \draw[Arrow] (0,-\R)    -- +( 0.01,0);
    \draw[Arrow] (0,-\H-\r) -- +( 0.01,0);
    \draw[Arrow] ( \d,0) -- +(0, 0.01);
    \draw[Arrow] (-\d,0) -- +(0,-0.01);
    \fill[red] (0,\H) circle (0.1*\r) (0,-\H) circle (0.1*\r);
    \draw[>=latex, <-] (0,\H) +(40:0.1*\r) [out=40,in=180] to +(20:2*\r)
        node[right] {$z=i$};
    \draw[>=latex, <-] (0,-\H) +(-40:0.1*\r) [out=-40,in=180] to +(-20:2*\r)
        node[right] {$z=-i$};
\end{tikzpicture}

\end{document}

@nm Taking the limit of t*y^2(t) = arctan(t) as t goes to zero, says 0*something=0.  That does not mean that the "something" is zero.

So, yes, the textbook's answer is wrong.  As to odetest, I suppose it's not designed to address tricky questions like this.

@acer You have made some instructive comments here.  For everyone's benefit, here is a worksheet that illustrates what you have described in words.

@acer Thanks for verifying the issue and submitting a bug report.

That's an odd question for at least two reasons:

1. Why would you ask someone else to write a code for you?  Isn't that your job?  What progress have you made?

2. Why would you ask for Matlab code in a Maple forum?  Matlab is not Maple.

It will help if you could upload your worksheet.

@segfault You don't need to make a new Maple executable script.  As I wrote in my answer, define an alias in your unix shell.  If your shell is bash, for instance, then in your ~/.bashrc insert the line

alias maple='maple -j 65536'

This will take effect in subsequently opened terminals.  If your shell is something like tcsh, then in your ~/.tcshrc insert the line

alias maple  maple -j 65536

I am assuming that you launch Maple through the comand-line in a terminal.

@acer Thanks for replacing the page with a readable version.  I had noticed the issue with displaying vectors in prior posts and I had assumed that's due to a quirk of this website rather than related to the recent versions of Maple.