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where n =[µ] + 1, and*
(k)
f n–µ (a)=0, k =0, 1, ... , n – 1. (16)
µ
Let µ > 0. We say that a function f(x) ∈ L 1 has an integrable fractional derivative D a+ f if
n–µ n
I a+ f(x) ∈ AC , where n =[µ]+1.
In other words, this definition introduces a notion involving only the first of the two condi-
µ
tions (15) and (16) describing the class I a+ (L 1 ).
Let µ > 0. In this case the relation
µ µ
D I ϕ(x)= ϕ(x) (17)
a+ a+
holds for any integrable function ϕ(x), and the relation
µ
µ
I D f(x)= f(x) (18)
a+
a+
holds for any function f(x) such that
µ
f(x) ∈ I (L 1 ). (19)
a+
If we replace (19) by the condition that the function f(x) ∈ L 1 (a, b) has an integrable deriva-
µ
tive D a+ f(x), then relation (18) fails in general and must be replaced by the formula
n–1 µ–k–1
(x – a) (n–k–1)
µ
µ
I D f(x)= f(x) – f n–µ (a), (20)
a+
a+
Γ(µ – k)
k=0
n–µ
where n =[µ] + 1 and f n–µ (x)= I a+ f(x). In particular, for 0 < µ < 1 we have
f 1–µ (a) µ–1
µ
µ
I D f(x)= f(x) – (x – a) . (21)
a+ a+
Γ(µ)
8.5-4. The Solution of the Generalized Abel Equation
Consider the Abel integral equation
x
y(t)

dt = f(x), (22)
(x – t) µ
a
where 0 < µ < 1. Suppose that x ∈ [a, b], f(x) ∈ AC, and y(t) ∈ L 1 , and apply the technique of
fractional differentiation. We divide Eq. (22) by Γ(1 – µ), and, by virtue of (1), rewrite this equation
as follows:
f(x)
1–µ
I a+ y(x)= , x > a. (23)
Γ(1 – µ)
1–µ
Let us apply the operator of fractional differentiation D a+ to (23). Using the properties of the
operators of fractional integration and differentiation, we obtain
1–µ
D a+ f(x)
y(x)= , (24)
Γ(1 – µ)
* From now on in Section 8.5, by f (n) (x) we mean the nth derivative of f(x) with respect to x and f (n) (a) ≡ f (n) (x) .
x=a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 451