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Carl Love

Carl Love

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12 years, 357 days
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Wayland, Massachusetts, United States
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My name was formerly Carl Devore.

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These are replies submitted by Carl Love

Example?...

Carl Love 28055
April 03 2020
0 0

Could you give a small example of a PDE which has two polynomial solutions, one of which is returned by PDEtools:-PolynomialSolutions and the other of which is the homogeneous polynomial that you want?

More details...

Carl Love 28055
April 02 2020
0 0

@notgoodatmaple This worksheet should help you understand many of the details from my (rather terse) Answer, although I haven't endeavored to explain every nuance of the syntax. Nonetheless, after you've experimented some with this worksheet, feel free to ask any specific questions about the syntax.
 

> 

restart:
> 

ode:= diff(theta(t),t) = 1+A+(A-1)*cos(theta(t));
Q:= dsolve({ode, theta(0)= -Pi/2}, numeric, parameters= [A]):
PlotQ:= proc(A)
    Q(parameters= [A]);
    plots:-odeplot(Q, args[2..], gridlines= false)
end proc
:
MultiPlot:= proc(
    As::list(realcons), trange::range(realcons):= -2..2,
    {Legend::truefalse:= false}
)
    local k, N:= nops(As)-1;
    plots:-display(
        seq(
            PlotQ(
                As[k+1], t= trange,
                color= COLOR(HUE, .85*k/N),
                `if`(Legend, legend= [A=As[k+1]], NULL)               
            ),
            k= 0..N
        ),
        legendstyle= [location= right], _rest
    )
end proc
:

diff(theta(t), t) = 1+A+(A-1)*cos(theta(t))

The procedure PlotQ produces the plot for a single value of A, which is its first argument:

> 

PlotQ(-2, t= -9..9, thickness= 3);

You can interogate what the current parameter value is:

> 

Q(parameters);

[A = -2.]

As as there is a numeric value for the parameter, you can request specific numeric values of theta(t):

> 

Q(-4), Q(4), Q(-9), Q(9), Q(99), Q(-99);

[t = -4., theta(t) = HFloat(1.9104993102616643)], [t = 4., theta(t) = HFloat(-1.9106294563198518)], [t = -9., theta(t) = HFloat(1.9106329863105198)], [t = 9., theta(t) = HFloat(-1.9106337351769633)], [t = 99., theta(t) = HFloat(-1.9106329949645597)], [t = -99., theta(t) = HFloat(1.910633435519435)]

> 

PlotQ(0, t= -5..5, thickness= 3, color= red);

Oviously, A=1 is a straight line:

> 

PlotQ(1, t= -5..5, thickness= 3, color= orange);

> 

PlotQ(2, t= -5..5, thickness= 3, color= yellow);

> 

PlotQ(4, t= -5..5, thickness= 3, color= green);

The procedure MultiPlot plots several curves on the same axes, using several specified values of A. The curves are color coded using the visible spectrum (red, orange, yellow, green, blue, violet and all continuous gradations between), so the first specified A is red, the last is violet, and the others are somewhere in between.

> 

MultiPlot([-2, -1, 0, 1, 2, 4], -5..5, thickness= 3, Legend);

The following does the same thing for -2 <= A <= 4, spaced by 0.05

> 

MultiPlot([seq(-2..4, 0.05)], thickness= 0);

To produce the animation shown in my Answer, I used 200 frames, each frame being a call to MultiPlot with 40 A values evenly spaced over an interval of width 2. Here's a smaller animation:

> 

plots:-animate(
    MultiPlot,
    [['seq'(A_red..A_red+1, .2)], thickness= 3],
    A_red= -3..3, frames= 60
);

> 

 


 

Download ODE_parameter_plot.mw

 

Warning or informative message would be ...

Carl Love 28055
April 02 2020
0 0

Considering how often Questions about fsolve returning unevaluated appear here on MaplePrimes, it would be appropriate if it returned a message akin to the "solutions may have been lost" that solve returns. Then the help page could advise on things to try when that message appears. This change can be made while still maintaining backwards compatibility.

Your initial conditions...

Carl Love 28055
April 02 2020
0 0

@mehdi jafari If you have initial conditions that make this system interesting, I'd be curious to see them.

Bug fixed...

Carl Love 28055
April 02 2020
0 0

@Carl Love The bug is fixed. Either download the code again, or simply change the exponent of -1 from j to k-j.

Small bug in code...

Carl Love 28055
April 02 2020
0 0

@mehdi jafari Thank you. There's a small bug in that code regarding the alternation of signs in computing the initial conditions. It'll probably take me a few minutes to fix.

Normalize?...

Carl Love 28055
April 01 2020
0 0

By "normalize", do you mean that you want to rescale the amplitude to [-1, 1]?

Polynomial interpolation...

Carl Love 28055
March 31 2020
0 0

Hint: What's equivalent to the remainder of p(x)/(x-a) for a polynomial p?

?evalapply...

Carl Love 28055
March 31 2020
0 0

@Thomas Dean As I said above, it's documented at ?evalapply.

Yes...

Carl Love 28055
March 31 2020
0 0

@Thomas Dean Yes, [f,g](a,b) becomes [f(a,b), g(a,b)]. It's as simple as that. In terms of a theory of operators, the function application operator distributes over the list-building operator.

?evalapply...

Carl Love 28055
March 30 2020
0 0

I think that he means the syntax by which [cos,sin](t) becomes [cos(t),sin(t)]. If that's what you want, it's documented at ?evalapply, but there isn't much to say about it: It works that way, and that's it.

Another possibility is the prefix-form list constructor `[]`. 

Can't do in 3D...

Carl Love 28055
March 29 2020
0 0

@goli  The command textplot3d does not support any directional or rotational options, even in Maple 2020.

However, the colored-dots Answer below can be easily adapted to 3D plots, even in Maple 15.

New Answer, using a different method...

Carl Love 28055
March 29 2020
0 0

@Carl Love I just posted a new Answer that doesn't use textplot.

No rotation option in Maple 15...

Carl Love 28055
March 29 2020
0 0

@goli Sorry, there's no rotation option for textplot in Maple 15. This is a situation where you'll need a newer version.

I can concoct a solution that uses colored symbols (that are not arrowheads) to indicate the direction of travel.

Execute the entire worksheet...

Carl Love 28055
March 29 2020
0 0

@goli Did you re-execute the entire worksheet by clicking on the !!! on the toolbar?

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