Mathematical Models and the Fractional Multiverse - Part-I.pdf
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About This Presentation
Mathematical Models and the Fractional
Multiverse: what is lost, what is gained, what is
transformed?
Slide Content
Mathematical Models and the Fractional
Multiverse: what is lost, what is gained, what is
transformed?
Sandro Rodrigues Mazorche
Departamento de Matematica-ICE
Universidade Federal de Juiz de Fora
Seminario periodico de Calculo Fracionario
https://www.youtube.com/watch?v=i7w4kreAG_Q
28/02/2023
Multiverse: what is lost, what is gained, what is
transformed?
Sandro Rodrigues Mazorche
Departamento de Matematica-ICE
Universidade Federal de Juiz de Fora
Seminario periodico de Calculo Fracionario
https://www.youtube.com/watch?v=i7w4kreAG_Q
28/02/2023
Prelude: Mathematical Models and Algorithms for
Optimization and Complementarity
Optimization Problem<=>Variational Inequalities Problems
<=>Nonlinear Complementarity Problems.
Algorithms: FDIPA, FAIPA(Prof. Jose Herskovits Norman);
FDA-NCP, FDA-MNCP
Obstacle Problem; Contact Problem;Dike Problem; Linear
Elasticity Problems; Signorini Problem; Elastic{plastic torsion;
Class of Moving Boundary Problems: In Situ Combustion;
Oxygen Diusion Problem; Cell Oxygenation in a Situation of
Cardiac Arrhythmia;
Problems Equilibrium: Stackelberg-Nash-Cournot;
Systems of Equations: Switched Complementarity linear
systems; Population Models of growth and mortality
structured by stages;
Optimization and Complementarity
Optimization Problem<=>Variational Inequalities Problems
<=>Nonlinear Complementarity Problems.
Algorithms: FDIPA, FAIPA(Prof. Jose Herskovits Norman);
FDA-NCP, FDA-MNCP
Obstacle Problem; Contact Problem;Dike Problem; Linear
Elasticity Problems; Signorini Problem; Elastic{plastic torsion;
Class of Moving Boundary Problems: In Situ Combustion;
Oxygen Diusion Problem; Cell Oxygenation in a Situation of
Cardiac Arrhythmia;
Problems Equilibrium: Stackelberg-Nash-Cournot;
Systems of Equations: Switched Complementarity linear
systems; Population Models of growth and mortality
structured by stages;
The Journey into the Fractional Universe
A random walk through Fractional Calculus through 2 Models,
in the company of Noemi Zeraick Monteiro and Matheus Tobias
Mendonca.
The Call to Begin the Journey;
Harmonic Oscillator.
The SIR Model - derived from Caputo;
The SIR Model - derived from Riemann-Liouville.
A random walk through Fractional Calculus through 2 Models,
in the company of Noemi Zeraick Monteiro and Matheus Tobias
Mendonca.
The Call to Begin the Journey;
Harmonic Oscillator.
The SIR Model - derived from Caputo;
The SIR Model - derived from Riemann-Liouville.
The Call to Begin the Journey
"It will lead to a paradox, from which one day useful consequences will be drawn"
Derived from Riemann-Liouville
Derived from Grunwald-Letnikov
Derived from Caputo-(1967*): Caputo-Djrbashian or Caputo-Dzhrbashjan.
There are many more denitions of FD: Weyl; Riesz; Marchaud; Hilfer; etc....
"An interesting Model to start in Fractional Calculus. (Just an Appetizer.)"
"A Powerful Tool: The Laplace Transform!"
\300 years later, everyone does it that way, but it could have been dierent."
"He had the solution and...."
"He had the solution and his friends had the tools..."
The father
The Queen
The Queen's children...
The friends...
The friends...
1896-1905: It had everything to be, but it just didn't.....
The Harmonic Oscillator: Almost 100 years later....
Sandro R. Mazorche and Matheus Tobias Mendonca, O Oscilador Harm^onico: ordem
inteira x ordem fracionaria, CNMAC 2022- XLI Congresso Nacional de Matematica
Aplicada e Computacional.
Takingy1(t) =E(!
t
) andy2(t) =tE;2(!
t
), and from the
theory of Second Order Linear Ordinary Dierential Equations:Ify1,y2
are two solutions of the dierential equationy
00
+P(t)y
0
+Q(t)y= 0
, whereP(t)andQ(t)are continuous functions on an open intervalI
and if there is a pointt02Iwhere the Wronskian ofy1andy2is
nonzero.
dy1
dt
=!
t
1
E;(!
t
) ;
d
2
y1
dt
2=!
t
2
E;1(!
t
)
dy2
dt
=E(!
t
) ;
d
2
y2
dt
2=!
t
1
E;(!
t
)
R. Garrappa. \Numerical evaluation of two and three parameter
Mittag-Leer functions".Em: SIAM J. Numer. Anal. 53 (3 2015), pp.
1350{1369.
inteira x ordem fracionaria, CNMAC 2022- XLI Congresso Nacional de Matematica
Aplicada e Computacional.
Takingy1(t) =E(!
t
) andy2(t) =tE;2(!
t
), and from the
theory of Second Order Linear Ordinary Dierential Equations:Ify1,y2
are two solutions of the dierential equationy
00
+P(t)y
0
+Q(t)y= 0
, whereP(t)andQ(t)are continuous functions on an open intervalI
and if there is a pointt02Iwhere the Wronskian ofy1andy2is
nonzero.
dy1
dt
=!
t
1
E;(!
t
) ;
d
2
y1
dt
2=!
t
2
E;1(!
t
)
dy2
dt
=E(!
t
) ;
d
2
y2
dt
2=!
t
1
E;(!
t
)
R. Garrappa. \Numerical evaluation of two and three parameter
Mittag-Leer functions".Em: SIAM J. Numer. Anal. 53 (3 2015), pp.
1350{1369.
The Harmonic Oscillator: integer order x fractional order
W(t) = [E(!
t
)]
2
+!
t
E;(!
t
)E;2(!
t
)
W
0
(t) =!
t
1
[E;1(!
t
)E;2(!
t
)E(!
t
)E;(!
t
)]
W2(t) =!
t
2
[!
t
[E;(!
t
)]
2
+E(!
t
)E;1(!
t
)]
W(t) = [E(!
t
)]
2
+!
t
E;(!
t
)E;2(!
t
)
W
0
(t) =!
t
1
[E;1(!
t
)E;2(!
t
)E(!
t
)E;(!
t
)]
W2(t) =!
t
2
[!
t
[E;(!
t
)]
2
+E(!
t
)E;1(!
t
)]
The Harmonic Oscillator: integer order x fractional order
P(t) =
W
0
(t)
W(t)
;Q(t) =
W2(t)
W(t)
;
P(t) =
W
0
(t)
W(t)
;Q(t) =
W2(t)
W(t)
;
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