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0 Fractional Derivative! 2024-08-09 第101回NearMe技術勉強会 Cyan

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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

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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

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3 Naive Guess

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4 Another Strategy

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5 Cauchy formula for repeated integration

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6 Let’s do combine two integrals first

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13 Integral + Derivative => Gamma Function

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15 Does this make sense?

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19 Riemann-Liouville Fractional Derivative

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20 Our result

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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

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22 Thank you