Variable order fractional derivatives Duarte Valrio 2017 / 9 / 7 - - PowerPoint PPT Presentation

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Variable order fractional derivatives Duarte Valrio 2017 / 9 / 7 - - PowerPoint PPT Presentation

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Variable order fractional derivatives Duarte Valrio 2017 / 9 / 7 9/7/2017 Instituto Superior Tcnico Get the lecture notes from our site They are an introduction to some papers With the notes, you can find the papers of which I

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Duarte Valério 2017 / 9 / 7

9/7/2017 Instituto Superior Técnico

Variable order fractional derivatives

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SLIDE 2

Get the lecture notes from our site

  • They are an introduction

to some papers

  • With the notes, you can

find the papers of which I am one of the authors

  • Look for the others in the

net

9/7/2017 Instituto Superior Técnico

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SLIDE 3

Let’s define Da(t)f(t) in four cases:

  • a(t) is natural
  • a(t) is integer
  • a(t) is real
  • a(t) is complex

9/7/2017 Instituto Superior Técnico

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SLIDE 4

We will also talk about:

  • Matlab implementations
  • an application to bone remodelling

dynamics

  • and other subjects too

9/7/2017 Instituto Superior Técnico

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SLIDE 5

Natural case

9/7/2017 Instituto Superior Técnico

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Switching after differentiating

9/7/2017 Instituto Superior Técnico

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SLIDE 7

9/7/2017 Instituto Superior Técnico

2 4 6 8 10 5 10 15 20

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Switching before differentiating

9/7/2017 Instituto Superior Técnico

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SLIDE 9

9/7/2017 Instituto Superior Técnico

2 4 6 8 10 5 10 15 20

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Switching both before and after

9/7/2017 Instituto Superior Técnico

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SLIDE 11

9/7/2017 Instituto Superior Técnico

2 4 6 8 10 5 10 15 20

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Integer case

9/7/2017 Instituto Superior Técnico

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Switching after differentiating

9/7/2017 Instituto Superior Técnico

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SLIDE 14

9/7/2017 Instituto Superior Técnico

2 4 6 8 10 10 20 30 40 50

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Switching before differentiating

9/7/2017 Instituto Superior Técnico

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SLIDE 16

9/7/2017 Instituto Superior Técnico

2 4 6 8 10 10 20 30 40 50

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Switching both before and after

9/7/2017 Instituto Superior Técnico

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9/7/2017 Instituto Superior Técnico

2 4 6 8 10 10 20 30 40 50

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Real case: constant orders

9/7/2017 Instituto Superior Técnico

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Real case: no memory

9/7/2017 Instituto Superior Técnico

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SLIDE 21

Real case: with memory

9/7/2017 Instituto Superior Técnico

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Real case: a different memory

9/7/2017 Instituto Superior Técnico

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Complex case: constant orders

  • GL definition works just the same

9/7/2017 Instituto Superior Técnico

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SLIDE 24

Complex case: no memory

9/7/2017 Instituto Superior Técnico

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Matlab toolboxes

  • Go to Matlab file exchange
  • Look for Duarte Valério
  • Look for Dominik Sierociuk

9/7/2017 Instituto Superior Técnico

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SLIDE 26

Bone remodelling

  • Osteoclasts destroy the bone
  • Osteoblasts rebuild the bone
  • The following scheme is from an

upcoming paper:

9/7/2017 Instituto Superior Técnico

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SLIDE 27

Bone remodelling

9/7/2017 Instituto Superior Técnico

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SLIDE 28

Healthy bone vs. tumourous bone

9/7/2017 Instituto Superior Técnico

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SLIDE 29

Other topics

  • Variable order control
  • Distributed order control
  • Viscoelasticity
  • Anomalous diffusion

9/7/2017 Instituto Superior Técnico