I'm a student in Engineering school without abundant math knowledge. And I am sorry if this is a silly question.
I know a little about ''self-similar'', and recently also encountered the concept of "self-affine". I cannot clearly describe what "self-affine" or "self-affinity" actually is and how they related to ''self-similar'' or ''self-similarity''? Specifically,
(1) Could you please show me an example of "self-affine solution" by illustrating how we can see a solution presents such a "self-affine''solution?
(2) In which way self-affinity differ from self-similarity? Or they are the same thing?
Thank you in advance!
Self-similar
Fractals, for example, are self-similar. If you zoom-in (or zoom-out), you will see a similar structure. Consider Mandelbrot set for example
If the objects is scaled by the same amount in all directions, we see similar pattern emerging again and again.
Self-affine
However, in Self-affine structure, you may not see similar pattern on scaling by same amount in all direction. On scaling at specific dispropotinatnate scale, we see similar pattern. The plot of position against time of a simple random walker is statistically self-affine. If you zoom out, the curve looks flat, if you zoom in, the curve look vertical.
In nature mountains for example are statistically self-affline. Pitch fluctuations $S_v(f)$ of music like Beethoven's 3rd Symphony are statistically self-similar. You can plot $S_v(f)=1/f$, and on zooming in or zooming out, the curve look identical.