分数阶微积分

Fractional Calculus
. .
Fractional Calculus
2014 年 4 月 19 日 1
Fractional Calculus
. 1.1The Basic Idea
. Theorem . 1.A(Fundamental Theorem of Classical Calculus) Let f : [a, b] → R be a continuous function,and let F : [a, b] → R be defined by ∫ x F (x) := f (t)dt.
a −∞
where ϕ1 (u) = and
{
un−1 for0 < u ≤ b − a, 0 { else, f (u) fora ≤ x ≤ b, 0 else.
ϕ2 (u) =
By consideration,ϕj ∈ L1 (R) for j ∈ {1, 2}.and thus by a classical result on Lebesgue integration the desired result follows. . 14
. Lemma . 1.2. Let m, n ∈ N,such that m > n,and let f be a function having a continuous nth derivatives on the interval [a, b].Then, .
m−n D n f = D m Ja f.
n f itself is almost everywhere x ∈ [a, b].Moreover,the function Ja
also an element of L1 [a, b]. .
13
Fractional Calculus . Proof. . We write the integral in questions as ∫ x ∫ +∞ n− 1 (x − t) f (t)dt = ϕ1 (x − t)ϕ2 (t)dt
a
Then,F is differentiable and . F ′ = f.
2
Fractional Calculus . Definition . 1.1. (a) By D,we denote the operator that maps a differentiable function onto its derivatives,i.e. Df (x) := f ′ (x). (b) By Ja ,we denote the operator that maps a function f ,assumed to be(Riemann) integrable on the compact interval [a, b],onto its primitive centered at a,i.e. ∫ Ja f (x) :=
a x
f (t)dt
for a ≤ x ≤ b. .
3
Fractional Calculus
. Definition . n to denote the n-fold (c) For n ∈ N we use the symbols Dn and Ja
1 := J , iterates of D and Ja , respectively,i.e. we set D1 := D, Ja a n := J J n−1 for n ≥ 2. and Dn = DDn−1 and Ja a a .
(n−1)
(x) = f
(n−1)
∫ (a) +
a
x
g (t)dt.
In this case we call g the (generalized) nth derivative of f,and we simply write g = f (n) . .
11
Fractional Calculus
m n Ja Ja ϕ(x)
1 = Γ(m)Γ(n)
∫
a
x
∫ ϕ(τ )
0
1
[(x − τ )(1 − s)]m−1
×[s(x − τ )]n−1 (x − τ )dsdτ . 16
Fractional Calculus . Proof. . 1 = Γ(m)Γ(n) ∫
a x
ϕ(x)(x − τ )
7
Fractional Calculus
. Definition . 1.3. Let 0 < µ < 1, k ∈ N and 1 ≤ p. Lp [a, b] := {f : [a, b] → R; f is measurable on [a, b] and L∞ [a, b] := {f : [a, b] → R; f is measurable and essentially bounded on[a, b]}, Hµ [a, b] := {f : [a, b] → R; ∃c > 0∀x, y ∈ [a, b] : |f (x) − f (y )| ≤ c|x − y |µ }, . ∫b
m+n−1
∫
0
1
(1 − s)m−1 sn−1 dsdτ
In view of the Theorem D.6, ∫1 m−1 sn−1 ds = Γ(m)Γ(n)/Γ(n + m), and thus 0 (1 − s) ∫ x 1 m n m+n Ja Ja ϕ(x) = ϕ(τ )(x − τ )m+n−1 dτ = Ja ϕ(x). Γ(m + n) a almost everywhere on [a, b]. Moreover,by the classical theorems on parameter integrals,if
a
|f (x)|p dt < ∞},
8
Fractional Calculus
. Definition . C k [a, b] := {f : [a, b] → R; f has a continuous kth derivative}, C [a, b] := C 0 [a, b], H0 [a, b] := C [a, b]. .
4
Fractional Calculus . Lemma . 1.1. Let f be Riemann integrable on [a, b].Then, for a ≤ x ≤ b and n ∈ N,we have .
n Ja f (x) =
1 (n − 1)!
∫
a
x
(x − t)n−1 f (t)dt.
m n Ja Ja ϕ(x)
1 = Γ(m)Γ(n)
∫
a
x
(x − t)
m−1
∫
a
t
(t − τ )n−1 ϕ(τ )dτ dt.
. 15
Fractional Calculus . Proof. . In view of Theorem 2.1,the integrals exist,and by Fubini’s theorem we may interchange the order of integration,obtaining ∫ x∫ x 1 m n Ja Ja ϕ(x) = (x − t)m−1 (t − τ )n−1 ϕ(τ )dtdτ Γ(m)Γ(n) a τ ∫ x ∫ x 1 = ϕ(τ ) (x − t)m−1 (t − τ )n−1 dtdτ. Γ(m)Γ(n) a τ The substitution t = τ + s(x − τ ) yields
10
Fractional Calculus
. Definition . 1.5. By An or An [a, b] we denote the set of functions with an absolutely continuous (n − 1)st derivative,i.e. the functions f for which there exists(almost everywhere) a function g ∈ L1 [a, b] such that f
5
Fractional Calculus . proof of lemma 1.2. . n f = f ,we have f = D m−n J m−n f. Applying the operator By Dn Ja a Dn to both sides of this relation and using the fact that Dn Dm−n = Dm ,the statement follows. . . Definition . 1.2 The function Γ : (0, ∞) → R, defined by ∫ ∞ Γ(x) := tx−1 e−t dt,
Fractional Calculus . Theorem . 2.2 Let m, n ≥ 0 and ϕ ∈ L1 [a, b].Then
m n m+n Ja Ja ϕ = Ja ϕ
holds almost everywhere on [a, b].If additionally ϕ ∈ C [a, b] or m . + n ≥ 1, the identity holds everywhere on [a, b]. . Proof. . We have
9
Fractional Calculus . Definition . 1.4 By H ∗ or H ∗ [a, b] we denote the set of functions f : [a, b] → R with the property that there exists some L > 0 such that |f (x + h) − f (x)| ≤ L|h|ln|h|−1 where |h| ≤ 1/2 and x, x + h ∈ [a, b]. . . Theorem . 1.B(Fundamental Theorem in Lebesgue Space) Let f ∈ L1 [a, b].Then,Ja f is differentiable almost everywhere in [a, b],and DJa f = f also holds almost everywhere oalled Euler’s function or Euler’s integral of the second kind.