Although I'm not very familiar with this topic, my guess is that you're more likely to find what you need in Maple's SignalProcessing package than in its inttrans package.

What's so bad about "having to type the equations manually into fsolve"? You need to somehow get the equations into Maple. It's also possible to send input to Maple via photo through Maple Calculator.

Rouben,

The title of this Question is precisely the example title given on the MaplePrimes "Ask a Question" form. I guess the OP's knowledge of English is such that that was the only title that they could come up with.

@ijuptilk You asked:

• Is it possible to have an array or list of u without defining u in the proc(xi) ?

Yes, it's easy, and commonly done. It's probably better to use a list rather than an array. You should make u a parameter of doCalc, like you originally had, something like this:

doCalc:= proc(xi, u::list(numeric))

And you can call it as (for example)

doCalc(3, [seq](0..20, 1))

@Hullzie16 You wrote:

• When we are telling maple that the discirminant is positive we are saying three real roots, but when doing what you described it only returns one, When it should be returning three. 

I get this:

restart;
f:= 2*y^3*z - y^2 - 2*m:
sols:= solve({f=0, discrim(f,y)>0, m<0, z>0}, {y}, parametric, real);
[[y = RootOf(2*_Z^3*z - _Z^2 - 2*m, index = real[1])], [y = RootOf(2*_Z^3*z - _Z^2 - 2*m, index = real[2])], [y = RootOf(2*_Z^3*z - _Z^2 - 2*m, index = real[3])]],  And(And(0 < z, -1/(54*z^2) < m), m < 0)

(For copy-and-paste to MaplePrimes, I've omitted the irrelevant "otherwise" branch of the piecewise, but it does appear in the worksheet display.) Notice the second arguments of the three RootOfs: index= real[...]. That's explicitly saying that there are three real roots. I know that you already know that; I'm pointing it out because it shows that Maple also knows it.

• Is the explanation similar to what I above and that there is always imaginry contributions?

Essentially, yes, although I wouldn't exactly call them "imaginary contributions". Rather, I'd say that it's often impossible to express the real roots as algebraic functions of the parameters without using I or sqrt(-1) or something equivalent. I'm using "algebraic function" here in its usual mathematical definition (the complement of transcendental function) rather than as Maple's type algebraic, which is much broader (compare help pages ?type,algebraic, ?type,algfun, ?Definition,algebraic). As @dharr explained, you can often get Maple to express them as trigonometric (transcendental) functions with no I. My opinion is that the RootOf form is the best, most-general, and usually most-compact way to express them.

Search for "color bar" in the MaplePrimes search. Many solutions have been posted over the years. There is currently no formal mechanism for these color bars.

@Hullzie16 You can include your inequalities and also use the options parametric, real:

solve({f=0, discrim(f,y)>0, m<0, z>0}, {y}, parametric, real)

This reduces the number of cases in the piecewise. In this instance, they're reduced to just 1 case (not counting the "otherwise", which doesn't satisfy the inequalities anyway). 

@jud Okay, given the different way that the labels are placed in the graph in Maple 2021 (raw integers rather than strings), the following will work for you:

restart:
Gp:= GroupTheory:
S4:= Gp:-SymmetricGroup(4):
L:= Gp:-GroupOrder~(Gp:-SubgroupLattice(S4, output= list)):
subsindets[3](Gp:-DrawSubgroupLattice(S4), specfunc(TEXT), k-> L[k], 2, applyop);

@jud I was using Maple 2019. I ran your worksheet in Maple 2019, and I get the correct graph. So I guess that there's a difference in Maple 2021, perhaps in the way that the numbers are represennted in the plot structure. I can't check Maple 2021 at the moment. However, if you execute the following commands and post the output, I may be able to diagnose the problem:

restart:
Gp:= GroupTheory:
S4:= Gp:-SymmetricGroup(4):
indets(Gp:-DrawSubgroupLattice(S4), specfunc(TEXT));

The output should look something like this:

{TEXT([-1.80739222823013, 1.], "2", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-1.73719807243076, 2.], "15", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-1.54919333848297, 1.], "3", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-1.46993683051833, 2.], "16", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-1.29099444873581, 1.], "4", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-1.20267558860591, 2.], "17", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-1.03279555898864, 1.], "5", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.894427190999916, 3.], "26", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.774596669241483, 1.], "6", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.668153104781061, 2.], "18", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.516397779494322, 1.], "7", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.447213595499958, 3.], "27", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.400891862868637, 2.], "19", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([-0.133630620956212, 2.], "20", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0., 0.], "1", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0., 1.], "8", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0., 3.], "28", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0., 4.], "30", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0.258198889747161, 1.], "9", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0.400891862868637, 2.], "21", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0.516397779494322, 1.], "10", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0.894427190999916, 3.], "29", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([0.935414346693485, 2.], "22", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.03279555898864, 1.], "11", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.20267558860591, 2.], "23", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.29099444873581, 1.], "12", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.46993683051833, 2.], "24", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.54919333848297, 1.], "13", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.73719807243076, 2.], "25", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.)), TEXT([1.80739222823013, 1.], "14", FONT("HELVETICA", "DEFAULT", 8), COLOUR(RGB, 0., 0., 0.))}

@Ronan All of the commands that you've shown are top-level, not part of LinearAlgebra or any other package.

Please use the green up arrow on the toolbar to upload your worksheet. Copy and paste isn't really human readable when used directly on Maple output. 

And do we really need several pages of author acknowledgements at the beginning? 

@LegendCero 

Re question 4:

To get any solution, it's mathematically necessary that there be two more equations. They don't necessarily need to be differential equations. Any 2 equations (independent from the others) that use the same function variables, or some subset thereof, would be mathematically sufficient.

From looking at the physics and geometry of the situation, I think that the last two equations should look similar to the first two. Here's why: Consider the first equation: It describes the interactions between x_1 and its immediate neighbors---which are x_2 and y_1. Likewise, the second equation describes the interactions between y_1 and its immediate neighbors---x_1 and y_2. But the third and fourth equations describe the interactions between x_k or y_k (respectively) and their three (each) immediate neighbors---x_(k-1), x_(k+1), and y_k or y_(k-1), y_(k+1), and x_k (respectively). But at the right-side column, x_50 and y_50 only have two neighbors each. Thus, I think that the last two equations should be

m*diff(x_50(t), t$2) + k_g*x_50(t) = k_c*(x_49(t) - x_50(t))^3 + k_e*(y_50(t) - x_50(t)),
m*diff(y_50(t), t$2) + k_g*y_50(t) = k_c*(y_49(t) - y_50(t))^3 + k_e*(x_50(t) - y_50(t))

What do you think about that?

Re question 3:

If we assume that the springs have no transferable potential energy at time 0, the then total energy at any future time will be the kinetic energy of x_1 from its initial velocity. We'll divide all the energies by that to get a color to plot.

Re question 5:

I think that each mass's deflection should be measured relative to its own initial position. Thus, no step size h should be used, and 0 should be used for all initial positions.

If you agree with all of this, then I may now have enough information to get a numeric solution and plots.

I think that I can help you if you can supply the following information:

1. In terms of the variables (and parameters) that you used in the system of equations, what is the horizontal axis "Oscillator index"?
2. In terms of the variables (and parameters) that you used in the system of equations, what is on the color-scale legend bar that goes from 0.0 to 0.8? (I'd guess it's what you called "normalized energy".)
3. Does "normalized" mean that they're divided by the maximum value? And, if so, is it the maximum value within the boundaries of the plot? or some theoretical maximum? 
4. There are 2*n+2 function variables x_1, ..., x_(n+1), y_1, ..., y_(n+1), but only 2*n equations. You need two more equations.
5. You need 4*n+4 initial conditions such as n+1 initial positions (both x(0) and y(0)) and n+1 initial velocities (both x'(0) and y'(0)).
6. Does the v0 = 0.06 m/s mentioned in the plot titles refer to sqrt(x'(0)^2 + y'(0)^2)?
7. Is the "energy" that you're interested in kinetic energy, thus energy = m*v^2/2 = m*(x'(t)^2 + y'(t)^2)/2?
8. What specific value of n do you want?

@BekkiRR In the final plot command, find the line that begins contours= ..., and replace it with

contours= [$0..4] * 25000,

@Christian Wolinski From reading many of your previous Answers, I suspect that you are using a very old version of Maple.

You wrote:

• I would suspect the reason is you are not using diff and mtaylor does.

Of course we have found out in this thread that diff is much faster than D, but series is even faster. In your Maple, it may be true that mtaylor uses diff, but that isn't true in modern Maple; it uses series. Other than checking for several possible error conditions, handling the rarely used options for variable weights and returnorder, and doing some simplifications of the output, mtaylor is equivalent to this very short procedure:

BySeries:= proc(f::algebraic, V::list(name= algebraic), Ord::posint:= Order)
local t, v:= lhs~(V), v0:= rhs~(V);
    subs(O(1)= 0, t= 1, v=~ v-~v0, series(subs(v=~ v*~t +~ v0, f), t, Ord+1))
end proc
:

CodeTools:-Usage(BySeries(1/(2+x*y^2), [x=0, y=0], 30));
memory used=19.97KiB, alloc change=0 bytes, cpu time=0ns, real time=3.00ms, gc time=0ns