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

djtak@dmi.uns.ac.rs

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!