@Mac Dude 

Simply showstat(`plots/display`);

There's no need to fear overwriting your Maple installation. Just save any changed procedures to your own directory.

@tomleslie 

At first I considered suggesting DynamicSystems:-BodePlot, but I can't see how to make it work with explicit data, which I think is what the OP has.

@Axel Vogt 

Good procedure, Axel. It may go a long way towards an algorithmic solution to the problem. I replaced your floor with ceil, and it made no difference in any example that I tried. But I wonder if there is any example where it makes the difference between an answer and no answer.

Kitonum: Your example h(n) shows a limitation in Maple's product, not really in limit. Of course, all infinite products and sums are limits of sequences, but the areas of Maple that are used are very different.

@Kitonum Yes, I know that the actual sequence is increasing. I am saying that your sequence of floating-point approximations produced at Digits = 10 is not monotonic, so that you can't say that the "accuracy falls". Actually, the accuracy varies erratically. 

@lham Sorry, I counted wrong. The equation has five variables (or one variable and four parameters), not four. So your question isn't as ridiculous as I first thought. The command to solve such equations is

solve(..., parametric);

But I wouldn't call it a "parametric equation"; I'd call it an "equation with parameters". Unfortunately, the above command only works for polynomial equations. See ?solve,parametric.

If you want help with MapleSim, it sure would be helpful if you selected MapleSim from the pull-down list in the editor. For the vast majority of us here who know Maple but know nothing of MapleSim, and who automatically try to interpret Questions in a Maple context, your question probably appears as utter jibberish. For example, the first thing that I think of seeing "CAD" is "Cylindrical Algebraic Decomposition". I only know to suspect MapleSim because I recognize your name as someone who often asks Questions about MapleSim.

@Kitonum 

Your sequence is not monotonic. It is just random floating-point effects because you have Digits set to 10, which isn't sufficient to accurately deal with your 11-digit integers.

@Kitonum 

That's fairly impressive considering that Maple's numeric summation capability is very weak. But that's no algorithm. I hope that that's not the way that Mathematica does it. The technique could be used to make a good guess at the answer, and a correct guess could be a substantial part of an analytic solution.

@lham 

So, you have one equation with four variables and no other conditions and you want to know what four-tuples satisfy the equation. Nothing can be said other than that they satisfy the equation; there's no other information. Perhaps the equation can be simplified a bit, and obviously it can be solved for phi[0], but that's all.

@Kanellopoulos 

No, to me it is not at all "obvious what will happen". It doesn't help that you don't show your Maple code for the above plots.

You say that you checked that the splines are solutions to the PDE system. How is it possible to do that without any variation in the t dimension? In other words, how did you approximate the derivatives with respect to t?

You say the orange curve is Maple and the blue curve is Mathematica. I admit to not knowing much about waves and PDEs, but the blue curve looks highly suspicious to me, and the orange curve looks reasonable. Why does the blue curve look more accurate to you?

The ::real is redundant---it is implied by the inequalities. So, the same bug is revealed by the simpler contrast of

is(a=b) assuming a^2=b^2, a>0, b>0;
is(a=b) assuming a^2=b^2, a>=0, b>=0;

I just point this out because it should make the bug easier to find/fix.

Now, given that Maple can correctly prove the second assertion above, shouldn't it be able to prove the following, which is much simpler (since it has half the number of variables) and is implied by the one above? Yet, it gets it wrong:

is(a=1) assuming a^2=1, a>=0;

     false

Weirder still, consider the negation of that assertion. One might guess that if Maple incorrectly answers false for an assertion, that it will answer true for the negation of that assertion. Yet

coulditbe(a <> 1) assuming a^2=1, a>=0;

     FAIL

The take-away from this is that it might be a good idea to also check the negation of an assertion. If one returns FAIL and the other doesn't, then the other can't be trusted.

@Kitonum Okay, you've changed my mind. I was thinking that the problem was in general too hard to be solved by an algorithm, but if Mathematica can do it, then Maple should be able to also.

@Mac Dude 

The with command is not needed to use a command in a module; any command in a module is accessible using :-. 

@Kitonum 

assume(n::posint);
limit(simplify(sin(convert(asympt(sqrt(n^2+n)*Pi, n, 1), polynom))^2), n= infinity);

     1

@tomleslie 

The Answer is satisfactory for me. In those cases where the sequence limit exists but the continuous doesn't, you need to use the positive integer property to do some sort of simplification.