Mathematical Models and the Fractional Multiverse - Part-II.pdf

The Harmonic Oscillator: integer order x fractional order

1. The Harmonic Oscillator: integer order x fractional order
P(t) =
W
0
(t)
W(t)
;Q(t) =
W2(t)
W(t)
;

2. The Harmonic Oscillator: integer order x fractional order
Int= 0 W(0) = 1 W
0
(0) = 0
t!0
+
W2(t) = +1.
Asymptotic behavior,jt

j ! 1, the expressionsW(t),W
0
(t) W2(t)
behave in the ways
1
t
2,
1
t
2+1and
1
t
2+2respectively according to the
asymptotic property of the Mittag-Leer Function. Thus, we can use the
Mellin Transform on these functionsW(t),W
0
(t) andW2(t). Doing
s=+ 1 we obtain a value that will be the equivalent of fractional
moments-(), for these functions.
M(s) =
Z
1
0
f(t)t
s1
dt
W=
Z
1
0
t

W(t)dt;W
0
=
Z
1
0
t

W
0
(t)dt;W2=
Z
1
0
t

W2(t)dt:
p=
W
0
W
; q=
W2
W
:

3. The Harmonic Oscillator: integer order x fractional order
P(t) =
W
0
(t)
W(t)
;p=
W
0
W
; Q(t) =
W2(t)
W(t)
;q=
W2
W
:

4. The Harmonic Oscillator: integer order x fractional order
P(t) =
W
0
(t)
W(t)
;p=
W
0
W
; Q(t) =
W2(t)
W(t)
;q=
W2
W
:

5. The Harmonic Oscillator: integer order x fractional order
In all cases we will useq= 1 and!= 1:
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0D

ty+y= 0

6. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0D

ty+y= 0

7. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0D

ty+y= 0

8. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0
D

ty+y= 0

9. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0
D

ty+y= 0

10. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0
D

ty+y= 0

11. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0
D

ty+y= 0

12. The Harmonic Oscillator: integer order x fractional order
EDO-(p;1) :y
00
+py
0
+y= 0EDF-:
C
0
D

ty+y= 0
EDO-(pf;1) :y
00
+pfy
0
+y= 0EDF-:
C
0
D

ty+y= 0

13. 1927 - The Year: "Essentially, all models are wrong, but
some are useful." (George Box)

14. March 6, 1927 - In Great Britain, around 1000 people die
each week thanks to a u epidemic.
W.O. Kermack & A.G. McKendrick. Contributions to the
mathematical theory of epidemics{I. Proceedins of the
Royal Society, Vol. 115A, pp. 700{721 (1927).

15. Epidemic Model - SIR and two guidelines
Classic Case with vital dynamics: [tempo]
1
dS(t)
dt
=N
S(t)
N
I(t)S(t)
dI(t)
dt
=
S(t)
N
I(t)I(t)I(t)
dR(t)
dt
= I(t)R(t)
N= S(t) +I(t) +R(t)
Construct a Fractional SIR model respecting the construction
of W.O. Kermack & A.G. McKendrick.
Dierent fractional orders in compartments.
Two types of Fractional Models: Caputo(99%) and
Riemann-Liouville(1%).

16. Model SIR-Caputo: S(t)+I(t)+R(t)=N
Correction of the units of the Model parameters.
Case-1: [tempo]

D

S(t) =N
S(t)
N
I(t)S(t)
D

I(t) =
S(t)
N
I(t)I(t)I(t)
D

R(t) = I(t)R(t)
Case-2: [tempo]

D

S(t) =

N
S(t)
N
I(t)

S(t)
D

I(t) =
S(t)
N
I(t)

I(t)

I(t)
D

R(t) =

I(t)

R(t)
Dokoumetzidis, Magin and Macheras, A commentary on
fractionalization of multi-compartmental models, Journal of
pharmacokinetics and pharmacodynamics, 2010.

17. Model SIR:16=26=3and N(t)= S(t)+I(t)+R(t)
D
1
S(t) =
1
S(t)
N
I(t)
D
2
I(t) =
2
S(t)
N
I(t)
2
I(t)
D
3
R(t) =
3
I(t)
Dierent ow between compartments and
dN(t)
dt
6= 0.
N(t) =N0+
Z
t
0

2(t)
21
(2)

1(t)
11
(1)

S()
N
I()d
+
Z
t
0

3(t)
31
(3)

2(t)
21
(2)

I()d
If1=2=3then
dN(t)
dt
= 0.

18. Parameters, constant population, ow direction

19. The sign of the fractional derivative does not imply
monotonicity

20. The sign of the fractional derivative does not imply
monotonicity
Diethelm, K.: Monotonicity of functions and sign changes of their
Caputo derivatives. Fract. Calc. Appl. Anal. 19(2), 561{566 (2016).

21. Limitations and applications in a fractional Barbalat's
Lemma

22. Barbalat's Classic Lemma.

23. Barbalat's lemma in integral form: Classical x Fractional.

24. Fractional Barbalat Lemma Limitation.