International GeoGebra Conference for
Southeast Europe
January 15-16, Novi Sad, Serbia
Fractional calculus and Geogebra
Đurđica Takači
Departman za matematiku i informatiku
Prirodno-matematički fakultet
Univerzitet u Novom Sadu
Gamma Function
Gamma
Gamma function
function extends
extends factorials
factorials to
to non-integer
non-integer
values
values
 ( n  1)  n!  n ( n  1)! ,
n N,
 ( a  1)  a  ( a )  a ( a  1)  ( a  1),

 (a ) 
x
a 1
0
GamaFunkcija
GamaFunkcija
x
e dx
a0
Convolution
Convolution of two
functions
Convolution.ggb
t
f  g (t ) 

0
f ( t   ) g ( ) d  ,
On the power (order) of the
derivative
f '( x) 
d
f ( x ),
D 
f ' ' ( x )  ( f ' ( x ))'
D 
2
dx
2
D  D
2
1/ 2
-- differential operator
dx
dx
d
d
,
a
D ,
aR
The origins of the fractional calculus go back to the end of the
17th century, when L'Hospital asked (in a letter) Leibniz about the
sense of the notation
n
D ,
n  1/ 2
the derivative of order 1/2. Leibniz's answer was
"An apparent paradox, from which one day useful consequences will
be drawn."
Fractional calculus is a natural
generalization of calculus
Many applications of fractional calculus in:
viscoelasticity and damping,
diffusion and wave propagation,
electromagnetism, chaos and fractals,
heat transfer, biology, electronics,
signal processing, robotics, system
identification, traffic systems,
genetic algorithms, percolation,
modeling and identification,
telecommunications, chemistry,
irreversibility, physics, control systems,
economy, and finance,...
t
J f (t ) 

f ( x ) dx ,
0
t
J ( J f ( t )) 
t
x
 J ( f ( x )) dx   ( 
0
0
0
f ( )) d  ) dx
Fractional calculus
Rimann-Liouville fractional integral operator of
order 
J


f ( x) 
J J

1
 ( )
t
 (t  x )
 1
f ( x ) dx ,
0

f ( x)  J J

Fractional-Integral
f ( x ),
  0,
f (x)  x ,
f ' ( x )  kx
k
f
(a)
( x) 
d
a
dt
a
d
1/ 2
dt
1/ 2
d
1/ 2
dt
1/ 2
x 
k
x
(
2

d
a
dt
a
f ( x) 
 ( k  1)
 ( k  a  1)
...,
( k  a )!
x
 (1  1 / 2  1)
x) 
,
k!
1
2
k 1
x
k a
,
a N
k a
x
1 1 / 2

2

x
 (1 / 2  1)
  (1 / 2  1 / 2  1)
x
1 / 2 1 / 2

2  (3 / 2 )

 (1)
x
0
Fractional derivatives in Caputo
sense
m
t


1
d
u ( x)
m
m  a 1
d t 
dx , m  1  a  m 
 (t  x )
m
m
D t
   (m  a ) 0
dx
,
m
dt
(m )


u
( t ),
am


mN
 Fractional Derivative
  0,
Fractional derivatives in Caputo
sense
D
1/ 2
u (t ) 
d
1/ 2
u (t )
dt

1
 (1  1 / 2 )

a
(t  x )
 (1  1 / 2 )
1 a
1 a
Fractional Derivative
t
1
t
0

2

 (t  x )
0
x
a
dx
y ' ( t )  y ( t ),
y  Ce
D y ( t )  y ( t ),
y (0)  A
a
D y (t ) 
a
1
 (1  a )
t
t

0
x y ' ( t  x ) dx
-a
Delta Functions
The Dirac delta function, or δ function
(introduced by Paul Dirac), is a generalized
function:

 ( x )  0,
x  0,
lim  ( x )   ,
x 0
  ( x ) dx


Delta sequences Academ.
Stevan Pilipovic
 n ( x) :


n
1
( x ) f ( t  x ) dx  f ( t ),
n 
References:
1. Caputo, M., Linear models of dissipation whose Q is almost
frequency independent- II, Geophys. J. Royal Astronom. Soc., 13,
No 5 (1967), 529-539
2. Mainardi, F., Yu. Luchko, Pagnini, G., The fundamental solution of
the space-time fractional diffusion equation Fractional Calculus
and its Application, 4,2. 2001, 153-192.
3. Mainardi, F., Pagnini, G., The Wright functions as the solutions of
time-fractional diffusion equation Applied Math. and Comp.
Vol.141, Iss.1, 20 August 2003, 51-62.
4. Podlubny, I., Fractional Differential Equations, Acad. Press, San
Diego (1999).
5. Ross, B., A brief history and exposition of fundamental theory of
fractional calculus, In: "Fractional Calculus and Its
Applications" (Proc. 1st Internat. Conf. held in New Haven,
1974; Lecture Notes in Math. 457, Springer-Verlag, N. York
(1975), pp. 1-37.
6. Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals
and Derivatives: Theory and Applications, Gordon and Breach
Sci. Publ., Switzerland (1993).
7.
Takači, Dj., Takači, A., On the approximate solution of
mathematical model of a viscoelastic bar, Nonlinear Analysis,
67 (2007), 1560-1569.
8. Takači, Dj., Takači, A., On the mathematical model of a
viscoelastic bar, Math. Meth. Appl. Sci. 2007; 30:1685-1695.
9. Yulita M. R., Noorani,M.S.M., Hashim, I. Variational iteration
method for fractional heat- and wave-like equations, Nonlinear
Analysis, 10 (2009), 1854-1869.
10. Odibat Z., M. Analytic study on linear systems of fractional
differential equations, Computers and Mathematics with
Applications
Delta Functions
The Dirac delta function, or δ function
(introduced by Paul Dirac), is a generalized
function:

 ( x )  0,
x  0,
lim  ( x )   ,
x 0
  ( x ) dx

Academician Stevan Pilipovic
Delta sequences

 n ( x) :


n
( x ) f ( t  x ) dx  f ( t ),
n 
1