Mathematical Physics Vol 1

A.6 Systems of Fractional Equations

467

L { f ( t ) } ( s )

f ( t )

√ kt

1 s √ s e 1 s ν e −

1 √ π t sh2

k / s

2 √ kt ) 2 √ kt )

k k

t t

ν − 1

k / s , ( ν > 0 )

2 J ν 2 I ν

− 1 ( − 1 ( 2

ν − 1

1 s ν e e − k

k / s , ( ν > 0 )

√ s ,

e − k

k 2 √ π t 3 erfc ( k 1 √ π t 2 q t − e ak e a e ak e a e − k

( k > 0 )

4 t

k √ s ,

1 s e −

( k ≥ 0 )

)

2 √ t

k √ s ,

2

1 √ s e − 1 s √ s e − √ s s ( a + √ s ) √ s √ s ( a + √ s ) s α p − β ( s α − λ ) p 1 ( s α − λ ) n ! ( s α − λ ) n + 1 n ! ( s α − λ ) n + 1 ae − k e − k

( k ≥ 0 ) ( k ≥ 0 )

4 t

k √ s ,

k 2 4 t − k erfc ( k 2 √ t ) ( a √ t + k 2 t erfc

π e −

)+ erfc ( k 2 √ t

, ( k ≥ 0 )

)

2 √ t

( a √ t + k

2 t erfc

, ( k ≥ 0 )

)

2 √ t

, ( ℜ ( β ) > 0 | λ s − α | < 1 ) t β − 1 E p α , β

( λ t α )

, ( | λ s − α | < 1 )

e λ t α

∂ ∂λ )

, ( | λ s − α | < 1 ) , ( | λ s − α | < 1 )

n e λ t α

(

t α n e λ z

α , n

A.6 Systems of Fractional Equations

Bellow we present the explicit solutions of the Riemann-Liouville and Caputo systems of linear fractional differential equations involving the following matrices A =   a 11 ... a 1 n .. . . . . .. . a n 1 ... a nn   , ¯ B ( x )=   b 1 ( x ) .. . b n ( x )   (A.80) These kind of systems are a important tool in many applied ar the application to the state-variable technique in control theory.

Problem

Solution ¯ Y h ( x )= e ¯ Y ( x )= e ¯ Y ( x )= e ¯ Y ( x )= e ¯ Y ( x )= E

A ( x − a ) α

¯ C , where ¯ C is a constant matrix

¯ Y ( x )= A ¯ Y ( x )

RL D α RL D α RL D α RL D α C D α C D α

a +

A ( x − a ) α A ( x − a ) α A ( x − a ) α

A ( x 0 − a ) α

¯ Y 0 , ( x 0 > a )

1 ¯ Y

a + Y ( x )= A Y ( x ) , ¯ Y ( x 0 )= a + Y ( x )= A Y ( x ) lim x a + ¯ Y ( x )= A ¯ Y ( x )+ B ¯ Y ( x ) a + Y ( x )= A Y ( x ) , ¯ Y ( a )= a + ¯ Y ( x )= A ¯ Y ( x )+ B ¯ Y ( x ) → a + [(

) −

( e

0

x − a ) 1 − α ¯ Y ( x )]= ¯ Y 0

¯ Y 0

¯ C + R α ( A ( x − a )

A ( x − ξ ) α

x a e α ) ¯ b

¯ B ( ξ ) d ξ

¯ b , ( ¯ b ∈ R n )

α ) C + R

A ( x − ξ ) α

x a e

¯ B ( ξ ) d ξ

Y ( x )= E α ( A ( x − a )

TableA.7: Systems of Fractional Equations.

where

infty ∑ k = 0

infty ∑ k = 0

z α k Γ ( k α + 1 )

z α k Γ ( k α + 1 )

A k

A k

E α ( A z )=

and E α ( A z )=

(A.81)

are, respectively, the natural matrix generalizations of the above mentioned Riemann-Liouville andCaputo σ -exponential. As to the issue of numerically solving fractional differential equations see [25].