SLIDE 1
s tr r r ss - - PowerPoint PPT Presentation
s tr r r ss - - PowerPoint PPT Presentation
s tr r r ss trrt rtrs trt r
SLIDE 2
SLIDE 3
■♥tr♦❞✉❝t✐♦♥
❊①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❞❡r✐✈❛t✐✈❡ (Dα
axf) (x) = d
dx x
a
f(t)(x − t)−α Γ(1 − α) dt α ∈ (0, 1) ❚❤❡♦r❡♠ ❬❆✳ ▼✳ ◆❛❦❤✉s❤❡✈✱ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✶✾✼✹✱ ✈♦❧✳ ✶✵❪ ▲❡t f(x) ∈ L(a, y)✱ f(x) ∈ Hλ[y − δ, y] (λ > α)✱ ❛♥❞ f(y) ≥ f(x) ∀x ∈ (a, y)✳ ❚❤❡♥ (Dα
axf) (y) ≥ f(y)(y − a)−α
Γ(1 − α) . ■♥ ❛❞❞✐t✐♦♥✱ ✐❢ f(y) > 0 t❤❡♥ (Dα
axf) (y) > 0✳
SLIDE 4
■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ◮ ▼✐①❡❞ t②♣❡ P❉❊ ◮ ▲♦❛❞❡❞ ✐♥t❡❣r❛❧ ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ◮ ❉❡❣❡♥❡r❛t❡ P❉❊ ◮ Pr♦❜❧❡♠s ✇✐t❤ s❤✐❢t ❢♦r P❉❊ ◮ ❋r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s
SLIDE 5
◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❉✐✛❡r✳ ❊q✉✳ ✶✾✼✹ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❉✐✛❡r✳ ❊q✉✳ ✶✾✼✺ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ✶✾✼✼ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❊q✉❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❇✐♦❧♦❣②✱ ▼♦s❝♦✇✱ ✶✾✾✺ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥✱ ▼♦s❝♦✇✱ ✷✵✵✸ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ Pr♦❜❧❡♠s ✇✐t❤ ❙❤✐❢t ❢♦r P❉❊✱ ▼♦s❝♦✇✱ ✷✵✵✻ ▲✉❝❤❦♦ ❨✉✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✵✵✾ ▼❛♠❝❤✉❡✈ ▼✳❖✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✵ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ▲♦❛❞❡❞ ❊q✉❛t✐♦♥s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥✱ ▼♦s❝♦✇✱ ✷✵✶✷ ❆❧✲❘❡❢❛✐ ▼✳ ❊❏◗❚❉❊ ✷✵✶✷ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✹ ❆❧✲❘❡❢❛✐ ▼✳✱ ▲✉❝❤❦♦ ❨✉✳ ❋❈❈❆ ✷✵✶✹ ❑❤✉❜✐❡✈ ❑✳❯✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✹ ▼❛s❛❡✈❛ ❖✳❑❤✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✹ ▼❛s❛❡✈❛ ❖✳❑❤✳ ❉✐✛❡r✳ ❊q✉✳ ✷✵✶✹ Ps❦❤✉ ❆✳❱✳ ❉✐✛❡r✳ ❊q✉✳ ✷✵✶✺ ▲✉❝❤❦♦ ❨✉✳✱ ❨❛♠❛♠♦t♦ ▼✳ ❋❈❈❆ ✷✵✶✻ ❈❛♦ ▲✳✱ ❑♦♥❣ ❍✳✱ ❩❡♥❣ ❙❤✳✲❉✳ ❏◆❙❆ ✷✵✶✼ ❊❢❡♥❞✐❡✈ ❇✳■✳ ▼❛t❤✳ ◆♦t❡s ✷✵✶✽ ❇♦r✐❦❤❛♥♦✈ ▼✳✱ ❑✐r❛♥❡ ▼✳✱ ❚♦r❡❜❡❦ ❇✳❚✳▼❛t❤✳ ▲❡tt✳ ✷✵✶✽ ❇♦r✐❦❤❛♥♦✈ ▼✳✱ ❚♦r❡❜❡❦ ❇✳❚✳ ▼❛t✳ ❩❤✳ ✷✵✶✽ ❑✐r❛♥❡ ▼✳✱ ❚♦r❡❜❡❦ ❇✳❚✳ ❋❈❈❆ ✷✵✶✾ ▲✉❝❤❦♦ ❨✉✳✱ ❨❛♠❛♠♦t♦ ▼✳ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ t✐♠❡✲❢r❛❝t✐♦♥❛❧ P❉❊s ❆♥❛t♦❧② ❑♦❝❤✉❜❡✐✱ ❨✉r✐ ▲✉❝❤❦♦ ✭❊❞s✳✮ ❍❛♥❞❜♦♦❦ ♦❢ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ✷ ✷✵✶✾
SLIDE 6
▼❛✐♥ r❡s✉❧ts ■♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs (Df) (x) = d dx x
a
k(x, t)f(t) dt, k : S → R f : (a, y) → R (Df) : (b, y) → R S = {(x, t) : b < x < y, a < t < x} a, b, y ∈ R ∪{−∞} −∞ ≤ a ≤ b < y f(t) ∈ C(a, y] ∩ L(a, y) [k(x, t)]x=y ∈ L(a, y) k(x, t), kx(x, t) ∈ C
- S \ {x = t}
SLIDE 7
▼❛✐♥ r❡s✉❧ts ❊①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❚❤❡♦r❡♠ ▲❡t [kx(x, t)]x=y ≤ 0 ❛♥❞ x−ε
a
[f(t) − f(x)] kx(x, t) dt ⇒ h0(x) d dx x−ε
a
k(x, t) dt ⇒ h1(x) [f(x − ε) − f(x)] k(x, x − ε) ⇒ 0 ε → 0 x ∈ (y − δ, y]
SLIDE 8
❚❤❡♦r❡♠ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♥ f(y) = sup
a<t<y f(t)
= ⇒ (Df) (y) ≥ f(y) (D 1) (y) (Df) (y) = f(y) (D 1) (y) ⇐ ⇒ f(t) ≡ const ♦r [kx(x, t)]x=y ≡ 0 ❛♥❞ f(t) ≡ const [kx(x, t)]x=y ≡ 0 f(y) (D 1) (y) ≥ 0 = ⇒ (D f) (y) > 0
SLIDE 9
Pr♦♦❢ s❦❡t❝❤ Kε(x) = x−ε
a
k(x, t) f(t) dt (ε > 0). Kε(x) = x−ε
a
[f(t) − f(x)] k(x, t) dt + f(x) x−ε
a
k(x, t) dt d dxKε(x) = x−ε
a
[f(t) − f(x)] kx(x, t) dt + f(x) d dx x−ε
a
k(x, t) dt (D f) (x) = lim
ε→0
d dxKε(x) = x
a
[f(t) − f(x)] kx(x, t) dt + f(x) (D 1) (x).
SLIDE 10
❘❡♠❛r❦ DRL = D I ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ t②♣❡ DC = I D ❈❛♣✉t♦ t②♣❡ DCu + Lu = F u ∈ C(Ω) DRLu + Lu = F u ∈ C(Ω) ϕ · u ∈ C(Ω)
SLIDE 11
❲❡✐❣❤t❡❞ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ▲❡t ϕ(t) ∈ C(a, y] ∩ L(a, y) ❛♥❞ f(t) = ϕ(t) g(t) g(t) ∈ C[a, y] ϕ(t) [kx(x, t)]x=y ≤ 0 ❛♥❞ x−ε
a
ϕ(t) [g(t) − g(x)] kx(x, t) dt ⇒ h0(x) d dx x−ε
a
ϕ(t) k(x, t) dt ⇒ h1(x) [g(x − ε) − g(x)] k(x, x − ε) ⇒ 0 ε → 0 x ∈ (y − δ, y]
SLIDE 12
❲❡✐❣❤t❡❞ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♥ g(y) = sup
a<t<y g(t)
= ⇒ (Df) (y) ≥ g(y) (D ϕ) (y) (Df) (y) = g(y) (D ϕ) (y) ⇐ ⇒ g(t) ≡ const ♦r ϕ(t) [kx(x, t)]x=y ≡ 0 ❛♥❞ g(t) ≡ const ϕ(t) [kx(x, t)]x=y ≡ 0 g(y) (D ϕ) (y) ≥ 0 = ⇒ (D f) (y) > 0
SLIDE 13
Pr♦♦❢ s❦❡t❝❤ ❇② f(t) = ϕ(t) g(t) ✇❡ ❣❡t (D f) (x) = d dx x
a
f(t) k(x, t) dt = d dx x
a
g(t) ˜ k(x, t) dt = ˜ D g
- (x),
✇❤❡r❡ ˜ k(x, t) = ϕ(t)k(x, t).
SLIDE 14
❲❡✐❣❤t❡❞ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ▲❡t α ∈ (0, 1) ❛♥❞ t1−αf(t) ∈ C[0, y] f(t) ∈ Hλ[y − δ, y] (λ > α, δ > 0) ❛♥❞ t1−αf(t) ≤ y1−αf(y) ∀t ∈ (0, y) ❚❤❡♥ (Dα
0xf) (y) ≥ 0
❛♥❞ (Dα
0xf) (y) = 0
⇐ ⇒ f(t) = const · tα−1
SLIDE 15
Pr♦♦❢ s❦❡t❝❤ ❚❛❦✐♥❣ k(x, t) = (x − t)−α Γ(1 − α) ❛♥❞ ϕ(t) = tα−1 ✇❡ ❣❡t g(t) = t1−αf(t) (D f) (x) = (Dα
0xf) (x)
❛♥❞ (Dα
0x ϕ) (x) = 0
❚❤❡r❡❢♦r❡ (Dα
0xf) (y) = (Dα 0x ϕg) (y) ≥ g(y) (Dα 0x ϕ) (y) = 0
SLIDE 16
❆♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡ Pr♦❜❧❡♠ st❛t❡♠❡♥t ∂α ∂yα − ∂2 ∂x2
- u(x, y) = f(x, y)
(0 < α < 1) Ω = {(x, y) : z1(y) < x < z2(y), 0 < y < T} u(z1(y), y) = ϕ1(y) u(z2(y), y) = ϕ2(y) 0 < y < T lim
y→0 y1−αu(x, y) = τ(x)
z1(0) < x < z2(0) y1−αu(x, y) ∈ C(Ω)
SLIDE 17
❆♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡ ❉♦♠❛✐♥ Ω = {(x, y) : z1(y) < x < z2(y), 0 < y < T} z1(y) ր z2(y) ց z1(y) < z2(y)
SLIDE 18
❆♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡ ❯♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s ▲❡t u(x, y) ❜❡ ❛ s♦❧✉t✐♦♥ ♦❢ ❤♦♠♦❣❡♥❡♦✉s ♣r♦❜❧❡♠ ✭f ≡ 0✱ τ ≡ 0✱ ϕi ≡ 0✮ ❛♥❞ v(x, y) = y1−αu(x, y) v(x, y) ∈ C(Ω) v(x, y) ≡ 0 = ⇒ ∃ (ξ, η) ∈ Ω : v(ξ, η) = sup
Ω
v(x, y) > 0 (♦t❤❡r✇✐s❡ v → −v) (ξ, η) ∈ ∂Ω\{y = T} = ⇒
- ∂2
∂x2v
- (ξ,η) ≤ 0
- ∂2
∂x2u
- (ξ,η) ≤ 0
- ∂α
∂yαu
- (ξ,η) > 0
♦r u(x, y) = A · yα−1 = ⇒ u ≡ 0
SLIDE 19
❈♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs ❈♦♥s✐❞❡r (Df) (x) ✇✐t❤ k(x, t) = k(x − t) ❛♥❞ a = 0 ✐✳❡✳ (Kf) (x) = d dx x
a
k(x − t)f(t) dt, ❬❆✳ ◆✳ ❑♦❝❤✉❜❡✐✱ ■♥t❡❣r❛❧ ❊q✉✳ ❖♣❡r✳ ❚❤❡♦r②✱ ✷✵✶✶✱ ✈♦❧✳ ✼✶❪ ❬❆✳ ◆✳ ❑♦❝❤✉❜❡✐✱ ●❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s✱ ■♥ ❍❛♥❞❜♦♦❦ ♦❢ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱
❱♦❧✉♠❡ ✶✱ ❇❡r❧✐♥✱ ✷✵✶✾❪
❬❨✉✳ ▲✉❝❤❦♦ ❛♥❞ ▼✳ ❨❛♠❛♠♦t♦✱ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ t✐♠❡✲❢r❛❝t✐♦♥❛❧ P❉❊s ■♥ ❍❛♥❞❜♦♦❦
♦❢ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s❀ ❱♦❧✉♠❡ ✷✱ ❇❡r❧✐♥✱ ✷✵✶✾❪
❬❨✉✳ ▲✉❝❤❦♦ ❛♥❞ ▼✳ ❨❛♠❛♠♦t♦✱ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ✷✵✶✻✱ ✈♦❧✳ ✶✾❪ ❬❆✳❱✳ Ps❦❤✉✱ ❘❡♣✳ ❆❞②❣✳ ✭❈❤❡r❦❡ss✳✮ ■♥t✳ ❆❝❛❞✳ ❙❝✐✳✱ ✷✵✵✶✱ ✈♦❧✳ ✺❪ ❬❆✳❱✳ Ps❦❤✉✱ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ♦❢ ❋r❛❝t✐♦♥❛❧ ❖r❞❡r✱ ▼♦s❝♦✇✱ ✷✵✵✺❪
SLIDE 20
❊①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❝♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs ▲❡t f(t) ∈ C(0, y] ∩ L(0, y) k(t) ∈ L(0, y) ∩ C1(0, y] k′(t) ≤ 0 ω(t) k′(t) ∈ L(0, y) lim
ε→0 ω(ε) k(ε) = 0
ω(t) = sup
t<x<y |f(x) − f(x − t)|
❚❤❡♥ f(y) ≥ f(t) ∀t ∈ (0, y) = ⇒ (K f) (y) ≥ f(y)k(y) ❛♥❞ (K u) (y) = f(y)h(y) ⇐ ⇒ f(t) ≡ const ♦r k(t) ≡ const Pr♦♦❢ (K 1) (x) = d
x
x
0 k(x − t) dt = k(x)
SLIDE 21
❋r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ ▲❡t α ∈ (0, 1)✱ g(t) ∈ C1[0, y]✱ ❛♥❞ g′(t) > 0 ∀t ∈ [0, y]
- Dα
g f
- (x) =
1 g′(x)Γ(1 − α) d dx x
a
f(t)g′(t) [g(x) − g(t)]αdt f(t) ∈ Hλ[y − δ, y] λ > α f(y) ≥ f(t) ∀t ∈ [a, y] = ⇒
- Dα
g f
- (y) ≥ f(y)[g(y) − g(a)]−α
Γ(1 − α) Pr♦♦❢
- Dα
g f
- (x) =
1 g′(x) (Df) (x) ✇❤❡r❡ k(x, t) = g′(t) Γ(1 − α) [g(x) − g(t)]−α
SLIDE 22