Fractional Derivative!

Transcript

1. 0 Fractional Derivative! 2024-08-09 第101回NearMe技術勉強会 Cyan
2. 1 I saw a video on the Youtube about ʻHalf Derivative’ It seems like I have also consider this possibility when I was in college. https://www.youtube.com/watch?v= 2dwQUUDt5Is&t=501s
3. 2 Recap of Calculus • Meaning of derivative ◦ Slope of a curve ◦ Speed of changing • Meaning of Integral ◦ Area • Integral cancel out derivative. => Fundamental theorem of calculus ◦ Derivative and Integral cancel each other
4. 3 Naive Guess
5. 4 Another Strategy
6. 5 Cauchy formula for repeated integration
7. 6 Let’s do combine two integrals first
14. 13 Integral + Derivative => Gamma Function
16. 15 Does this make sense?
20. 19 Riemann-Liouville Fractional Derivative
21. 20 Our result
22. 21 The properties and applications of fractional derivatives • Fractional derivatives is ʻnon-local’ ◦ A normal derivative depends only on the points around the place we would like to calculate slope ◦ Fractional derivatives use the integration as intermediate steps and hence inherit this properties. • Systems involving dissipation ◦ Fractional Schrödinger equation in quantum theory
23. 22 Thank you