@Reshu Gupta That has nothing to do with your original Question or the title of this thread. Please post it as a new Question.

And you haven't made any "mistake". The problem is due to limitations of Maple's BVP solver for boundary-layer problems. Getting solutions to converge often requires sophisticated adjustments to the dsolve options.

@mylikes If you still have access to those output files, it would be helpful in diagnosing the situation if you would report the last 5 or so lines of the form "memory used=..., alloc=..., time=..." before receiving the error message. Please post this information for each error message.

The message "Execution Stopped: Unhandled signal caught (UNKNOWN: 1)" looks like it comes directly from the operating system rather than from Maple. Have you received this message more than once? Is it reproducible in the sense that if you repeat the calculation with exactly the same input then you'll always get this message?

My guess at this point is that they are all running out of memory, but I'd like to see those "alloc" numbers.

As far as I know, Maple doesn't offer any way to make a graceful recovery from a memory allocation error (including stack limit errors). This is very unfortunate. In other words, there is no datalimit command that works analogously to the timelimit command, which allows you to set the amount of cputime used at which you want to make a graceful recovery. So memory errors tend to produce a "hard crash" with messages that aren't very informative.

@Arif Ullah khan The solution by Preben Alsholm appears above, immediately under your Question. I repeat it here:

A1 := dsolve(subs(para,rho=2, {bc, eq1, eq2}), numeric,method = bvp[midrich],
   initmesh=1024,maxmesh=12500, output=Array([seq( 0.01*i, i=0..100*N)]),abserr=1.5e-2):

So, I ask you for the third time, by how much does this solution differ from the solution that you want?

I'll send you an email.

@Arif Ullah khan The following statements that you made about Preben's solution make no sense to me when taken together. You said to Preben:

• [A1] Thanks for your reply. I check it and it does not give me the required results.

So then I asked you:

• [C1] Is Preben's solution far off? Or is it only off in the 2nd decimal place? 

To which you replied: 

• [A2] Sorry @Carl Love I did not know about Preben's solution.

Your statement A1 implies that you were able to compare Preben's solution to some other solution deemed correct, presumably from that paper, and the solutions differed. So, did the two solutions differ by more than the 2nd decimal place? Or did they differ by less?

I can't get that paper from that linked website without paying them US$30.

@Arif Ullah khan I will attempt a shooting solution. I'll be busy for the next 10 hours or so. These BVPs often have multiple valid solutions, and mine may converge on the same solution as Preben's, or it may not converge at all. I'll look for guidance from the paper. It may be better to use the approxsoln option.

@Arif Ullah khan These boundary-layer fluid-flow ODE BVPs are quite tricky, and Questions about them are one of the most common topics here on MaplePrimes. Unfortunately, at this time, there is no "stock" solution (as in method= bvp[...]). Rather, there is just a "bag of tricks" that we use on an ad hoc basis, fine-tuning the numerous options to the dsolve command.

Is Preben's solution far off? Or is it only off in the 2nd decimal place? Knowing that will give me an idea how to proceed.

@Carl Love Example:

plot(
    sin(x), x= -Pi/2..Pi/2, legend= [HPM_case3], 
    labels= [xi, f(xi)],
    axis[2]= [tickmarks= [seq](-1..1, 0.1)], axesfont= [HELVETICA, BOLD, 10], 
    gridlines, axes= boxed
);

The font size small enough to avoid crowding (I used 10 above) might be different on your display than on my display. I don't know what the default is, but it was too big on my QHD+ display.

1. The proposed solution contains a variable varepsilon that doesn't appear in the input and isn't otherwise defined. Thus, this can't be the solution, or at least it can't be the complete solution.

2. Shouldn't the S__2 at the end of the proposed solution actually be S__1?

3. Although it's not an error to do so, you shouldn't use the quotes with `S__2` because of the possibility of enclosing an extra space in them (which you did do), which is also not an error, but could easily lead to one.

@Bohdan I will try to make some estimates of that. Before I go too far with it: Is it intentional that some gamma have two indices and some have three?

@Bohdan As I've been trying to tell you, it's the problem itself that's infeasible. The infeasibility is not due to the method used to solve it. There does not exist---and there never will exist---any method that's able to solve this in any feasible amount of time or memory.

The system of equations in your worksheet is mathematically equivalent to the inverse problem. Therefore, it's merely a different method, not a different problem.

Note that by "infeasible" I do not mean that the problem has no solution. Its solution definitely "exists" in some abstract mathematical sense. But that solution can never be achieved,

@mylikes Maple has no practical inherent limitation on the size of such a computation. The limitation is the amount of memory you have, which Maple has no control over.

@Bohdan I don't know the details of the computational complexity of computing the Moore-Penrose for a symbolic matrix, but my vague guess is that it's far more complicated than the regular inverse.

I'll say it again: It's not worth the effort required to even think about it.

@acer Having read the extensive worksheet that the OP posted with the new Question (I still have the worksheet open), I wouldn't even call it a followup, let alone a close followup.

Just to show you some examples of the times of computations with your matrix, I started doing some computations with numerical evaluations of it. I ran into a problem that I totally wasn't expecting simply because I figured that you would've already checked this possibility yourself before even considering inverting it. Your matrix is **very** rank deficient, its rank being only 9. So, it has no inverse, not even close. I'm guessing that this is due to a simple error that you made in constructing it rather than a theoretical problem, lest your whole theory is nonsense.

Probably I was a fool to waste my time even to begin discussing inverting such a matrix. So, it's no surprise that no-one else participated in this inversion discussion.

@faizfrhn Ah, I see how that can be quite confusing because the 2 has nothing to do with the algebra at hand; it only pertains to the meta-algebra. The 2 specifies the 2nd operand of e. For an equation, such as e, the 2nd operand is the right side (the part after the equals sign). It's not possible to use command subsop without knowing the operand numbers that you want to act on. Likewise with Kitonum's applyop: His 2 refers to the 2nd operand, and his [2, 3] refers to the 3rd operand of the 2nd operand. Since the 3rd operand of a term is far more nebulous and ad hoc than the 2nd operand of an equation, I don't understand why someone gave his Answer a Vote Up but not mine! The term in question has 5 operands.