Page 467 - Handbook Of Integral Equations

8.5. Method of Fractional Differentiation

8.5-1. The Definition of Fractional Integrals

A function f(x) is said to be absolutely continuous on a closed interval [a, b] if for each ε > 0 there
exists a δ > 0 such that for any finite system of disjoint intervals [a k , b k ] ⊂ [a, b], k =1, ... , n, such
that n (b k – a k )< δ the inequality n |f(b k ) – f(a k )| < ε holds. The class of all these functions is
k=1 k=1
denoted by AC.
n
Let AC , n =1,2, ... , be the class of functions f(x) that are continuously differentiable on [a, b]
up to the order n – 1 and for which f (n–1) (x) ∈ AC.
Let ϕ(x) ∈ L 1 (a, b). The integrals

1 x ϕ(t)
I ϕ(x) ≡ dt, x > a, (1)
µ
a+ 1–µ
Γ(µ) a (x – t)
1 b ϕ(t)
I ϕ(x) ≡ dt, x < b, (2)
µ
b– 1–µ
Γ(µ) x (t – x)
where µ > 0, are called the integrals of fractional order µ. Sometimes the integral (1) is called
µ µ
left-sided and the integral (2) is called right-sided. The operators I a+ and I are called the operators
b–
of fractional integration.
The integrals (1) and (2) are usually called the Riemann–Liouville fractional integrals.
The following formula holds:
b b

µ µ
ϕ(x)I ψ(x) dx = ψ(x)I ϕ(x) dx, (3)
a+ b–
a a
which is sometimes called the formula of fractional integration by parts.
Fractional integration has the property
µ β
µ
β
I I ϕ(x)= I µ+β ϕ(x), I I ϕ(x)= I µ+β ϕ(x), µ >0, β > 0. (4)
a+ a+ a+ b– b– b–
Property (4) is called the semigroup property of fractional integration.

8.5-2. The Definition of Fractional Derivatives
It is natural to introduce fractional differentiation as the operation inverse to fractional integration.
For a function f(x)defined on a closed interval [a, b], the expressions

1 d x f(t)
µ
D f(x)= dt, (5)
a+ µ
Γ(1 – µ) dx a (x – t)
µ 1 d b f(t)
D f(x)= – dt (6)
b– µ
Γ(1 – µ) dx x (t – x)
are called the left and the right fractional derivative of order µ, respectively. It is assumed here that
0< µ <1.
The fractional derivatives (5) and (6) are usually called the Riemann–Liouville derivatives.
Note that the fractional integrals are defined for any order µ > 0, but the fractional derivatives
are so far defined only for 0 < µ <1.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 449