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Oct 18, 2023 315 likes •517 views

r trr tr qts rst rst rt st t t s s

SLIDE 1

❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❢ ✜rst ❦✐♥❞

❡♥♥❛❞✐ ❱❛✐♥✐❦❦♦

❯♥✐✈❡rs✐t② ♦❢ ❚❛rt✉✱ ❊st♦♥✐❛

SLIDE 2

(Vϕu)(t) = ˆ t 1 tϕ s t

u(s)ds, (Vϕ,au)(t) =

ˆ t 1 tϕ s t

a(t, s)u(s)ds, 0 < t ≤ T

❈❖◆❚❊◆❚ ✶✳ ❙♣❛❝❡s Cm ❛♥❞ Cm,r

⋆

✷✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs Vϕ ❛♥❞ Vϕ,a✿ s♦♠❡ ♠❛♣♣✐♥❣ ♣r♦♣❡rt✐❡s ✸✳ ❘❡❞✉❝t✐♦♥ ♦❢ ❡q✳ Vϕ,au = f t♦ s❡❝♦♥❞ ❦✐♥❞ ❡q✳ u = V −1

ϕ Vϕ,bu + V −1 ϕ f1

✹✳ ■♥✈❡rs✐♦♥ ♦❢ Vϕ✱ ❝❛s❡ ´ 1

0 xr |ϕ(x)| dx < ∞,

´ 1

0 xr+1 |ϕ′(x)| dx < ∞

✺✳ ■♥✈❡rs✐♦♥ ♦❢ Vϕ✱ ❝❛s❡ ´ 1

0 xr |ϕ(x)| dx < ∞✱

´ 1

0 xr+1(1 − x) |ϕ′(x)| dx < ∞

✻✳ ❈♦♠♣❛❝t♥❡ss ❝♦♥❞✐t✐♦♥s ❢♦r Vϕ,b ✇✐t❤ b(t, t) ≡ 0 ✼✳ ■♥✈❡rs✐♦♥ ♦❢ Vϕ,a ✭t❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤❡ t❛❧❦✮ ✽✳ ❍♦✇ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ Vϕu = f❄ ✾✳ P♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❡q✉❛t✐♦♥ Vϕ,au = f ✶✵✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❡q✉❛t✐♦♥ Vϕ,au = f ✶

SLIDE 3

❘❊❋❊❘❊◆❈❊❙ ✭❚❤❡ t❛❧❦ ✐s ❜❛s❡❞ ♦♥ ❬✶✕✺❪ ❛♥❞ ♦♥ ❛ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ ❬✺❪✮✳ ✶✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✵✾✮✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✶✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✸✵✱ ✶✶✹✺✕✶✶✼✷✳ ✷✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✶✵✮✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✷✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✸✶✱ ✶✾✶✕✷✶✾✳ ✸✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✶✵✮✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛✲ t✐♦♥s✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✸✶✱ ✸✶✸✕✸✸✽✳ ✹✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✶✶✮✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥✲✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ◆✉♠✳ ❋✉♥❝t✳ ❆♥✳ ❖♣t✐♠✳✱ ✸✷✱ ✽✸✕✶✵✾✳ ✺✳ ●✳ ❱❛✐♥✐❦❦♦ ✭s✉❜♠✐tt❡❞✮✳ ❋✐rst ❦✐♥❞ ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✶✳ ✻✳ ❚✳ ❉✐♦❣♦✱ P✳ ▲✐♠❛ ✭✷✵✵✽✮✳ ❙✉♣❡r❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❛ ❝❧❛ss ♦❢ ✇❡❛❦❧② s✐♥❣✳ ❱♦❧t❡rr❛ ✐♥t✳ ❡q✳ ❏✳ ❈♦♠♣✉t✳ ❆♣♣❧✳ ▼❛t❤✳✱ ✷✶✽✱ ✸✵✼✕✸✶✻✳ ✼✳ ❙✳ ●✳ ▼✐❦❤❧✐♥ ✭✶✾✻✶✮✳ ▲✐♥❡❛r ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s✳ ❘♦✉t❧❡❞❣❡ P✉❜❧✳ ✽✳ ❑✳ ❊✳ ❆t❦✐♥s♦♥ ✭✶✾✼✹✮✳ ❆♥ ❡①✐st❡♥❝❡ t❤❡♦r❡♠ ❢♦r ❆❜❡❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ❙■❆▼ ❏✳ ▼❛t❤✳ ❆♥❛❧✳✱ ✺✱ ✼✷✾✕✼✸✻✳ ✷

SLIDE 4

✶✳ ❙♣❛❝❡s Cm ❛♥❞ Cm,r

⋆

❋♦r m ≥ 0✱ Cm = Cm[0, T] ✐s t❤❡ st❛♥❞❛r❞ s♣❛❝❡ ✇✐t❤ t❤❡ ♥♦r♠ u Cm= max

0≤k≤m max 0≤t≤T | u(k)(t) | .

❋♦r m ≥ 0✱ r ∈ R✱ s♣❛❝❡ Cm,r

⋆

= Cm,r

⋆

(0, T] ❝♦♥s✐sts ♦❢ ❢✉♥❝t✐♦♥s u ∈ Cm(0, T] s✉❝❤ t❤❛t ✜♥✐t❡ ❧✐♠✐ts limt→0 tk−ru(k)(t)✱ k = 0, ..., m✱ ❡①✐st❀ u Cm,r

⋆

:= max

0≤k≤m sup 0<t≤T

tk−r | u(k)(t) | . ■t ❤♦❧❞s Cm,r

⋆

= trCm,0

⋆

✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r u ∈ Cm✱ t❤❡ ❢✉♥❝t✐♦♥ tru(t) ❜❡❧♦♥❣s t♦ Cm,r

⋆

❜✉t s✉❝❤ ❢✉♥❝t✐♦♥s ❞♦ ♥♦t ❝♦✈❡r ✇❤♦❧❡ Cm,r

⋆

✳ ■t ❤♦❧❞s C = C0 = C0,0

⋆ ✱ ❛♥❞ Cm ⊂ Cm,0✱ Cm = Cm,m ⋆

⊕ Pm−1 ❢♦r m ≥ 1✳ ❈❧❡❛r❧②✱ Cm′,r′

⋆

⊂ Cm,r

⋆

❢♦r m′ ≥ m ≥ 0✱ r′ ≥ r✳ ✸

SLIDE 5

✷✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs

❋♦r ✏❝♦r❡✑ ϕ ∈ L1(0, 1), t❤❡ ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ♦♣❡r❛t♦r Vϕ ∈ L(C) ✐s ❞❡✜♥❡❞ ❬✶❪ ❛s (Vϕu)(t) = ˆ t 1 tϕ s t

u(s)ds =

ˆ 1 ϕ(x)u(tx)dx, 0 ≤ t ≤ T, u ∈ C. ❈❧❡❛r❧②✱ ✐t ❤♦❧❞s t❤❛t Vϕ L(C)≤ ´ 1

0 |ϕ(x)| dx✳ ❋♦r u ∈ Cm✱

(Vϕu)(k)(t) = ˆ 1 ϕ(x)xku(k)(tx)dx, k = 0, ..., m, ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❛t Vϕ ∈ L(Cm)✱ Vϕ L(Cm)≤ ´ 1

0 |ϕ(x)| dx✳ ❋♦r t❤❡ ♦♣❡r❛t♦r

(Vϕ,au)(t) = ˆ t 1 tϕ s t

a(t, s)u(s)ds

✇✐t❤ t❤❡ ❝♦❡✣❝✐❡♥t ❢✉♥❝t✐♦♥ a ∈ Cm(△T)✱ △T =

(t, s) : 0 ≤ s ≤ t ≤ T
✱ ✇❡ st✐❧❧

❤❛✈❡ Vϕ,a ∈ L(Cm)✳ ✹

SLIDE 6

▲❡ss ♦❜✈✐♦✉s ❢❛❝ts ❬✶✱✷❪ ❝♦♥❝❡r♥ t❤❡ s♣❡❝tr❛✿ σL(Cm)(Vϕ) = {0} ∪ { ϕ(k) : k = 0, ..., m − 1} ∪ { ϕ(λ) : λ ∈ C, ❘❡λ ≥ m}, σL(Cm)(Vϕ,a) = a(0, 0)σL(Cm)(Vϕ), ✇❤❡r❡

ϕ(λ) =

ˆ 1 ϕ(x)xλdx, λ ∈ C, ❘❡λ ≥ 0, ✐s t❤❡ ✭s❤✐❢t❡❞✮ ▼❡❧❧✐♥ tr❛♥s❢♦r♠ ♦❢ ϕ✳ ❍❡♥❝❡ Vϕ ∈ L(Cm) ❛♥❞ Vϕ,a ∈ L(Cm) ❛r❡ ♥♦♥❝♦♠♣❛❝t ❢♦r ϕ = 0✱ a(0, 0) = 0✳ ■t ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❢❛❝t t❤❛t Vϕvλ = ϕ(λ)vλ ✇❤❡r❡ vλ(t) = tλ, 0 < t ≤ T, ❘❡λ ≥ 0. ■♥ ♣❛rt✐❝✉❧❛r✱ Vϕ ♠❛♣s ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ s❛♠❡ ♦r ❧❡ss ❞❡❣r❡❡✳ ✺

SLIDE 7

❚♦ tr❡❛t t❤❡ ♦♣❡r❛t♦rs Vϕ ❛♥❞ Vϕ,a ✐♥ s♣❛❝❡s Cm,r

⋆

✱ ✇❡ ❛ss✉♠❡✱ ✐♥st❡❛❞ ♦❢ ϕ ∈ L1(0, 1)✱ t❤❛t ´ 1

0 xr | ϕ(x) | dx < ∞. ❋♦r u ∈ Cm,r ⋆

✱ 0 < t ≤ T ✇❡ ❤❛✈❡ (Vϕu)(t) = ˆ 1 ϕ(x)u(tx)dx, (Vϕu)(k)(t) = ˆ 1 ϕ(x)xku(k)(tx)dx, tk−r(Vϕu)(k)(t) = ˆ 1 ϕ(x)xr(tx)k−ru(k)(tx)dx, lim

t→0 tk−r(Vϕu)(k)(t) =

ˆ 1 ϕ(x)xrdx lim

t→0(tk−ru(k)(t)), k = 0, ..., m.

❲❡ ♦❜s❡r✈❡ t❤❛t Vϕ ♠❛♣s Cm,r

⋆

✐♥t♦ ✐ts❡❧❢✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t a ∈ Cm(△T) ❛♥❞ ´ 1

0 xr | ϕ(x) | dx < ∞ ❢♦r ❛♥ r ∈ R✳ ❚❤❡♥

Vϕ, Vϕ,a ∈ L(Cm,r

⋆

), Vϕ L(Cm,r

⋆

)≤

ˆ 1 xr | ϕ(x) | dx, σL(Cm,r

⋆

)(Vϕ) = {0}∪{

ϕ(λ) : λ ∈ C, ❘❡λ ≥ r}, σL(Cm,r

⋆

)(Vϕ,a) = a(0, 0)σL(Cm,r

⋆

)(Vϕ).

✻

SLIDE 8

✸✳ ❘❡❞✉❝t✐♦♥ ♦❢ ❡q✳ Vϕ,au = f t♦ ❛ s❡❝♦♥❞ ❦✐♥❞ ❡q✳

❲❡ st✉❞② t❤❡ ✉♥✐q✉❡ s♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ✜rst ❦✐♥❞ ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ❡q✉❛t✐♦♥ Vϕ,au = f, ♦r ˆ t 1 tϕ s t

a(t, s)u(s)ds = f(t), 0 < t ≤ T,

✭✶✮ ❛♥❞ ❞✐s❝✉ss ♠❡t❤♦❞s ♦❢ s♦❧✈✐♥❣✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥✳ ❊①❛♠♣❧❡s✳ ✶✳ ❋♦r ϕ(x) = x−α✱ α ∈ R✱ ❡q✉❛t✐♦♥ ✭✶✮ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ❉✐♦❣♦✬s ❡q✉❛t✐♦♥ tα−1 ´ t

0 s−αa(t, s)u(s)ds = f(t)✱ ♦r

´ t

0 a(t, s)u(s)ds = f(t)✳ ❈❢✳ ❬✻❪✳

✷✳ ❋♦r ϕ(x) = x−α(1 − xγ)−ν✱ α ∈ R✱ γ > 0✱ ν ∈ (0, 1)✱ ❡q✉❛t✐♦♥ ✭✶✮ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ t❤❡ ❆❜❡❧ ❡q✉❛t✐♦♥ ❬✼✱✽❪ tα+γν−1 ˆ t (tγ−sγ)−νs−αa(t, s)u(s)ds = f(t), ♦r ˆ t (tγ−sγ)−νa(t, s)u(s)ds = f(t). ✸✳ ϕ(x) = x−αg(x)✱ ϕ(x) = x−α(1 − xγ)−νg(x) ✇✐t❤ g ∈ C[0, 1]✱ g′ ∈ L1(0, 1) ❜♦✉♥❞❡❞ ✐♥ ❛ ✈✐❝✐♥✐t② ♦❢ x = 0 ❛♥❞ ✐♥ ❛ ✈✐❝✐♥✐t② ♦❢ x = 1✳ ✼

SLIDE 9

❚❤❡ st✉❞② ♣r♦❣r❛♠✳ ❆ss✉♠✐♥❣ t❤❛t a(t, t) = 0 ❢♦r 0 ≤ t ≤ T✱ r❡✇r✐t❡ ✭✶✮ ❛s ˆ t 1 tϕ s t

u(s)ds −

ˆ t 1 tϕ s t 1 − a(t, s) a(t, t)

u(s)ds = f(t)

a(t, t), ♦r ❛s Vϕu − Vϕ,bu = f1, b(t, s) = 1 − a(t, s) a(t, t), f1(t) = f(t) a(t, t), ♦r ❛s u = V −1

ϕ Vϕ,bu + V −1 ϕ f1.

◆♦t✐❝❡ t❤❛t b(t, t) ≡ 0✳ ❲❡ ❤❛✈❡ t♦ s♦❧✈❡ ✸ s✉❜♣r♦❜❧❡♠s✿ ✶✳ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ V −1

∈ L(Y, X✮ ❢♦r s✉✐t❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s X ❛♥❞ Y ❀ ✷✳ ♣r♦✈❡ t❤❛t Vϕ,b ∈ L(X, Y ) ✐s ❝♦♠♣❛❝t❀ ✸✳ ♣r♦✈❡ t❤❛t u ∈ X✱ u = V −1

ϕ Vϕ,bu =

⇒ u = 0 ✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ✇❡ ♦❜t❛✐♥ t❤❛t ❡q✉❛t✐♦♥ Vϕ,au = f ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ u ∈ X ❢♦r ❛♥② f s✉❝❤ t❤❛t f1 ∈ Y ✳ ■♥ t❤❡ ♣r❡s❡♥t t❛❧❦ ✇❡ r❡❛❧✐③❡ t❤✐s ♣r♦❣r❛♠ ❢♦r X = Cm✱ Y = Cm+1 ❛♥❞ ❢♦r X = Cm,r

⋆

✱ Y = Cm+1,r

⋆

✭t❤✐s ❝❤♦✐❝❡ ♦❢ s♣❛❝❡s ✐s r❡❧❛t❡❞ ✇✐t❤ ❝♦♥❞✐t✐♦♥ ϕ(1) = 0✮✳ ❆❜♦✉t t❤❡ ❝❤♦✐❝❡ X = Cm,r

⋆

✱ Y = Cm+k,r

⋆

s❡❡ ❬✺❪✳ ✽

SLIDE 10

✹✳ ■♥✈❡rs✐♦♥ ♦❢ Vϕ✱ ❝❛s❡

ˆ 1 xr | ϕ(x) | dx < ∞, ˆ 1 xr+1 | ϕ′(x) | dx < ∞. ✭✷✮ ❚❤❡♦r❡♠ ✷✳ ❆ss✉♠❡ ✭✷✮✳ ❚❤❡♥ Vϕ ∈ L(Cm,r

⋆

, Cm+1,r

⋆

) ❢♦r m ≥ 0✳ ❚❤❡ ✐♥✈❡rs❡ V −1

∈ L(Cm+1,r

⋆

, Cm,r

⋆

) ❡①✐sts ✐❢ ❛♥❞ ♦♥❧② ✐❢ ϕ(1) = 0 ❛♥❞ ϕ(λ) := ˆ 1 ϕ(x)xλdx = 0 ❢♦r ∀λ ∈ C ✇✐t❤ ❘❡λ ≥ r. ✭✸✮ ❯♥❞❡r ❝♦♥❞✐t✐♦♥s ✭✷✮ ❛♥❞ ✭✸✮✱ V −1

L(Cm+1,r′

⋆

,Cm,r′

⋆

)≤ cr ❢♦r r′ ≥ r (cr ✐♥❞❡♣❡♥❞❡♥t ♦❢ r′).

❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ❡q✉❛t✐♦♥s (D⋆ − αI)Vϕu = (D⋆ − αI)f ❛♥❞ Vϕu = f ❢♦r α < r + 1✱ ✇❤❡r❡ (D⋆u)(t) = (tu(t))′ = tu′(t) + u(t)✳ ❊q✉❛t✐♦♥ Vϕu = f t❛❦❡s t❤❡ ❢♦r♠ ϕ(1)u = Vψαu + (D⋆ − αI)f ✇✐t❤ ψα(x) = xϕ′(x) + αϕ(x)✳ ✾

SLIDE 11

✺✳ ■♥✈❡rs✐♦♥ ♦❢ Vϕ✱ ❝❛s❡

ˆ 1 xr | ϕ(x) | dx < ∞, ˆ 1 xr+1(1 − x) | ϕ′(x) | dx < ∞, ✭✹✮

ϕ(r) :=

ˆ 1 xrϕ(x)dx > 0, d dx

xαϕ(x)
≥ 0 (0 < x < 1) ❢♦r ❛♥ α < r + 1.

✭✺✮ ❚❤❡♦r❡♠ ✸✳ ❯♥❞❡r ❝♦♥❞✐t✐♦♥s ✭✹✮ ❛♥❞ ✭✺✮✱ Vϕ ✐s ✐♥❥❡❝t✐✈❡ ✐♥ C0,r

⋆ ✱ ✐t ❤♦❧❞s t❤❛t

Vϕ(Cm,r

⋆

) ⊃ Cm+1,r

⋆

✱ V −1

∈ L(Cm+1,r

⋆

, Cm,r

⋆

)✱ ❛♥❞ V −1

L(Cm+1,r′

⋆

,Cm,r′

⋆

)≤

D⋆ − αI L(Cm+1,r′

⋆

,Cm,r′

⋆

)

ϕ(r)(r + 1 − α)

≤ cr ❢♦r m ≥ 0, r′ ≥ r. ❚❤❡♦r❡♠s ✷ ❛♥❞ ✸ ❤❛✈❡ t❤❡✐r ❝♦✉♥t❡r♣❛rts ❢♦r s♣❛❝❡s Cm ❛ss✉♠✐♥❣ ④✭✷✮✱✭✸✮⑥ ♦r ④✭✹✮✱✭✺✮⑥ ❢♦r r = 0 ✭❛♥❞ ❤❡♥❝❡ α < 1 ✐♥ ✭✺✮✮✳ ■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ V −1

L(Cm+1,Cm)≤ 1

ϕ(0)(1 − α) D⋆ − αI L(Cm+1,Cm) ❢♦r m ≥ 0.

✶✵

SLIDE 12

✻✳ ❈♦♠♣❛❝t♥❡ss ♦❢ Vϕ,b ✐❢ b(t, t) ≡ 0

▲❡♠♠❛ ✶ ✭❬✷❪✮✳ ▲❡t ϕ ∈ L1(0, 1) ❛♥❞ a ∈ Cm(△T) ❢♦r ❛♥ m ≥ 0✳ ❚❤❡♥ Vϕ,a ∈ L(Cm)❀ ✐❢ ✐♥ ❛❞❞✐t✐♦♥ a(0, 0) = 0 t❤❡♥ Vϕ,a ∈ L(Cm) ✐s ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✹✳ ❆ss✉♠❡ ✭✹✮ ❢♦r r = 0 ❛♥❞ ❧❡t b ∈ Cm+1(△T) ❢♦r ❛♥ m ≥ 0✳ ❋✉rt❤❡r✱ ❧❡t b(t, t) = 0 ❢♦r 0 ≤ t ≤ T✳ ❚❤❡♥ Vϕ,b ∈ L(Cm, Cm+1)❀ ✐❢ ✐♥ ❛❞❞✐t✐♦♥ ∂b(t, s)/∂t

t=s=0= 0 t❤❡♥ Vϕ,b ∈ L(Cm, Cm+1) ✐s ❝♦♠♣❛❝t✳

❚❤❡♦r❡♠ ✺✳ ❆ss✉♠❡ ✭✹✮ ❢♦r ❛♥ r ∈ R ❛♥❞ b ∈ Cm+1(△T) ❢♦r ❛♥ m ≥ 0✳ ❋✉rt❤❡r✱ ❧❡t b(t, t) = 0 ❢♦r 0 ≤ t ≤ T✳ ❚❤❡♥ Vϕ,b ♠❛♣s Cm,r

⋆

✐♥t♦ Cm+1,r+1

⋆

✱ ❛♥❞ Vϕ,b ∈ L(Cm,r

⋆

, Cm+1,r

⋆

) ✐s ❝♦♠♣❛❝t✳ ▼♦r❡♦✈❡r✱ Vϕ,b L(Cm,r′

⋆

,Cm+1,r′

⋆

)→ 0 ❛s r′ → ∞✳

✭❈♦♥❞✐t✐♦♥ ∂b(t, s)/∂t

t=s=0= 0 ♦❢ ❚❤❡♦r❡♠ ✹ ✐s ♥♦t ♥❡❡❞❡❞ ♥♦✇✦✮

■♠♣❧✐❝❛t✐♦♥ u = V −1

ϕ Vϕ,bu✱ u ∈ C0,r ⋆

= ⇒ u = 0 ❡❛s✐❧② ❢♦❧❧♦✇s ✉s✐♥❣ ❡st✐♠❛t❡ V −1

L(Cm+1,r′

⋆

,Cm,r′

⋆

)≤ cr ❢♦r r′ ≥ r✳ ❲❡ ♦❜t❛✐♥ t❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤❡ t❛❧❦✿

✶✶

SLIDE 13

✼✳ ■♥✈❡rs✐♦♥ ♦❢ Vϕ,a ✭♠❛✐♥ r❡s✉❧ts✮

❘❡❝❛❧❧✿ ˆ 1 xr | ϕ(x) | dx < ∞, ˆ 1 xr+1 | ϕ′(x) | dx < ∞, (2) ϕ(1) = 0 ❛♥❞ ϕ(λ) = 0 ❢♦r ∀λ ∈ C ✇✐t❤ ❘❡λ ≥ r, (3) ˆ 1 xr | ϕ(x) | dx < ∞, ˆ 1 xr+1(1 − x) | ϕ′(x) | dx < ∞, (4)

ϕ(r) :=

ˆ 1 xrϕ(x)dx > 0, d dx

xαϕ(x)
≥ 0 (0 < x < 1) ❢♦r ❛♥ α < r + 1. (5)

❚❤❡♦r❡♠ ✻✳ ❆ss✉♠❡ ❡✐t❤❡r ④✭✷✮✱✭✸✮⑥ ♦r ④✭✹✮✱✭✺✮⑥✳ ❋✉rt❤❡r✱ ❛ss✉♠❡ t❤❛t a ∈ Cm+1(△T) ❛♥❞ a(t, t) = 0 ❢♦r 0 ≤ t ≤ T✳ ❚❤❡♥ Vϕ,a ✐s ✐♥❥❡❝t✐✈❡ ✐♥ C0,r

⋆ ✱

Vϕ,a(Cm,r

⋆

) ⊃ Cm+1,r

⋆

✱ ❛♥❞ V −1

ϕ,a ∈ L(Cm+1,r ⋆

, Cm,r

⋆

)✳ ❚❤❡♦r❡♠ ✼✳ ❆ss✉♠❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✻ ❢♦r r = 0. ❚❤❡♥ Vϕ,a ✐s ✐♥❥❡❝t✐✈❡ ✐♥ C ✱ Vϕ,a(Cm) ⊃ Cm+1✱ ❛♥❞ V −1

ϕ,a ∈ L(Cm+1, Cm)✳

❲❡ ❣❡t ∂b(t, s)/∂t

t=s=0= 0 ❜② t❤❡ tr✐❝❦ a(t, s)u(s) = ω(s)a(t, s) u(s)

ω(s)✳

✶✷

SLIDE 14

❊①❛♠♣❧❡✿ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❆❜❡❧ ❡q✉❛t✐♦♥✳ ❋♦r ϕ(x) = x−α(1−xγ)−ν✱ α ∈ R✱ γ > 0✱ ν ∈ (0, 1)✱ r > α − 1, ❝♦♥❞✐t✐♦♥s ✭✹✮ ❛♥❞ ✭✺✮ ❛r❡ ❢✉❧✜❧❧❡❞✳ ❆ss✉♠❡ t❤❛t a ∈ Cm+1(△T)✱ a(t, t) = 0 ❢♦r 0 ≤ t ≤ T✳ ❚❤❡♥ V −1

ϕ,a ∈ L(Cm+1,r ⋆

, Cm,r

⋆

) ❡①✐sts ❜② ❚❤❡♦r❡♠ ✻✳ ❆s ♠❡♥t✐♦♥❡❞✱ ❡q✉❛t✐♦♥ Vϕ,au = f ❤❛s t❤❡ ❢♦r♠ ♦❢ ❆❜❡❧ ❡q✉❛t✐♦♥ tα+γν−1 ˆ t (tγ − sγ)−νs−αa(t, s)u(s)ds = f(t), ✭❆✶✮ ♦r ˆ t (tγ − sγ)−νa(t, s)u(s)ds = f(t), ✭❆✷✮ ✇❤❡r❡ f(t) = t1−α−γνf(t)✱ u(s) = s−αu(s)✳ ❖♥ t❤❡ ❜❛s✐s ♦❢ ❚❤❡♦r❡♠ ✻ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t♦ f ∈ Cm+1,r

⋆

t❤❡r❡ ❝♦rr❡s♣♦♥❞s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ u ∈ Cm,r

⋆

♦❢ ✭❆✶✮✳ ❍❡♥❝❡✱ t♦ f ∈ Cm+1,r+1−α−γν

⋆

t❤❡r❡ ❝♦rr❡s♣♦♥❞s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ u ∈ Cm,r−α

⋆

♦❢ ✭❆✷✮✱ ♦r t♦ f ∈ Cm+1,r

⋆

t❤❡r❡ ❝♦rr❡s♣♦♥❞s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ u ∈ Cm,r+γν−1

⋆

✳ ❚❤✐s ✐s ✐♥ ❛ ❣♦♦❞ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡ r❡s✉❧t ♦❢ ❆t❦✐♥s♦♥ ❬✽❪❀ ✇❡ r❡❞✉❝❡❞ t❤❡ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥ a ∈ Cm+2(△T) ♦❢ ❬✽❪ t♦ ♠♦r❡ ♥❛t✉r❛❧ a ∈ Cm+1(△T)✳ ✶✸

SLIDE 15

✽✳ ❍♦✇ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ Vϕu = f❄

❆ss✉♠❡ ❡✐t❤❡r ④✭✷✮✱✭✸✮⑥ ♦r ④✭✹✮✱✭✺✮⑥✳ ▲❡t f ∈ C1,r

⋆ ❀ t❤❡♥ f = tr ¯

f ✇✐t❤ ¯ f ∈ C1,0

⋆ ✳

❆♣♣r♦①✐♠❛t❡ ¯ f ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ¯ fN = N

n=0 dntn ✭✉s✐♥❣ ❡✳❣✳ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥

♦r ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥✮ s♦ t❤❛t ¯ f − ¯ fN C1,0

⋆ ≤ δ✳ ❚❤❡♥ ❢♦r fN := N

n=0 dntn+r

✇❡ ❤❛✈❡ f − fN C1,r

⋆ ≤ crδ✱ ❛♥❞ uN := V −1

ϕ fN = N n=0 dn

ϕ(n+r)tn+r ❛♣♣r♦①✐♠❛t❡s

u = V −1

ϕ f ✇✐t❤ t❤❡ ❛❝❝✉r❛❝②

u − uN C0,r

⋆ = V −1

ϕ (f − fN) C0,r

⋆ ≤ V −1

L(C1,r

⋆ ,C0,r ⋆ ) crδ.

▼♦r❡♦✈❡r✱ ✐❢ f ❝❛♥ ❜❡ ❡①♣❛♥❞❡❞ ✐♥t♦ ❛ ♣♦✇❡r s❡r✐❡s f(t) = ∞

n=0 dntn+r, t❤❡♥

u = V −1

ϕ f = ∞ n=0 dn

ϕ(n+r)tn+r✱ ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛❞✐✐ ♦❢ t❤❡ t✇♦ s❡r✐❡s ❛r❡ ❡q✉❛❧✳

■♥ ❝❛s❡ ④✭✷✮✱✭✸✮⑥ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❛♣♣❧② s♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s t♦ ❡q✉✐✈❛❧❡♥t s❡❝♦♥❞ ❦✐♥❞ ❡q✉❛t✐♦♥ ϕ(1)u = Vψαu + (D⋆ − αI)f✱ ✇❤❡r❡ ψα(x) = xϕ′(x) + αϕ(x)✱ α < r + 1✳ ❯♥❢♦rt✉♥❛t❡❧②✱ s♦♠❡t✐♠❡s ❬✸❪ t❤❡ ♥♦♥✲❝♦♠♣❛❝t♥❡ss ♦❢ Vψα ❝❛✉s❡s ❛ ♥♦♥✲❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ t❤❡ s♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ♦♥ t❤❡ ✜rst st❡♣s✳ ✶✹

SLIDE 16

✾✳ P♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❡q✉❛t✐♦♥ Vϕ,au = f

❆ss✉♠❡ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✼✱ ✐✳❡✳ a ∈ Cm+1(△T)✱ a(t, t) = 0 ❢♦r 0 ≤ t ≤ T✱ ❛♥❞ ❡✐t❤❡r ④✭✷✮✱✭✸✮⑥ ♦r ④✭✹✮✱✭✺✮⑥ ❛r❡ ❢✉❧✜❧❧❡❞ ✇✐t❤ r = 0✳ ❚❤❡♥ ❡q✉❛t✐♦♥ Vϕ,au = f ❤❛s ❢♦r f ∈ Cm+1 ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ u ∈ Cm✳ ■t ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ t❤❡ ❡q✉✐✈❛❧❡♥t ❡q✉❛t✐♦♥ u = V −1

ϕ Vϕ,bu + V −1 ϕ f1, f1(t) = f(t)

a(t, t), b(t, s) = 1 − a(t, s) a(t, t), ✭✻✮ ✐♥tr♦❞✉❝❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳ ❙✐♥❝❡ Vϕ ❛♥❞ V −1

♠❛♣ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ N ✐♥t♦ ♣♦❧②♥♦♠✐❛❧s ♦❢ t❤❡ s❛♠❡ ❞❡❣r❡❡✱ ♣♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s ❛r❡ r❛t❤❡r ♥❛t✉r❛❧ ❢♦r s♦❧✈✐♥❣ ❡q✉❛t✐♦♥ ✭✻✮✳ ❚❤❡ ♦♣❡r❛t♦r ❢♦r♠ ♦❢ t❤❡ ❈❤❡❜②s❤❡✈ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ t❛❦❡s t❤❡ ❢♦r♠ uN = V −1

ϕ ΠNVϕ,buN + V −1 ϕ ΠNf1,

✶✺

SLIDE 17

✇❤❡r❡ t❤❡ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❥❡❝t♦r ΠN ✐s ❞❡✜♥❡❞ ❜② ❝♦♥❞✐t✐♦♥s ΠNv ∈ PN, (ΠNv)(ti) = v(ti), ti = T 2

1 + cos

2i + 1 2(N + 1)

, i = 0, 1, ..., N.

❙❧✐❣❤t❧② str❡♥❣t❤❡♥✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥s ✭❝❢✳ ❬✷❪✮✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ s♣❡❡❞ ✐s uN − u C≤ cT mN −m(1 + log N) u(m) C, ✇❤❡r❡ t❤❡ ❝♦♥st❛♥t c ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ N, T ❛♥❞ f✳ ❆ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ♠❡t❤♦❞ ❬✷❪ ❝♦♥✈❡r❣❡s ✇✐t❤ t❤❡ s♣❡❡❞ uN − u C≤ cT mN −m(1 + log N)2 u Cm . ❚❤❡ ✏❝♦♠♣❛❝t✐✜❝❛t✐♦♥ tr✐❝❦✑ ❢♦r Vϕ,b ∈ L(Cm, Cm+1) ✭❞❡s❝r✐❜❡❞ ✐♥ ❙❡❝t✐♦♥ ✼✮ ♣r✐♦r t♦ t❤❡ s♦❧✈✐♥❣ ❡q✉❛t✐♦♥ Vϕ,au = f s❡❡♠s t♦ ❜❡ ♥❡❝❡ss❛r② ✐♥ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♠❡t❤♦❞s✳ ❚❤❡ s❛♠❡ ❝♦♥❝❡r♥s t❤❡ ❢♦❧❧♦✇✐♥❣ ❙❡❝t✐♦♥ ✶✵✳ ✶✻

SLIDE 18

✶✵✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r t❤❡ ❡q✉❛t✐♦♥ Vϕ,au = f

❆ss✉♠❡ ✭✷✮✱✭✸✮ ❢♦r r = 0✳ ❯s✐♥❣ t❤❡ ✐❞❡❛ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❡♥❞ ♦❢ ❙❡❝t✐♦♥ ✹✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❡q✉❛t✐♦♥ Vϕ,au = f ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ s❡❝♦♥❞ ❦✐♥❞ ❝♦r❞✐❛❧ ❡q✉❛t✐♦♥ ϕ(1)u = Vψαu + D⋆Vϕ,bu − αVϕ,bu + D⋆f1 − αf1 ✭✼✮ ✇❤✐❝❤✱ ❞✐✛❡r❡♥t❧② ❢r♦♠ ✭✻✮✱ ❞♦❡s ♥♦t ✉s❡ t❤❡ ✐♥✈❡rs✐♦♥ ♦❢ Vϕ. ■♥ ✭✼✮✱ (D⋆Vϕ,bu)(t) = ˆ t 1 tϕ s t

t∂b(t, s)

∂t u(s)ds − ˆ t 1 tϕ′ s t s t b(t, s)u(s)ds ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s ❛♥❞ t❤❡✐r ❞✐s❝r❡t❡ ✈❡rs✐♦♥s ♦❢ ❛❝❝✉r❛❝② O(N −m) ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ✭✼✮✱ ♣r♦✈✐❞❡❞ t❤❛t ❛ s♣❡❝✐❛❧ ❛♣♣❧✐❝❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ❬✸❪ ✐s ❢✉❧✜❧❧❡❞ ❜② t❤❡ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞ ❢♦r t❤❡ s✐♠♣❧❡ ♥♦♥❝♦♠♣❛❝t ♠❛✐♥ ♣❛rt ϕ(1)u = Vψαu + g ♦❢ t❤❡ ❡q✉❛t✐♦♥✳ ❋♦r ♠♦r❡ ❞❡t❛✐❧s ❛♥❞ t❤❡ ♠❛tr✐① ❢♦r♠ ♦❢ t❤❡ ♠❡t❤♦❞s s❡❡ ❬✸✱✹❪✳ ✶✼

SLIDE 19

■♥ ❝❛s❡ ♦❢ ❝♦♥❞✐t✐♦♥s ✭✹✮✱✭✺✮✱ r = 0✱ t❤❡ s✐t✉❛t✐♦♥ ✐s s♦♠❡✇❤❛t ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✳ ❲❡ ❝❛♥ ♣r❡s❡♥t ❢♦r ❡q✉❛t✐♦♥ Vϕ,au = f ❛ s✉✐t❛❜❧❡ ❛♣♣r♦①✐♠❛t❡ ❡q✉❛t✐♦♥ ♦❢ t②♣❡ ✭✼✮ r❡♣❧❛❝✐♥❣ ϕ ❜② t❤❡ ❝♦r❡ ϕε(x) =

ϕ(x),

0 < x ≤ 1 − ε ϕ(1 − ε)(1 − ε)αx−α, 1 − ε < x ≤ 1 , 0 < ε ≤ ε0, ✇❤✐❝❤ s❛t✐s✜❡s ✭✷✮ ❛♥❞ ✭✺✮✳ ▼♦r❡♦✈❡r✱ ´ 1

0 |ϕε(x) − ϕ(x)| dx → 0 ❛s ε → 0✱ t❤❛t

❡♥❛❜❧❡s t♦ ❤♦❧❞ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ✉♥❞❡r ❝♦♥tr♦❧ ✇❤❡♥ t❤❡ s♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s ♦r t❤❡✐r ❞✐s❝r❡t❡ ✈❡rs✐♦♥s ❛r❡ ❛♣♣❧✐❡❞ t♦ ❡q✉❛t✐♦♥ ✭✼✮ ✇✐t❤ ϕε ✐♥ t❤❡ r♦❧❡ ♦❢ ϕ✳ ❚❍❆◆❑ ❨❖❯ ❱❊❘❨ ▼❯❈❍ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆✦ ✶✽