Mathematical Physics Vol 1

Chapter A. Fractional Calculus: A Survey of Useful Formulas

464

RL D α

a + f ( x ) , ℜ ( α ) ≥ 0

function - f ( x ) , x > a

Fractional Derivative -

p a

1 2 , p > x

1 p − x

for a = 0 ∧ α = 1 2

( p − x ) − xe µ x

π x

x E x ( − α , µ )+ α E x ( 1 − α , µ ) , for a = 0 E x ( µ − α , ν ) , for a = 0

E x ( µ , ν ) , µ > − 1 x ( µ , ν ) , λ + µ > − 1 x E x ( µ , ν ) , µ > − 2 ( x − a ) − α − 1 sin ( 2 λ ( x − a )) ( x − a ) − α − 1 cos ( 2 λ ( x − a )) x λ , E

Γ ( λ + µ + 1 ) x λ + µ − α Γ ( µ + 1 ) Γ ( λ + µ + 1 − α ) 2

F 2 ( λ + µ + 1 , µ + 1 , λ + µ − α + 1; ν x ) for a = 0

x E x ( µ − α , ν )+ α E x ( µ − α + 1 , ν ) , for a = 0

√ π ( x − a √ π ( x − a

1 2 ) sin ( λ ( x − a )) J 1 2 ) cos ( λ ( x − a )) J

( α +

( λ ( x − a )) ( λ ( x − a ))

2 λ ) − 2 λ ) −

1 2 )

− ( α + − ( α +

( α +

1 2 )

e ( λ ( z − a )) α

( λ ( x − a )) α

λ e

( x )

TableA.3: Analytical Expressions of Some Fractional Derivatives.

L D α

function - f ( x ) , ℜ ( λ ) > 0 ∧ µ > 0 ( b − ax ) γ − 1 ℜ ( γ − α ) < 1 ∧ a > 0 ∧ ax < b

Fractional Derivative -

+ f ( x )

Γ ( 1 + α − γ ) Γ ( 1 − γ ) a − α λ α e λ x

( b − ax ) γ − 1 − α , for ℜ ( α ) ≥ 0

e λ x

µ α sin ( µ x + µ α cos ( µ x + πα 2 ) , for ℜ ( α ) > − 1 πα 2 ) , for ℜ ( α ) > − 1 ( λ 2 + µ 2 ) α / 2 e λ x sin ( µ x + α arctg µ λ ) ( λ 2 + µ 2 ) α / 2 e λ x cos ( µ x + α arctg µ λ )

sin ( µ x ) cos ( µ x )

e λ x sin ( µ x ) e λ x cos ( µ x )

TableA.4: Analytical Expressions of Some Fractional Derivatives.

where is a ∈ ( − ∞ , ∞ ) .

=   0 ,

( x − p ) − α − n − 1 Γ ( − α − n )

, if x > p ≥ a ,

d n δ ( x − p ) d x n

RL D α

(A.73)

a +

if a ≤ x ≤ p ∨ p < a .

where is n ∈ N , a ∈ ( − ∞ , ∞ ) . A.5 Laplace and Fourier Transforms

It is well known that the Laplace and Fourier transforms, for suitable functions, are given by

+ ∞ Z 0

e − st ϕ ( t ) d t

L ϕ ( s )=( L ϕ ( t ))( s )= ˆ ϕ ( s )=

(A.74)

+ ∞ Z

e i κ x ϕ ( x ) d x .

F ϕ ( κ )=( F ϕ ( x ))( κ )= ˆ ϕ ( κ )=

(A.75)

− ∞

A.5.1 Some properties

In connection with the fractional operators we have the following properties for ℜ ( α ) > 0, ℜ ( α ) ∈ ( n − 1 , n ] and suitable functions

n − 1 ∑ k = 0

α 0 + f ( s )= s

α

s n − k − 1 D k RL I n − α

L RL D

L f ( s ) −

0 + f ( 0 ) .

(A.76)