❙❡♠✐❣r♦✉♣s ❢♦r ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦♥✈♦❧✉t✐♦♥ ♠❡♠♦r② t❡r♠s ◆✳ ❆✳ ❘❛✉t✐❛♥ ❛♥❞ ❱✳ ❱✳ ❱❧❛s♦✈ ▲♦♠♦♥♦s♦✈ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❖❚❑❘✲✷✵✶✾✱ ❉❡❝❡♠❜❡r ✶✾✲✷✷✱ ✷✵✶✾ ✶ ✴ ✷✺
■♥tr♦❞✉❝t✐♦♥ ▼❛♥② ✐♥t❡r❡st✐♥❣ ♣❤②s✐❝❛❧ ♣❤❡♥♦♠❡♥❛ ✭s✉❝❤ ❛s ✈✐s❝♦❡❧❛st✐❝✐t②✱ ♣♦♣✉❧❛t✐♦♥ ❞②♥❛♠✐❝s ♦r ❤❡❛t ✢♦✇ ✐♥ r❡❛❧ ❝♦♥❞✉❝t♦rs✱ t♦ ♥❛♠❡ s♦♠❡✮ ❛r❡ ♠♦❞❡❧❧❡❞ ❜② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇❤✐❝❤ ❛r❡ ✐♥✢✉❡♥❝❡❞ ❜② t❤❡ ♣❛st ✈❛❧✉❡s ♦❢ ♦♥❡ ♦r ♠♦r❡ ✈❛r✐❛❜❧❡s✿ t❤❛t ❛r❡ s♦✲❝❛❧❧❡❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♠❡♠♦r②✳ ❚❤❡ ♠❛✐♥ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❛♥❛❧②s✐s ♦❢ ❡q✉❛t✐♦♥s ♦❢ t❤✐s ❦✐♥❞ ❧✐❡s ✐♥ t❤❡✐r ♥♦♥❧♦❝❛❧ ❝❤❛r❛❝t❡r✱ ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ♠❡♠♦r② t❡r♠s ✭✐♥ ❣❡♥❡r❛❧✱ t❤❡ t✐♠❡ ❝♦♥✈♦❧✉t✐♦♥ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ❛❣❛✐♥st ❛ s✉✐t❛❜❧❡ ♠❡♠♦r② ❦❡r♥❡❧s✮✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❝♦rr❡❝t ♠♦❞❡❧❧✐♥❣ ♦❢ ♠❛t❡r✐❛❧s ✇✐t❤ ♠❡♠♦r② ❤❛s ❛❧✇❛②s r❡♣r❡s❡♥t❡❞ ❛ ♠❛❥♦r ❝❤❛❧❧❡♥❣❡ t♦ ♠❛t❤❡♠❛t✐❝✐❛♥s✳ ❚❤❡ ♦r✐❣✐♥s ♦❢ ♠♦❞❡r♥ ✈✐s❝♦❡❧❛st✐❝✐t② ❛♥❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ♦❢ t❤❡ s♦✲❝❛❧❧❡❞ ❤❡r❡❞✐t❛r② s②st❡♠s tr❛❞✐t✐♦♥❛❧❧② tr❛❝❡ ❜❛❝❦ t♦ t❤❡ ✇♦r❦s ♦❢ ▲✉❞✇✐❣ ❇♦❧t③♠❛♥♥ ❛♥❞ ❱✐t♦ ❱♦❧t❡rr❛✱ ✇❤♦ ✜rst ✐♥tr♦❞✉❝❡❞ t❤❡ ♥♦t✐♦♥ ♦❢ ♠❡♠♦r② ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ❛♥❛❧②s✐s ♦❢ ❡❧❛st✐❝ ♠❛t❡r✐❛❧s✳ ❚❤❡ ❦❡② ❛ss✉♠♣t✐♦♥ ✐♥ t❤❡ ❤❡r❡❞✐t❛r② t❤❡♦r② ♦❢ ❡❧❛st✐❝✐t② ❝❛♥ ❜❡ st❛t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✳ ❋♦r ❛♥ ✈✐s❝♦❡❧❛st✐❝ ❜♦❞② ♦❝❝✉♣②✐♥❣ ❛ ❝❡rt❛✐♥ r❡❣✐♦♥ B ⊂ R N ❛t r❡st✱ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠ ❛t ❛♥② ♣♦✐♥t x ∈ B ✐s ❛ ❢✉♥❝t✐♦♥ ❜♦t❤ ♦❢ t❤❡ ✐♥st❛♥t❛♥❡♦✉s str❡ss ❛♥❞ ♦❢ ❛❧❧ t❤❡ ♣❛st str❡ss❡s ❛t x ✳ ✷ ✴ ✷✺
❋♦r ❡①❛♠♣❧❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ●✉rt✐♥✲P✐♣❦✐♥ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ � t u tt ( x, t ) = u xx ( x, t ) − K ( t − τ ) u xx ( x, τ ) dτ + f ( x, t ) 0 ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡ ♣r♦❝❡ss ♦❢ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥ ✐♥ ♠❡❞✐❛ ✇✐t❤ ♠❡♠♦r②✱ ♣r♦❝❡ss ♦❢ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥ ✐♥ t❤❡ ✈✐s❝♦❡❧❛st✐❝ ♠❡❞✐❛ ❛♥❞ ❛❧s♦ ❛r✐s✐♥❣ ✐♥ t❤❡ ♣r♦❜❧❡♠s ♦❢ ♣♦r♦✉s ♠❡❞✐❛ ✭❉❛r❝② ❧❛✇✮✳ ❙❡❡ ●✉rt✐♥ ▼✳ ❊✳✱ P✐♣❦✐♥ ❆✳ ❈✳ ●❡♥❡r❛❧ t❤❡♦r② ♦❢ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ✇✐t❤ ✜♥✐t❡ ✇❛✈❡ s♣❡❡❞✳ ✴✴ ❆r❝❤✐✈❡ ❢♦r ❘❛t✐♦♥❛❧ ▼❡❝❤❛♥✐❝s ❛♥❞ ❆♥❛❧②s✐s✳ ✖ ✶✾✻✽✳ ✖ ✸✶ ✳ ✖ P✳ ✶✶✸✕✶✷✻✳ ✸ ✴ ✷✺
❲❡ ❝♦♥s✐❞❡r ❛❧s♦ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❞②♥❛♠✐❝❛❧ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ♦❢ ✈✐s❝♦❡❧❛st✐❝ ✐s♦tr♦♣✐❝ ♠❡❞✐❛ u tt ( x, t ) = ρ − 1 [ µ ∆ u ( x, t ) + ( µ + λ ) / 3 · ❣r❛❞ ( ❞✐✈ u ( x, t ))] − � t K ( t − τ ) ρ − 1 µ [∆ u ( x, τ ) + 1 / 3 · ❣r❛❞ ( ❞✐✈ u ( x, τ ))] dτ − − 0 � t Q ( t − τ ) ρ − 1 λ [1 / 3 · ❣r❛❞ ( ❞✐✈ u ( x, τ ))] dτ + f ( x, t ) − ✭✶✮ 0 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✜❡❞✳ ❙❡❡ ❆✳❆✳ ■❧②✉s❤✐♥✱ ❇✳❊✳ P♦❜❡❞r②❛ ❋♦✉♥❞❛t✐♦♥s ♦❢ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r② ♦❢ t❤❡r♠♦✈✐s❝♦❡❧❛st✐❝✐t②✳ ✲ ▼✳ ◆❛✉❦❛✱ ✶✾✼✵✱ ❘✳▼✳ ❈❤r✐st❡♥s❡♥ ❚❤❡♦r② ♦❢ ✈✐s❝♦❡❧❛st✐❝✐t②✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥✳ ✲ ❆❝❛❞❡♠✐❝ Pr❡ss ◆❡✇ ❨♦r❦ ❛♥❞ ▲♦♥❞♦♥✱ ✶✾✼✶ ✹ ✴ ✷✺
❊①❛♠♣❧❡s ♦❢ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ t❤❡ ♣r♦❜❧❡♠s ♦❢ ❤❡❛t ♣r♦♣❛❣❛t✐♦♥ ✐♥ ♠❡❞✐❛ ✇✐t❤ ♠❡♠♦r② ❛♥❞ ✈✐s❝♦❡❧❛st✐❝✐t②✿ m m � � K ( t ) = a k R k ( t ) , Q ( t ) = b k R k ( t ) ✭✷✮ k =1 k =1 ■✮ R k ( t ) = e − β k t ✱ ✇❤❡r❡ β k +1 > β k > 0 ✱ l ∈ N ✱ β k → + ∞ ( j → + ∞ ) ✳ ■■✮ R k ( t ) = t α − 1 e − β k t ✱ Γ( α ) ∞ ( − β k ) n t nα � ■■■✮ R k ( t ) = t α − 1 Γ[( n + 1) α ] ✭❘❛❜♦t♥♦✈ ❢✉♥❝t✐♦♥s✮ n =0 ✭■■✮✱ ✭■■■✮✕ ❢r❛❝t✐♦♥❛❧✲❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s✱ ✇❤❡r❡ 0 < α < 1 , Γ( · ) ✕ ❣❛♠♠❛✲ ❢✉♥❝t✐♦♥✱ 0 < β k < β k +1 , ✱ k ∈ N ✱ β k → + ∞ ✱ k → + ∞ ✳ ✺ ✴ ✷✺
� ∞ e − tτ dµ k ( τ ) , R k ( t ) = 0 ✇❤❡r❡ dµ j ✐s ❛ ♣♦s✐t✐✈❡ ♠❡❛s✉r❡✳ ❲❡ ✐❞❡♥t✐❢② t❤✐s ♠❡❛s✉r❡ ✇✐t❤ ✐ts ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s µ j ✱ s♦ µ j ✐s ✐♥❝r❡❛s✐♥❣✱ ❝♦♥t✐♥✉♦✉s ❢r♦♠ t❤❡ r✐❣❤t✱ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❙t✐❡❧t❥❡s ✐♥t❡❣r❛❧✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ ❘❛❜♦t♥♦✈ ❢✉♥❝t✐♦♥ ✭■■■✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ γ + iR + ∞ � � e λt dλ e − tτ dτ 1 = sin πα R k ( t ) = lim k τ − α , λ α + β k τ α + 2 β k cos πα + β 2 2 πi π R → + ∞ γ − iR 0 ✇❤❡r❡ λ α (0 < α < 1) ✐s t❤❡ ♠❛✐♥ ❜r❛♥❝❤ ♦❢ ♠✉❧t✐✈❛❧✉❡ ❢✉♥❝t✐♦♥ f ( λ ) = λ α ✱ λ ∈ C ✇✐t❤ ❛ ❝✉t ♦♥ ♥❡❣❛t✐✈❡ r❡❛❧ s❡♠✐❛①✐s t❤❛t ✐s λ α = | λ | α e iα arg λ ✱ − π < arg λ < π. ✻ ✴ ✷✺
Recommend
More recommend