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Alexey Ivanov

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11 years, 297 days

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Source of interest in this topic.  There is a problem with drawing a cylinder. I draw a cylinder using the arrow function as follows:

cylinder := seq(arrow(`<,>`((L[1][i]+L[4][i])*(1/2), (L[2][i]+L[5][i])*(1/2), (L[3][i]+L[6][i])*(1/2)), `<,>`(-(L[1][i]+L[4][i])*(1/2)+(L[7][i]+L[10][i])*(1/2), -(L[2][i]+L[5][i])*(1/2)+(L[8][i]+L[11][i])*(1/2), -(L[3][i]+L[6][i])*(1/2)+(L[9][i]+L[12][i])*(1/2)), width = 1, color = green, head_length = 0, head_width = 0, length = VectorNorm(`<,>`((L[1][i]+L[4][i])*(1/2)-(L[7][i]+L[10][i])*(1/2), (L[2][i]+L[5][i])*(1/2)-(L[8][i]+L[11][i])*(1/2), (L[3][i]+L[6][i])*(1/2)-(L[9][i]+L[12][i])*(1/2)), 2), transparency = 0.), i = 1 .. N+1):

(L [j], j = 1..12 - the coordinates of the red points on the cylinder).


But, as you can see, because of the faces, everything does not turn out very smoothly. Is there a way to display the cylinder smoothly, but so that without too much computing resources, ie no more resource than a "arrow"?

Example of Duffing equation with boundary conditions.
y'' + 0.2y' + y^3 - 0.3cos(s) = 0;
y(0) = y (2Pi);
y'(0) = y'(2Pi);
For convenience, we replace the original equation with a system of two first order equations:
--------------------------------------------------------------------------
x1'(t) = 2*Pi*x2(t);
x2'(t) = - 0.4*Pi*x2(t) - 2*Pi*x1(t)^3 +0.6*Pi*cos(2*Pi*t);
x1(0) = x1(1);
x2(0) = x2(1);
--------------------------------------------------------------------------
I have long wanted to apply an optimization package to solve a boundary value problem for ODE. The decision helped procedure for solving ODE, written by forum member vv.
It seems to me that two solutions have been found and that the solutions are weakly sensitive to the initial approximations. These are two closed trajectories. For example, these are points that belong to these solutions:
(0.5966963,  1.0482816) , ( - 0.3132584, 0.0664941).
I am wondering: are the solutions right, and how justified is the use of optimization methods for such tasks?
At the end of the program, the solution is checked on the original Duffing equation using standard Maple functions.   Duffing_equation_BC.mw

(In the figures, the trajectory bypass occurs three times.)

    My profile picture was formerly animation and looked like this: 


  It would be interesting to paint a triangle on a sphere. How can I do that?

      Inspired by the theme
http://www.mapleprimes.com/questions/219995-Finding-A-Convinient-Parametrization-Of-Surfaces
Examples in the Mathematica did Alexander Bannikov.
It is equidistant radius 0.1 to the surface

   (x1 ^ 2 + x2 ^ 2-0.4) ^ 2 + (x3 + sin (x1 * x2 + x3)) ^ 4-0.1 = 0;

https://vk.com/doc7819263_439405418?hash=af46d61d8aad95f70b&dl=9f245f5b6b68b47075

and an example of parameterization the same surface

https://vk.com/doc7819263_439432143?hash=36cf31d52c97e2e373&dl=7e4fa17a771dffb331

As I have understood from the words of Alexander Bannikov, parameterization was performed using the functions: RegionFunction, ContourPlot3D, ClippingPlanes.

It turns out that Maple functions inferior?

     It is known that ODE boundary value problem is similar to the problem of solving systems of nonlinear equations. Equations are the boundary conditions, and the variables are the values of the initial data.
For example:

y '' = f (x, y, y '), 0 <= x <= 1,

y (0) = Y0, y (1) = Y1;

Where y (1) = Y1 is the equation, and Z0 is variable, (y '(0) = Z0).

     solve () and fsolve () are not directly suitable for such tasks. Directly should work the package of optimization in relation to a system of nonlinear equations. (Perhaps it has already been implemented in Maple.)
Personally, I am very small and unprofessional know Maple and cannot do it. Maybe there is someone who would be interested, and it will try to implement this approach to solving ODE boundary value problems?  

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