Glorious GIFs

30.03.10

While doing research on the Schrödinger equation, I've come across some beautiful plots that I'd like to share.



To watch the animation, just click on the image.

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This animation shows the error of a numerical approximation to the Schrödinger equation with starting value u(x,0) = exp(ix). The number of nodes in space and time starts at 2 and goes to 100.

(4.3 MB)

This animation shows how the equation behaves while the parameter lambda is increased. I believe the plot shows the interval [0,10 000]. It almost looks chaotic before the solution settles down into neat wave packets.

(2.5 MB)

This last animation shows how the solution forms when the number of nodes are increased. Note that this is just the real part of the solution (the imaginary part looks almost the same).

(2.6 MB)

More animations like these are sure to come in the future. Until then, take care! :)

Happy Pi Day!

14.03.10

Today is March 14th, or 3/14. Since 3.14 are the first three digits of the mathematical constant pi, this date is named pi-day and is considered an important holiday for mathematicians around the world. It also happens to be the birthday of Albert Einstein.

Now how about that.

If you find this one day to be insufficient for your pi-celebration-requirements, you can always celebrate pi-approximation day which is 22/7 (approximately 3,1428...). Pure mathematicians (and those who know pi to more than three digits) will however frown upon this date since it is a poor approximation indeed (only valid for the three first digits).

The first two hounded (or so) digits of pi are as follows

3.141592653589793238462643383279502884197169399375105820974944592
30781640628620899862803482534211706798214808651328230664709384460
95505822317253594081284811174502841027019385211055596446229489549
303819644288109756

Since pi is irrational there is no way of knowing what digit which will follow the next, so out of curiosity I therefore checked to see if any interesting sequences appeared in the first 200 million decimals. I was able to locate the following:

- My phone number (eight digits)
- My date of birth (eight digits)
- My social security number (five digits)
- 12345678
- 7182818 (seven first digits of Euler's number)

So, bake a pie, draw circles (and calculate the circumference / area) and reflect upon the great achievements of Albert Einstein on this fine day.

Numerous Numerics

01.03.10

What you are looking at is the approximation to the famous Schrödinger equation with a cubic nonlinearity.



I am currently studying this equation for my term project in the subject TMA4212, so more pictures of this manner are sure to appear in the near future.

^____^

Edit: 30.03
After studying the equation some more I have to conclude that the above plot is not a solution to the equation. Still, it looks neat.