tr ss trrt - PowerPoint PPT Presentation

❙♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ ✐♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r✐s✐♥❣ ✐♥ ✈✐s❝♦❡❧❛st✐❝✐t② ◆✳ ❆✳ ❘❛✉t✐❛♥✱ ❱✳ ❱✳ ❱❧❛s♦✈ ✭▲♦♠♦♥♦s♦✈ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t②✮ ❖❚■◆❉✲✷✵✶✻✱ ❉❡❝❡♠❜❡r ✶✼✕✷✵✱ ✷✵✶✻ ✶ ✴ ✹✷

■♥tr♦❞✉❝t✐♦♥ ❲❡ st✉❞② ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ❝♦❡✣❝✐❡♥ts ✐♥ ❍✐❧❜❡rt s♣❛❝❡✳ ❚❤❡ ♠❛✐♥ ♣❛rt ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s ✐s ❛♥ ❛❜str❛❝t ❤②♣❡r❜♦❧✐❝ ❡q✉❛t✐♦♥s✱ ❞✐st✉r❜❡❞ ❜② t❤❡ t❡r♠s ❝♦♥t❛✐♥✐♥❣ ❛❜str❛❝t ✐♥t❡❣r❛❧ ❱♦❧t❡rr❛ ♦♣❡r❛t♦rs✳ ❚❤❡ ❡q✉❛t✐♦♥s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡ ❛r❡ t❤❡ ❛❜str❛❝t ❢♦r♠ ♦❢ t❤❡ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ ●✉rt✐♥✲P✐♣❦✐♥ ✭s❡❡ ❜✐❜❧✐♦❣r❛♣❤② ❝✐t❡❞ ❜❡❧♦✇ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮ ❞❡s❝r✐❜✐♥❣ t❤❡ ♣r♦❝❡ss ♦❢ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥ ✐♥ ♠❡❞✐❛ ✇✐t❤ ♠❡♠♦r②✱ ♣r♦❝❡ss ♦❢ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ t❤❡ ✈✐s❝♦✲❡❧❛st✐❝ ♠❡❞✐❛✱ ❛♥❞ ❛❧s♦ ❛r✐s✐♥❣ ✐♥ t❤❡ ♣r♦❜❧❡♠s ♦❢ ♣♦r♦✉s ♠❡❞✐❛ ✭❉❛r❝✐ ❧❛✇✮✳ ❲❡ ♦❜t❛✐♥ ❝♦rr❡❝t s♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r t❤❡ ❞❡s❝r✐❜❡❞ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s ♦♥ t❤❡ ♣♦s✐t✐✈❡ s❡♠✐❛①✐s✳ ❲❡ ❛♥❛❧②s❡ s♣❡❝tr❛❧ ♣r♦❜❧❡♠s ❢♦r t❤❡ ♦♣❡r❛t♦r✕✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ t❤❡ s②♠❜♦❧s ♦❢ t❤❡s❡ ❡q✉❛t✐♦♥s✳ ▼♦r❡♦✈❡r ✇❡ st✉❞② t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ❛❜str❛❝t ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ ●✉rt✐♥✲P✐♣❦✐♥ t②♣❡✳ ✷ ✴ ✹✷

▲❡t ✉s H ❜❡ ❛ s❡♣❛r❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ A ❜❡ ❛ s❡❧❢✲❛❞❥♦✐♥t ♣♦s✐t✐✈❡ ♦♣❡r❛t♦r A ∗ = A � κ 0 ✭ κ 0 > 0 ✮ ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H ❛♥❞ ❤❛✈✐♥❣ ❛ ❝♦♠♣❛❝t ✐♥✈❡rs❡ ♦♣❡r❛t♦r✳ ▲❡t ✉s B ❜❡ ❛ s②♠♠❡tr✐❝ ♦♣❡r❛t♦r ( Bx, y ) = ( x, By ) ✱ ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H ❤❛✈✐♥❣ t❤❡ ❞♦♠❛✐♥ Dom ( B ) ✭ Dom ( A ) ⊆ Dom ( B ) ✮✳ ▼♦r❡♦✈❡r B ❜❡ ❛ ♥♦♥♥❡❣❛t✐✈❡ ♦♣❡r❛t♦r t❤❛t ✐s ( Bx, x ) � 0 ❢♦r ❛♥② x, y ∈ Dom ( B ) ❛♥❞ s❛t✐s❢②✐♥❣ t♦ ✐♥❡q✉❛❧✐t② � Bx � � κ � Ax � ✱ 0 < κ < 1 ❢♦r ❛♥② x ∈ Dom ( A ) ❛♥❞ I ❜❡ t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ❛❝t✐♥❣ ✐♥ t❤❡ s♣❛❝❡ H ✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ❢♦r ❛ s❡❝♦♥❞✲♦r❞❡r ✐♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦♥ t❤❡ s❡♠✐❛①✐s R + = (0 , ∞ ) ✿ � t � t d 2 u ( t ) + Au ( t )+ Bu ( t ) − K ( t − s ) Au ( s ) ds − Q ( t − s ) Bu ( s ) ds = dt 2 0 0 = f ( t ) , t ∈ R + , ✭✶✮ u (1) (+0) = ϕ 1 . u (+0) = ϕ 0 , ✭✷✮ ✸ ✴ ✹✷

❆ss✉♠❡ t❤❛t t❤❡ s❝❛❧❛r ❢✉♥❝t✐♦♥s K ( t ) ❛♥❞ Q ( t ) t❤❛t ❛r❡ t❤❡ ❦❡r♥❡❧s ♦❢ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ❛❞♠✐ts t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥s✿ ∞ ∞ � � a k e − γ k t , b k e − γ k t , K ( t ) = Q ( t ) = ✭✸✮ k =1 k =1 ✇❤❡r❡ a k > 0 ✱ b k � 0 ✱ γ k +1 > γ k > 0 ✱ k ∈ N ✱ γ k → + ∞ ( k → + ∞ ) ✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ tr✉❡✿ ∞ ∞ a k b k � � < 1 , < 1 . ✭✹✮ γ k γ k k =1 k =1 ❚❤❡ ❝♦♥❞✐t✐♦♥s ✭✹✮ ♠❡❛♥s t❤❛t K ( t ) , Q ( t ) ∈ L 1 ( R + ) ✱ � K � L 1 < 1 ✱ � Q � L 1 < 1 ✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭✹✮ ❛r❡ s✉♣♣❧❡♠❡♥t❡❞ ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s ∞ ∞ � � K (0) = a k < + ∞ , Q (0) = b k < + ∞ . ✭✺✮ k =1 k =1 t❤❡♥ t❤❡ ❦❡r♥❡❧s K ( t ) ❛♥❞ Q ( t ) ❜❡❧♦♥❣ t♦ t❤❡ s♣❛❝❡ W 1 1 ( R + ) ✳ ✹ ✴ ✹✷

❚❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛♥ ❛❜str❛❝t ❢♦r♠ ♦❢ ❞✐♥❛♠✐❝❛❧ ✈✐s❝♦❡❧❛st✐❝ ✐♥t❡❣r♦❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✇❤❡r❡ ♦♣❡r❛t♦♣rs A ❛♥❞ B ❛r❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t✐❛❧ ❡①♣r❡ss✐♦♥s � ∆ u + 1 � B = − 1 A = − ρ − 1 µ 3 ρ − 1 λ · ❣r❛❞ ( ❞✐✈ u ) , 3 ❣r❛❞ ( ❞✐✈ u ) , u ( x, t ) ∈ R 3 ✐s ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r ♦❢ ✈✐s❝♦❡❧❛st✐❝ ❤❡r❡❞✐t❛r② ❤❡r❡ u = � ✐s♦tr♦♣✐❝ ♠❡❞✐❛ t❤❛t ✜❧❧ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ⊂ R 3 ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r②✱ ∂ Ω ✱ ρ ✐s ❛ ❝♦♥st❛♥t ❞❡♥s✐t②✱ ρ > 0 ✱ ▲❛♠❡ ♣❛r❛♠❡t❡rs λ, µ ❛r❡ t❤❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥ts✱ K ( t ) ✱ Q ( t ) ❛r❡ t❤❡ r❡❧❛①❛t✐♦♥ ❢✉♥❝t✐♦♥s ❝❤❛r❛❝t❡r✐③✐♥❣ ❤❡r❡❞✐t❛r② ♣r♦♣❡rt✐❡s ♦❢ ♠❡❞✐❛✳ ❖♥ t❤❡ ❞♦♠❛✐♥ ❜♦✉♥❞❛r② ∂ Ω t❤❡ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥ u | ∂ Ω = 0 . ✭✻✮ ✐s s❛t✐s✜❡❞✳ ❚❤❡ ❍✐❧❜❡rt s♣❛❝❡ H ❝❛♥ ❜❡ r❡❛❧✐③❡❞ ❛s t❤❡ s♣❛❝❡ ♦❢ t❤r❡❡ ❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r✲❢✉♥❝t✐♦♥s L 2 (Ω) ✳ ❚❤❡ ❞♦♠❛✐♥ Dom ( A ) ❜❡❧♦♥❣s t♦ t❤❡ ❙♦❜♦❧❡✈ s♣❛❝❡ W 2 2 (Ω) ♦❢ ✈❡❝t♦r ❢✉♥❝t✐♦♥s s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✻✮✳ ✺ ✴ ✹✷

■♥ ❝❛s❡ ♦♣❡r❛t♦r B = 0 ✱ ♣♦s✐t✐✈❡ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r A ❝❛♥ ❜❡ r❡❛❧✐③❡❞ ❛s ♦♣❡r❛t♦r Ay = − y ′′ ( x ) ✱ ✇❤❡r❡ x ∈ (0 , π ) ✱ y (0) = y ( π ) = 0 ✱ ♦r t❤❡ ♦♣❡r❛t♦r Ay = − ∆ y ✇✐t❤ ❉✐r✐❝❤❧❡t ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Q ⊂ R n ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r② ✭ H = L 2 ( Q ) ✮ ♦r ♠♦r❡ ❣❡♥❡r❛❧ ❡❧❧✐♣t✐❝ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦rs ✐♥ t❤❡ s♣❛❝❡ L 2 ( Q ) ✳ ❚❤❡ ❡q✉❛t✐♦♥ ✭✶✮ ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛♥ ❛❜str❛❝t ❢♦r♠ ♦❢ t❤❡ ●✉rt✐♥✲P✐♣❦✐♥ ❡q✉❛t✐♦♥ t❤❛t ❞❡s❝r✐❜❡s ❤❡❛t tr❛♥s❢❡r ✐♥ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r② ✇✐t❤ ✜♥✐t❡ s♣❡❡❞✳ ✶✮ ●✉rt✐♥ ▼✳ ❊✳✱ P✐♣❦✐♥ ❆✳ ❈✳ ●❡♥❡r❛❧ t❤❡♦r② ♦❢ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ✇✐t❤ ✜♥✐t❡ ✇❛✈❡ s♣❡❡❞ ✴✴ ❆r❝❤✳ ❘❛t✳ ▼❡❝❤✳ ❆♥❛❧✳✱ ✶✾✻✽✱ ❱✳ ✸✶✱ P✳ ✶✶✸✕✶✷✻✳ ✷✮ Pr✉ss ❏✳ ❊✈♦❧✉t✐♦♥❛r② ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✴✴ ▼♦♥♦❣r❛♣❤s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✶✾✾✸✱ ❱✳✽✼✱ ❇✐r❦❤❛✉s❡r ❱❡r❧❛❣✳ ❇❛s❡❧✲❇❛st♦♥✲❇❡r❧✐♥✳ ✸✮ ❆♠❡♥❞♦❧❛ ●✳✱ ❋❛❜r✐③✐♦ ▼✳✱ ●♦❧❞❡♥✱ ❏✳ ▼✳ ❚❤❡r♠♦❞②♥❛♠✐❝s ♦❢ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r②✿ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✱ ✷✵✶✷✳ ✻ ✴ ✹✷