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

$: ALA 20210 On the operational solution of the system of fractional differential equations Đurđica Takači Department of Mathematics and Informatics Faculty$
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

[email protected]

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 elements of the Mikusinski operator field are

convolution quotients of continuous functions,0,, CC gf

g

f


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


, 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 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


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



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 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 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!

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

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