ala 20210 on the operational solution of the system of fractional differential equations Đurđica...

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 gf

g

f

The Mikusinski operator field

The Wright function

The character of the operational function

e s 1

t ,0 t | | 21 , 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

aij a ij1I aij

2, #

bi bi1I bi

2p , #

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 47 234

5 11 2 3 3 9 27 81 243 7291

( )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

aij a ij1I aij

2, #

bi bi1I bi

2p , #

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

mmdxu

tm

mt

txu

t

m

mm

m

m

t

txutxuD

0

Fractional calculus

Basic 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 calculus

Relations between fractional integral and differential operators

1( )

!0

( ) ( );

( ) ( ) (0 ) .k

mk t

kk

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

x

txu

t

txu

0

2

2 10

1( , ) , 0 1,

(1 ) ( )( , )

1( , ) , 1 2.

(1 ) ( )

t

t

du x

tu x t

t du x

t

x R, 0 t T

The time-fractional diffusion equation

with the conditions

),,0(),()0,( lxxxu 0 1

),,0(),()0,(

),()0,( lxxt

xuxxu

1 2

u0, t ft, u1, t gt, t 0, #

,)()(

))()((,),( 1

)(0)1(

1

xsxus

xxsuxutd

t

2

2 1

2 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,

nn

tx

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 example

The 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

n

ex 2k 1s /2 k 0

n

e x 2ks /2

2e3 3 i 0

p

i 2e3 k 0

n

e x 2k 1s /2 k 0

n

ex 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 d

dt dt dt dt

, , 1, ,ii

i

ri n

m

1 2

( )( ), (0) [ (0) (0) (0)] , 0T

n

d x tBX t x x x x t a

dt

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 nnn n

s X s x X

Xs 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!