@Carl Love pp is notation and stand for pressure profile.  Below is the picture that I take from a paper of this.

In Mathematica file (pdf) written a simplified form after putting u=diff(psi,y) and its simplified form is gievn in equation 62 and its BCs are also given Mathematica file.

firstly Need to solve for  eq 62 for psi against y and then use the solution of psi to get the solution for u against and similarly use the solution of psi to plot diff(p,x) against x and integrate the solution of diff(p,x) and find solution for p against x .  In mathematica file pp stand for solution of p. I am not understanding mathematics code and how to transform that in Maple?

@acer , sir, thanks for the correction. Is it possible to create one more column for the answer check? Means residual

@mmcdara Sir, can you help with my problem even though I lost the file you uploaded

Help_Numeric_solution.mw

@mmcdara Dear Sir, unfortunately, my laptop got damaged, and could not access the website. Managed a new laptop, found all accounts, logged in to the website, and saw your responses on my question again. I even do not see your reply due to my laptop issue.  I am sorry for the inconvenience. Could you please upload your response(Maple worksheet) to my problem? Sir waiting for your response.

@mmcdara is it possible to check that our obtained solution is correct or not? means compression of solution with any other method available in Maple? and why did you make Hypothesis := p(y) = a+b*y for diff(p(x),x)? even in original problem,we have .diff(p(x), x).but you pleased diff(p(y), y)how could I plot for diff(p(x),x), rather than p(y)?

@mmcdara thanks for your help. The results look like this

1. for u(y) against y for different values of k like 0.1,0.3,0.5,0.7,0.9 ( we show only odd values not all )

2. and the results for diff(p(x), x) against x expected to be look like as 

Note: you plot diff(p(y), y), which wrong

3. Finally the results for T(y) against y look like as

@mmcdara sir thanks for your response, I chekced my problem again and here are response to your observation.

[1]
What does diff(p(x), x)  mean in equation 5?
Wouldn't it be diff(p(y), y)  instead? But if it is so you cannot solve equation 5 numerically as p(y) is undefined.

Ans. diff(p(x), x) stands for pressure grandeint in equation 5 and is unknown so its difficult to solve it numerically, So we took the derivative of Eq. 5  and obtained Eq. 6, which is free from diff(p(x), x) .

[2]
You have 2 odes (equations 5 and 6) that psi(y) must verify... you must decide the one you want to keep unless they both have the same solution, whixh is highly unlikely given thefirst one depends
on p.

Ans: Actually, our main problem or  Odes are given in (Eqs. 1 and 2). I want to solve Eq. 1 for u(y) against y but it has diff(p(x), x), which is unknown. So, I define u(y):=diff(psi(y),y) (Eq. 4) and transform my Eq. 1 in the form of psi. So, firstly, we need to solve Eq. 6 for psi by using Bcs given in Eq. 7. Then use this solution of psi into u(y):=diff(psi(y),y) (Eq. 4) and plot for u(y) against y for different values of k and fixing other parameters.

Now, use the solution of u(y) into Eq. 2 and solve it  for T(y) against y with the help of Bcs given in Eq. 3.

Finally, use the solution of u(y) into Eq. 1 and solve for diff(p(x), x).

[3]
Your bc are given at 3 points (-1, 0, 1) which is not supported by Maple.
Either you formulate the problem differently by considering a 1st problem in [-1..0], a second in [0..+1] and interface continuity equations... or you remove on point, for instance 0:

Ans: I checked the BCs and found correct.

@acer streamline means countour plots for the given ODE for the given values parameters. for refernce please have look on the figure. Beside this sir, is it possible to plot  the tabuler values for Analytic and Numeric solution against y togetherly  in a single graph and Abs.error versus  y as a separate graph?

@acer Thanks a lot, sir for your help. My series solution has a lot of other parameters due to some assumptions, which may be doubtful. To check the error or accuracy of my series solution, I solved it numerically. Sir, I have the two following: 

1: How do I check the convergence or validity of the numeric method?

2: Is it possible to draw a streamline pattern or countour plot directly from the given system of ODEs because I always draw by considering stream functions, and it takes a lot of computation and steps.

@acer I obtained series solutions (analytic solutions) for my ODE, and for simplicity, I do not give the details here. I want to compare my analytic solution with the numeric method to check the accuracy or error, which means the absolute error of the analytic solution and the numeric solution. The error is zero zero, how to find absolute error. Is it possible to write or check the convergence of the numeric method, or how could we check the convergence of the method for this problem?

@acer thanks sir. By the way, when I run same file in Maple 22, the view is not similar to what you uploaded. Is this a setting problem?

@acer Thanks, sir. phi is the main function for which I plotted the contour and then converted into 2D by p1...........Is it possible view 3D as like in the picture uploaded below?

@mmcdara you right my mistake about use of code. I have run simple code on same expression but the answer is different. Please have look on attached file. Beside this, is it possible to fixed all parameters and chnage only one means ( fix= N, beta, m] and change Br=0.1.0.2..0.9, for every value of Br the value of x0 is different which is given in the file you uploaded. 

Help.mw

@mmcdara sir, I have changed the expression and followed the same procedure that you defined in the above file, but there is an error in my attempt. And is it possible to check errors for all obtained answers in tabulated form rather than apply them one by one?

Help.mw

@mmcdara thanks for help. Let me try for some other expressions.