Wolfram Mathematica

3:58 AM

@xzczd Yes it is related to the problem : "Position of discontinuous coefficent influences the solution of PDE" and other previous related problems.

4:14 AM

But it is not really a problem. In the case I present above, NDSolve gives the right solution. The "problem" is that I don't understand how it is possible : how NDSolve guess that that Ex=ETangential should be continuous and Ey=EOthogonal should have a step (in the first image). In the second image, I have simply verified that the directions Ox and Oy are treated the same way (answer : yes, fortunately. The treatment is isotrope).

@xzczd I think I have not still fully seen the consequences of user21's answer here

@xzczd I go back to bed (We are in the middle of the night here)

4:48 AM

@andre Oh, so you mean NDSolve managed to solve the equation as desired but you failed to implement it yourself? (I haven't tested your code, I'm still in v9 as you know. )

@xzczd I'm back here (can't sleep)

@xzczd Yes NDSolve manage to solve the equation

I don't try to implement it myself (not at all !)

@andre I can't give a concrete proof at the moment, but I have a feeling that "tangential component of E at the dielectric interface must be continous" is equivalent to "Curl[E, {x,y,z}] is continuous" and "Epsilon[x,y]*E must be continous" is equivalent to "Div[epsilon.Grad[E[x, y], {x, y}], {x, y}] is continuous".

(end of my comment above) I just want to understand when NDSolve decide to preserve continuity and when it doesn't. And even more precisely: when it doesn't, how does it choose the magnitude of the step. I think that mathematicaly, all possibilities are acceptables. And physically it makes sense (just imagine a layer of charges at the dielectric interface, or a heat source etc...)

5:16 AM

@andre Though not clearly mentioned in user21's answer, I guess the answer is Grad[ u] and (-c Grad[ u]- alpha u+gamma ) is continuous.

Maybe we should ask user21 to clarify this in his answer?

I will not ask user21 now. Maybe later. I want to cogitate first.

If I ever ask user21, your participation is welcome.

OK :)

2 hours later…

7:46 AM

I am looking at WRI video, and do not understand how to type the command shown in my notebook. The command on the video is in traditional format. Does any one knows how to write the command in standard notations? Here is screen shot, and also link to video

Here is link to the video, it is at 30:30 time. youtube.com/…

Reflections on Teaching Computational Physics and Mathematics

@Nasser is this what is missing: \[Function]?

This is what I tried:
FixedPoint[
y[x] :>
y[x] - Assuming[Element[n, Integers],
Integrate[
Integrate[(1 - x^2) y''[x] - 2 x y'[x] + n (n + 1) y[x], x],
x]] + O[x]^10, 1]

@Kuba I think something is missing, but I have no idea. The video do not show it :(

yep, it is function, like Function[y, whatever].

so instead of :> you need \[Function].

I do not like traditional format Hard to read/ i.e to translate to normal Mathematica code, since it hides some things.

well, I can;t get it to work. This is what I tried

FixedPoint[
Function[y[x],
y[x] - Integrate[
Integrate[(1 - x^2) y''[x] - 2 x y'[x] + n (n + 1) y[x], x], x] +
O[x]^10], 1]

I give up, going to sleep :)

@Nasser so first arrow is a function but I'm confused how they are able to get dy/dx with y.

7:53 AM

@Kuba Yes. May be this is how traditional format. I do not use Traditional format.

@Nasser

FixedPoint[
 y \[Function]
  y - Integrate[
    Integrate[(1 - x^2) D[y, {x, 2}] - 2 x D[y, x] + n (n + 1) y, x],
    x] + O[x]^10,
 1]

@Nasser I was confused because in the first iteration all D are 0. But later y inside are replaced with (n x^2)/2+(n^2 x^2)/2 etc so it makes sense now :)

8:28 AM

@Kuba thanks! But I am confused, how come integrate works now with y having no dependency on x, as in y[x] in there? i.e. normally one write D[y[x], {x, 2}] not D[y, {x, 2}], but it seems to work ok in this case. There seems to be something more going on. I really never used fixedpoint like this example shows.

    This method does not seem to work for removable singularity in the ODE. Only for ordinary points. For example, this gives series solution OK:

    FixedPoint[
     Function[y,
      y - Integrate[Integrate[ D[y, {x, 2}] + x  D[y, x] + y, x], x] +
       O[x]^10], 1]

But this hangs

FixedPoint[
 Function[y,
  y - Integrate[Integrate[ D[y, {x, 2}] + 1/x  D[y, x] + y, x], x] +
   O[x]^10], 1]

@Nasser before Integrateevaluates y is injected because this is a function: Function[y, something]. First iteration, the argument is just 1 so all D[y, x]->D[1,x] -> 0, but later the result is (n x^2)/2+(n^2 x^2)/2 and D[(n x^2)/2+(n^2 x^2)/2,x] isn't 0 anymore.

I was hoping it will work for Bessel ODE and give the series the solution also.

@Kuba I see. thanks.

Mathematica is still missing feature in DSolve to give series solution as an option. This feature is standard in Maple for years. I'd like to do this in M one day:

2 hours later…

10:44 AM

@Kuba @Nasser I believe they type the traditional-form-code by first typing input/standard-form-code first and pressing Ctrl+Shift+T then.

11:09 AM

@xzczd that is probably the case, sure.

2 hours later…

J. M.

1:09 PM

@Nasser If it's a linear DE, you can always use Series[] on DifferentialRoot[].