## 6996 Reputation

16 years, 214 days

## @mmcdara The purpose of my question...

@mmcdara The purpose of my question is pedagogical.  In your solution you refer to L2 and Linfinity norms.  It seems to me that Otttimor does not know what those terms mean.  That''s why I asked him/her for an explanation.

## How do you measure error/deviation?...

@Otttimor Let's see if this helps.  Consider two lists of numbers:

```vals1 := [1,3,5,6,6,6,7,5,3];
vals2 := [1,3,4,5,6,8,7,4,4];```

Can you explain how you would want to measure the error/deviation between these lists? Your answer to that will determine the strategy to use in your problem.

By the way, you may plot that data through

`dataplot([vals1,vals2], color=[red,blue]);`

## @Jonas111996 The coefficients of x1...

@Jonas111996 The coefficients of x1(t) and x2(t) on the right-hand sides of your differential equations are things like sin(t*sqrt(-3)) which are explicit functions of t.  That makes your system nonautonomous as Carl has already pointed out, so a phase portrait makes no sense.

Furthermore, due to the presence of the sqrt(-3), those coefficients are complex, making a 2-dimensional phase portrait impossible.

I suspect that you have gone off in the wrong direction in the process of arriving at your differential equations.  If you explain where those equations come from, someone should be able to orient you in the right direction.

## Write to the author...

You may want to ask the first author whose email is given in the article:

rxyao@sjtu.edu.cn

## @acer I agree.  I wouldn't...

@acer I agree.  I wouldn't be surprised if a close look at the algorithm reveals better ways of achieving whatever that program is aiming to achieve.

## @zenterix Here is what I got when r...

@zenterix Here is what I got when running your file on my Linux machine.

In the first attempt Maple crashed after running for some half an hour or so.  I montitoried the system's status during the execution and observed that my swap partition, which is a measly 0.8 GB, got filled and apparently that was the cause of the crash.

Then I added a largish (100 GB) swap file and ran the program again.  I didn't time it, but I am guessing that it took about an hour or so to produce the sphere with an arrow.  It turned out that 100 GB of swap space was much more than necessary.  Only a small fraction of it was used.  As I played the animation, however, the memory usage went up until it nearly ate up all of the 64 GB RAM of my computer's memory and the machine started swapping the memory to disk.   At that point I killed the program because the machine had become almost unresponsive.

With your 256GB RAM you may have better luck.

The lesson learned here is you need a descent sized swap space to run your program.  Type "free" on your Linux command-line to see the size of your swap area.  (The number shown is in kilobytes).  It looks like you will need something like 20 GB of swap space to run your program.

Normally a swap partition is created during the installation of Linux.  If your swap partition is too small, you may temporarily add a swap file to enlarge it.  It's not difficult.  Let me know if you need help with doing that.  Perhaps your Mac machine has a large swap partition and that's why things work smoothly there.

## Thanks!...

@Carl Love Your post came just in the right time. I was looking for a way for doing just that.  Vote up!

## Fourier series of parametric curves...

In an attempt to understand what you are asking, I searched for "Bart Simpson, epicycles" on Google and found a YouTube Video which I think is related to your question.  Let me summarize.

We have a line drawing in the complex plane.  We parametrize the drawing as F(t), where t goes from 0 to 2*Pi.  Then we expand F(t) into the complex Fourier series, and interpret the partial sums of the series as depicting a planet, a moon, a moon of that moon, and so on.  Thus, the Fourier series approximation of F(t) may be viewed as the orbit of the last moon in the sequence of the moons.

In the attached Maple worksheet I show how to animate the moons as they trace the curve corresponding to F(t), which, for the sake of illustration I have taken to be a square.  The animation below corresponds to picking 7 terms of the Fourier series, and consequently the target square (drawn in green) is rendered with rounded corners (rendered in red).

You may replace the square with any other drawing provided that you can obtain a parametric representation for it.  I understand that you are looking for a method of extracting the data for a given curve from an image, and then parametrizing it.  That's a rather complex operation.  I haven't given it much thought.  The video quoted above suggests using a traveling salesman algorithm to connect points extracted from a graph.

Here I have replaced the square with an F-shape:

## Issue solved...

@ecterrab I updated to the latest version of the Physics package and the trouble went away.  I have added a postscript to my earlier post indicating that pdsolve() was not at fault.

## Trouble with subscripted arbitraryfuncti...

@ecterrab Hi Edgardo, I have narrowed down the problem to the subscripted arbitraryfunctions business.

Specifically, in my mapleinit file I have:
`pdsolve/arbitraryfunctions` := "subscripted":

If I remove that line, then
pdsolve({pde1, pde2, ic1, ic2});
works just fine.  Otherwise I get the error that I had noted.

The attached worksheet was executed with the troublesome line commented out in mapleinit and it shows that pdsolve works in that case.  Further down in the worksheet I insert
`pdsolve/arbitraryfunctions` := "subscripted":
after which we see that pdsolve fails.

## Version dependency...

@astroverted My original worksheet works in Maple 2021 and 2022.  It fails in earlier versions.  Here is an updated worksheet that works in Maple 2017 onward.

In this worksheet I have set the grid sizes m and n to small values to speed up the calculations.  Increase the values for greater accuracy (but then you will have to be patient while you wait for the results).

## @imparter You're welcome. ...

@imparter You're welcome.  I have reported this bug to Maplesoft and I hope that it will be corrected in the next release.

## The boundary values...

dharr's assertion regarding the boundary conditions is correct.  The paper that you have cited provides four initial/boundary conditions in equations (15) through (18).  But (16) and (18) are not arbitrary; they are derived from (15) and (17) and the PDEs as I will explain in the Answer that I am going to post next.  The paper assumes that's obvious and does not state that fact explicitly.

## @romanrieme As acer suggested you n...

@romanrieme As acer suggested, you need display(F2(t),F3(t)):

`animate(display, [`if`(t < 0, F2(t), display(F1(t), F3(t))), scaling = constrained], t = -1 .. 1, frames = 140);`

Alternatively, you may do it as I suggested:

`display([animate(F2, [t], t = -1 .. 0), animate([F1, F3], [t], t = 0 .. 1)], insequence)`

## Terminology...

@mmcdara I use "extrapolation" for that purpose, as in "φ(x,y) extrapolates the boundary data into the rectangle".  I have seen people refer to that as "Dirichlet lift" or just "lift", which is closer to the French "relèvement".

 1 2 3 4 5 6 7 Last Page 2 of 84
﻿