@tomleslie

Tom, your version of the ODEs contains an error carried over from the OP's code: In the last line, F is used as a coefficient rather than as a function. 

@aoakindele Please don't waste my time (and it takes much longer to write these posts on my phone) by posting worksheets that still contain errors that I just told you how to correct. The worksheet above still contains the Pr and alpha errors.

By focusing on the explicit error messages, one often ignores the true errors, which often don't produce error messages. 

@aoakindele 

Please read the extensive additions that I just made to my Answer above.

I have no Internet connection right now, and I'm posting from my phone. That means that I can't view the worksheet that you just posted. If you post it inline (i.e., explicitly displayed), I'll likely be able to answer your question. 

@pik1432

ToInert(expr) might give the expression transversal that you're looking for, although I usually find it to be excessively detailed, and I wish that one could selectively expand just the subparts that one was interested in. And beginning every operator's name with the 7 characters _Inert_ makes it hard to read and is so unnecessary.

A more readable (but not programmatically friendly) alternative is dismantle(expr). However, this is an older command, and some of its information is out of date. 

The commands addressof and disassemble do allow for selective expansion of subparts, but all the information is numeric: memory addresses and expression-type key numbers (called dagtags).

@fatemeh1090 All I did was make a numeric comparison (evalf) of the two sides of the equation for a few points u and nu using 1000 instead of infinity as the upper limit of summation. 

I do not know the mathematics needed to answer your other questions. 

I don't have Internet access at the moment. I'm posting this from my phone, which of course doesn't have Maple.

@mmcdara The difference is just an issue of the design; it doesn't have any mathematical significance. If the first argument to solve is in a set or list, then the returned results (if any) include the variable(s). If there's only a single item being solved, and it isn't in a set or list, then the solution is returned without the variable. This capriciousness is quite a nuisance. Compare:

solve(x-1);
solve({x-1});

It seems that the real parts of both sides of the red-boxed equation match (for real nu and u), but the right side has a huge imaginary component also. This may be due to a difference in the way the Legendre functions are defined in Maple and in your source reference; I don't know. It appears to me that the I*Pi/2*LegendreP(n - 1/2, cosh(nu)) contributes only to the imaginary component. Perhaps you should get rid of it.

For nu=1, u=1, I get an imaginary component approximately 10^435.

@Arif Ullah khan To quote directly from the paper, here are the algorithm steps that I object to:

• Step 1: "Select a value of B."
• Steps 2-4: [I have no problem with these steps.]
• Step 5: "Check to see if Eq. (2.19) is satisfied; if not iterate on B until satisfied."

I don't think that that should be acceptable in a scientific journal, Physics of Fluids.

@Arif Ullah khan The author's algorithm is not well specified. The last step is, essentially, "If the results are not sufficiently close, pick different initial values and try again." There's no indication of what's "sufficiently close" or how to converge to correct initial values. 

If it's your job to referee this paper, I would reject it until this last step is specified more precisely. To me, this algorithm is the only significant contribution that the paper makes. 

The majority of papers that I've read on boundary-layer ODE BVPs (several of which I've discussed here on MaplePrimes) have similar flaws. It's a very shoddy area of mathematics.

Also, don't assume infinity = 6. Continue integrating until the results are sufficiently stable. The fact that this can be easily done is a significant improvement over other BVP methods for systems with a boundary at infinity.

@Carl Love It is somewhat easier to count partial orders (relations that are reflexive, transitive, and antisymmetric) because there are far fewer of them for any given n. There is a recursive formula to count the number of transitive relations given the number of partial orders. For n= 0..7 the number of partial orders are 1, 1, 3, 19, 219, 4231, 130023, 6129859. The details can be found by following the links that I gave above.

@tomleslie The union of two intervals is not necessarily an interval, yet your procedure can only return an interval for the union.

Your title "Plotting in Maple" sidesteps the main issue. Once any functional equation is solved, plotting it is usually trivial. Any list of numeric [x,y] pairs is sufficient. The means of generating the pairs depends on the form of the solution.

But, AFAIK there's no stock method in Maple for solving fractional differential equations. You need to use an ad hoc method, i.e., one that's tailored to the particular equation. So, to get further help, I think that you'll need to post the equation itself.

So, I changed the title.

@Mo_Jalal From the code that I see in your subsequent Question, I think that you've already discovered that fnormal is the solution for that "nearly zero" issue.

@janhardo That's not at all surprising to me. I've never seen a system of mathematical typesetting where the double quote character was a derivative symbol.

It's a tough problem for which no general solution is known. There is some relatively recent research in this area. Brute force counting is out of the question since there are 2^(7^2) = 6.x10^14 relations on 7 elements. According to The Online Encyclopedia of Integer Sequences https://oeis.org/A006905 the counts for n= 0..7 are 1, 2, 13, 171, 3994, 154303, 9415189, 878222530.

I suggest that you start researching with the Wikipedia article "Transitive relation", and follow the research links in the section "Counting transitive relations".

I changed the title of your Question from "Discrete mathematics in Maple" to "Counting transitive relations".