Fractional calculus is a generalization of differentiation and integration to non-integer orders. Some key points: 1) Fractional differentiation allows for differentiation of arbitrary non-integer order, not just integers like 1, 2, 3. 2) The concept was first explored in the 17th century but several mathematicians contributed definitions and approaches over subsequent centuries. 3) Fractional integration involves taking the nth integral of a function where n is any real number, not just integers. This can be done using the Riemann-Liouville definition or the Laplace transform.

1. Fractional Optimal Control
p
Problems: A Simple Application
in Fractional Kinetics
Vicente Rico-Ramirez
Vi t Rico-R i
Ri
Department of Chemical Engineering
p g g
Instituto Tecnologico de Celaya
Mexico
M i

2. 1 Introduction
What i Fractional
h is i l
Calculus?

3. Fractional Calculus
• Fractional calculus is a generalization of ordinary
differentiation and integration to arbitrary NON INTEGER
order.
order.
d f Ordinary differentiation:
O di diff ti ti
n dx Integer n=1
d f
n
dx Non-
Non-integer n
d1 2 f
12
= ? Fractional differentiation
dx

4. A Bit of History: 1695 (Igor Podlubny)
Podlubny)
dn f
dt n ?
What if the order
will be n=1/2 ?
It will lead to a paradox
from which one day
useful consequences will
be drawn
L’Hopital
p
Leibniz (1661-
(1661-1704)
(1646-
(1646-1716)

5. A Bit of History
Several mathematicians have contributed with alternative
approaches to fractional order differentiation:
differentiation:
n mx
XVII Century: Leibniz d e
= m n e mx
dx n
XVIII Century: Euler
d n xm
= m(m − 1)K (m − n + 1) x m − n
dx n
XIX Century
Lagrange, L l
L Laplace, F i
Fourier
Riemann-
Riemann-Liouville
Caputo, 1967

6. Fractional Integration
F(t) is obtained back through nth-integration
nth-
of Y(t)
d n F (t ) t t n −1 t1
dt n
= Y (t ) F (t ) = ∫∫
0 0
... ∫ Y ( t 0 ) dt 0 ... dt n − 2 dt n −1
0
1 t 1 Using Laplace
F (t ) = J Y (t ) = ∫0 (t − τ )1−n Y (τ )dτ
n
(n − 1)! Transform
Non
integer 1 t 1 Riemann-
Riemann-Liouville
∫0 (t − τ )1−α Y (τ )dτ
−α
values of n 0 D Y (t ) =
Γ(α ) Definition
t
( renamed
as α)
t α −1
−α
0 D Y (t ) = * Y (t )
Γ(α )
t

7. Fractional Derivation
Fractional differentiation or order α is expected to be the
inverse ope ation of f actional integ ation:
in e se operation fractional integration
integration:
Riemann-Liouville Definition (Left)
1 d t 
 ∫ (t − τ ) Y (τ )dτ 
α −α
a Dt Y (t ) =
Γ(1 − α ) dt  a 