ala 20210 on the operational solution of the system of fractional differential equations
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ALA 20210 On the operational solution of the system of fractional differential equations. Đurđica Takači Department of Mathematics and Informatics Faculty of Science, Univer sity of Novi Sad Novi Sad, Serbia [email protected]. The Mikusinski operator field. - PowerPoint PPT PresentationTRANSCRIPT
ALA 20210
On the operational solution of the system of fractional differential equations
Đurđica TakačiDepartment of Mathematics and Informatics
Faculty of Science, University of Novi SadNovi Sad, Serbia
The Mikusinski operator field
The set of continuous functions with supports in with the usual addition and the multiplication given by the convolution
is a commutative ring without unit element. By the Titchmarsh theorem, it has no divisors of
zero; its quotient field is called the Mikusinski operator
field
C
0
( ) ( ) ( ) , 0t
f g t f t g d t
0, ,
C
The Mikusinski operator field The elements of the Mikusinski operator field
are convolution quotients of continuous functions
,0,, CC gfgf
The Mikusinski operator fieldThe Wright function
The character of the operational function
e s 1t ,0 t | |
2 1 , 0 1 , #
, z n 0
zn
n! n . #
se
The matrices with operators
, square matrix, is a given vector, is the unknown vector
AX B, #
A n n B
X x 1 x 2 x nT
a ij a ij1I aij2, #
bi bi1 I bi2p , #
x i P iQi
, i 1,2, ,n,
Example
1 1 2
2 1
2 1 2
x 1
x 2
x 3
1
2
2
,
2
2 3
2
2 3
3 2
2 3
4 15 11 2 3
7 2 25 11 2 3
2 11 2 85 11 2 3
.X
22 3 4 5
1 2 3
4 1 1 11 70 587 5209 47 2345 11 2 3 3 9 27 81 243 729
1 ( )3
x
I
2 3 411 70 587 5209 47 234( )9 27 81 2 243 3! 729 4!
t t tt
The matrices with operators
, square matrix, is a given vector, is the unknown vector
AX B, #
A n n B
X x 1 x 2 x nT
a ij a ij1I aij2, #
bi bi1 I bi2p , #
x i P iQi
, i 1,2, ,n,
The matrices with operators The exact solution of
The approximate solution
Xm x 1m x 2m x nm T, x im k 1
m
x ikk 1 , #
X x 1 x 2 x nT, x i k 1
x ikk 1 #
A X B
Fractional calculus The origins of the fractional calculus go back to the
end of the 17th century, when L'Hospital asked in a letter to Leibniz about the sense of the notation
the derivative of order
Leibniz replied: “An apparent paradox, from which one day useful
consequences will be drawn"
,n
nDDx
n 1/2 1/2
Fractional calculus
The Riemann-Liouville fractional integral operator of
order
Fractional derivative in Caputo sense
,0,)()()(
1)(0
1
dftxfJ
),()( xfJJxfJJ
0,),(),(1,),()(
)(1
,),(0
1
mmdxutm
mt
txu
t
m
mm
m
mttxutxuD
0
Fractional calculusBasic properties of integral operators
J J ft J ft , 0;
J J ft J J ft;
J t c c 1 c 1
t c,
#
Fractional calculusRelations between fractional integral
and differential operators
1( )
!0
( ) ( );
( ) ( ) (0 ) .km
k tk
k
D J f t f t
J D f t f t f
fxfJ )(
1
0
1
1
0
1
)0,()(
)0,()(),(
m
k
kk
k
m
k
kmk
kmmm
sxut
xus
sxut
xustxuD
Relations between the Mikusiński and the fractional calculus
On the character of solutions of the time-fractional diffusion equation
to appear in Nonlinear Analysis Series A:
Theory, Methods & Applications
Djurdjica Takači, Arpad Takači, Mirjana Štrboja
The time-fractional diffusion equation
,),(),(2
2
xtxu
ttxu
0
2
2 10
1 ( , ) , 0 1,(1 ) ( )( , )
1 ( , ) , 1 2.(1 ) ( )
t
t
du xtu x t
t du xt
x R, 0 t T
The time-fractional diffusion equation
with the conditions
),,0(),()0,( lxxxu 0 1
),,0(),()0,(),()0,( lxxtxuxxu
1 2
u0, t ft, u1, t gt, t 0, #
,)()(
))()((,),( 1)(0
)1(1
xsxus
xxsuxutd
t
2
2 12 21
(2 ) ( )0
2
( , ) , ( ( ) ( ) ( ))
( ) ( )) ( ))
td
tu x s u x s x x
s u x s x s x
,10
1 2
The time-fractional diffusion equation
The time-fractional diffusion equation
In the field of Mikusinski operators the time-fractional diffusion equation has the form
,10
))()()(
),())()(
xsxusxu
xuxsxus
2
2
)()()()(
),()())()(
xsxsxusxu
xuxsxsxus
,21
u0 f, u1 g, #
The time-fractional diffusion equation The solution is
The character of operational functions The Wright function
/ 2 / 2
1 1( ) ( ),xs xspu x C e C e u x
exs ,
.)(!
)(1)(1
00,
nntx
txt
te
nn
n
xs
0,
The time-fractional diffusion equation The exact solution
ux up0 f k 0
ex 2k 1s /2
k 0
e x 2ks /2
up1 g k 0
e x 2k 1s /2
k 0
ex 2k 1s /2 upx.
#
A numerical example
The exact solution
In the Mikusinski field
ux, t t
2ux, tx 2
2ext2
3 t2ex, 0 1 #
ux, 0 0, # u0, t t2 , u1, t et2 , #
x 0,1,
0 t T.
ux, t t2ex,
ux s ux 2ex3 23 ex . #
u0 23 , u1 2e3 , #
The solution has the form
A numerical example
ux C1exs /2 C2e xs
/2 upx,
upx 2ex3 3 i 0
i . #
A numerical exampleThe exact solution
ux 23 3 i 0
i 23
k 0
ex 2k 1s /2
k 0
e x 2ks /2
2e3 3 i 0
i 2e3
k 0
e x 2k 1s /2
k 0
ex 2k 1s /2
2ex3 3 i 0
i.
#
A numerical example
ũxn 23 3 i 0
p
i 23 k 0
nex 2k 1s /2
k 0
ne x 2ks /2
2e3 3 i 0
p
i 2e3 k 0
ne x 2k 1s /2
k 0
nex 2k 1s /2
2ex3 3 i 0
p
i.
#
A numerical example
The system of fractional differential equationsInitial value problem (IVP)
1 2
1 2, , ,
n
n
d d d ddt dt dt dt
, , 1, ,ii
i
r i nm
1 2( ) ( ), (0) [ (0) (0) (0)] , 0T
nd x t BX t x x x x t adt
Caputo fractional derivative, order
1 2 1 2 , 1[ ] , [ ], [ ]
0 1, 1,...,
T n n nn n ij i j
i
x x x x B a R R
i n
1 1
2 2
11 1 1
122 2
1
(0)
(0)
(0)n n nn n
s X s x XXs X s x
B
Xs X s x
1
2
11 12 1
21 22 2
1 2 2
,
n
n
n
n n n
a s a a
a a s a
A
a a a s
AX B
1
2
11
12
1
(0)
(0)
(0)nn
s x
s x
B
s x
0 1
1 2 ,
( ) ( ) (0)! ( 1)
1 , ( , ,..., ), , 1, ,
i
ckpnijk c
j ii k i
in i
i
A tx t t x
k ck c
rc m lcm m m m i n
m m
The initial value problem (IVP) has a unique continuous solution x
References Caputo, M., Linear models of dissipation whose Q is
almost frequency independent- II, Geophys. J. Royal Astronom. Soc., 13, No 5 (1967), 529-539 (Reprinted in: Fract. Calc. Appl. Anal.,11, No 1 (2008), 3-14.)
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.
Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999).
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; Ed. B. Ross), Lecture Notes in Math. 457, Springer Verlag, N. York (1975), pp. 1-37.
Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999).
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; Ed. B. Ross), Lecture Notes in Math. 457, Springer-Verlag, N. York (1975), pp. 1-37.
Thank you for your attention!