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❆❞❛♣t✐✈❡ ❚✐♠❡ ❉✐s❝r❡t✐③❛t✐♦♥ ❢♦r ❘❡t❛r❞❡❞ P♦t❡♥t✐❛❧s ❙t❡❢❛♥ ❙❛✉t❡r ✱ ❯♥✐✈❡rs✐tät ❩ür✐❝❤ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❡①❛♥❞❡r ❱❡✐t ✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦ ❘■❈❆▼✱ ▲✐♥③ ◆♦✈❡♠❜❡r ✶✵✱ ✷✵✶✻

❖✉t❧✐♥❡ ■♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥ ❉✐s❝r❡t✐③❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ✐♥ t✐♠❡ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❢♦r t❤❡ ❛❞❛♣t✐✈❡ ❛❧❣♦r✐t❤♠

■♥tr♦❞✉❝t✐♦♥ Ω u i u e Ω i n u e Γ u e ❋✐❣✉r❡✿ ❙❝❛tt❡r✐♥❣ ♣r♦❜❧❡♠ ✐♥ ✉♥❜♦✉♥❞❡❞ ❡①t❡r✐♦r ❞♦♠❛✐♥s✳

■♥tr♦❞✉❝t✐♦♥ ▲❡t Ω ⊂ R ✸ ❜❡ ❛ ▲✐♣s❝❤✐t③ ❞♦♠❛✐♥ ✇✐t❤ ❜♦✉♥❞❛r② Γ ✳ ■♥ s❝❛tt❡r✐♥❣ ♣r♦❜❧❡♠s Ω ✐s t②♣✐❝❛❧❧② ❛♥ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ✇❛✈❡ ❡q✉❛t✐♦♥ ∂ ✷ t u − ∆ u = ✵ ✐♥ Ω × [ ✵ , T ] ✇✐t❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s u ( · , ✵ ) = ∂ t u ( · , ✵ ) = ✵ ✐♥ Ω ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s u = g ♦♥ Γ × [ ✵ , T ] ♦♥ ❛ t✐♠❡ ✐♥t❡r✈❛❧ [ ✵ , T ] ❢♦r s♦♠❡ T > ✵✳

❚❤❡ ❡①t❡♥s✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳ ❚❤❡r❡❢♦r❡ t❤❡ ✉♥❦♥♦✇♥ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ✈✐❛ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦♥ ✵ ✹ ■♥t❡❣r❛❧ ❋♦r♠✉❧❛t✐♦♥ ❲❡ ❡♠♣❧♦② ❛♥ ❛♥s❛t③ ❛s ❛ s✐♥❣❧❡ ❧❛②❡r ♣♦t❡♥t✐❛❧ u ( x , t ) : = ( S φ )( x , t ) � t � δ ( t − τ − � x − y � ) = φ ( y , τ ) d Γ y d τ ✹ π � x − y � ✵ Γ φ ( y , t − � x − y � ) � = d Γ y , ( x , t ) ∈ Ω × [ ✵ , T ] . ✹ π � x − y � Γ

■♥t❡❣r❛❧ ❋♦r♠✉❧❛t✐♦♥ ❲❡ ❡♠♣❧♦② ❛♥ ❛♥s❛t③ ❛s ❛ s✐♥❣❧❡ ❧❛②❡r ♣♦t❡♥t✐❛❧ u ( x , t ) : = ( S φ )( x , t ) � t � δ ( t − τ − � x − y � ) = φ ( y , τ ) d Γ y d τ ✹ π � x − y � ✵ Γ φ ( y , t − � x − y � ) � = d Γ y , ( x , t ) ∈ Ω × [ ✵ , T ] . ✹ π � x − y � Γ ❚❤❡ ❡①t❡♥s✐♦♥ x → Γ ✐s ❝♦♥t✐♥✉♦✉s✳ ❚❤❡r❡❢♦r❡ t❤❡ ✉♥❦♥♦✇♥ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ φ ✐s ❞❡t❡r♠✐♥❡❞ ✈✐❛ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ � φ ( y , t − � x − y � ) d Γ y = g ( x , t ) ♦♥ Γ × [ ✵ , T ] . ✹ π � x − y � Γ

❲❡ ✉s❡ ❛ ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡ ✐♥ ♦r❞❡r t♦ ❞✐s❝r❡t✐③❡ t❤✐s ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✳ ❚❤✉s ✇❡ ♠❛❦❡ t❤❡ ❞✐s❝r❡t❡ ❛♥s❛t③ ●❛❧❡r❦✐♥ ✶ ✶ ❱❛r✐❛t✐♦♥❛❧ ❋♦r♠✉❧❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t✐③❛t✐♦♥ ❆ ❝♦❡r❝✐✈❡ s♣❛❝❡✲t✐♠❡ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ ❋✐♥❞ φ s✳t✳ � T ˙ � � φ ( y , t − � x − y � ) ζ ( x , t ) d Γ y d Γ x dt ✹ π � x − y � Γ Γ ✵ � T � = g ( x , t ) ζ ( x , t ) d Γ x dt ˙ ✵ Γ ❢♦r ❛❧❧ ζ ✳

❱❛r✐❛t✐♦♥❛❧ ❋♦r♠✉❧❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t✐③❛t✐♦♥ ❆ ❝♦❡r❝✐✈❡ s♣❛❝❡✲t✐♠❡ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ ❋✐♥❞ φ s✳t✳ � T ˙ � � φ ( y , t − � x − y � ) ζ ( x , t ) d Γ y d Γ x dt ✹ π � x − y � Γ Γ ✵ � T � = g ( x , t ) ζ ( x , t ) d Γ x dt ˙ ✵ Γ ❢♦r ❛❧❧ ζ ✳ ❲❡ ✉s❡ ❛ ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡ ✐♥ ♦r❞❡r t♦ ❞✐s❝r❡t✐③❡ t❤✐s ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✳ ❚❤✉s ✇❡ ♠❛❦❡ t❤❡ ❞✐s❝r❡t❡ ❛♥s❛t③ N M � � α j φ ●❛❧❡r❦✐♥ ( x , t ) = i ϕ j ( x ) b i ( t ) . i = ✶ j = ✶

❲❡ ❝♦♥str✉❝t s♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ✉s✐♥❣ t❤❡ P❛rt✐t✐♦♥ ♦❢ ❯♥✐t② ▼❡t❤♦❞ ✳ ■♥s❡rt✐♥❣ t❤✐s ❛♥s❛t③ ✐♥ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❧❡❛❞s t♦✿ ❋✐♥❞ α j i , i = ✶ , . . . , N , j = ✶ , . . . , M s✉❝❤ t❤❛t N M � � ϕ j ( y ) ϕ l ( x ) � � α j ✹ π � x − y � ψ k , i ( � x − y � ) d Γ y d Γ x i Γ Γ i = ✶ j = ✶ � T � = g ( x , t ) ϕ l ( x ) b k ( t ) d Γ x dt ˙ Γ ✵ ❢♦r k = ✶ , . . . , N ❛♥❞ l = ✶ , . . . , M ❛♥❞ � T ˙ ψ k , i ( r ) := b i ( t − r ) b k ( t ) dt ✵ ✇❤❡r❡ r ∈ R > ✵ ✳

■♥s❡rt✐♥❣ t❤✐s ❛♥s❛t③ ✐♥ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❧❡❛❞s t♦✿ ❋✐♥❞ α j i , i = ✶ , . . . , N , j = ✶ , . . . , M s✉❝❤ t❤❛t N M � � ϕ j ( y ) ϕ l ( x ) � � α j ✹ π � x − y � ψ k , i ( � x − y � ) d Γ y d Γ x i Γ Γ i = ✶ j = ✶ � T � = g ( x , t ) ϕ l ( x ) b k ( t ) d Γ x dt ˙ Γ ✵ ❢♦r k = ✶ , . . . , N ❛♥❞ l = ✶ , . . . , M ❛♥❞ � T ˙ ψ k , i ( r ) := b i ( t − r ) b k ( t ) dt ✵ ✇❤❡r❡ r ∈ R > ✵ ✳ ❲❡ ❝♦♥str✉❝t s♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ✉s✐♥❣ t❤❡ P❛rt✐t✐♦♥ ♦❢ ❯♥✐t② ▼❡t❤♦❞ ✳

❆ s♠♦♦t❤ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ✶ ✵ τ ✶ τ ✷ τ ✸ τ ✹ τ ✺ τ ✻ τ ✼ Θ ✶ Θ ✸ Θ ✺ Θ ✼ Θ ✷ Θ ✹ Θ ✻ Θ ✽ ❋✐❣✉r❡✿ P❛rt✐t✐♦♥ ♦❢ ✉♥✐t② { ϕ i } ❢♦r ✽ t✐♠❡st❡♣s✳

❇❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ❲❡ ❝♦♥s✐❞❡r ❛ ✭❧♦❝❛❧❧②✮ q✉❛s✐✲✉♥✐❢♦r♠ t✐♠❡ ♠❡s❤ ✇✐t❤ t✐♠❡st❡♣s t i s✳t✳ ✵ = t ✵ < t ✶ < t ✷ < . . . < t N − ✷ < t N − ✶ = T . ❲❡ ♦❜t❛✐♥ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ❜② ♠✉❧t✐♣❧②✐♥❣ t❤❡ ♣❛rt✐t♦♥ ♦❢ ✉♥✐t② ❢✉♥❝t✐♦♥s ✇✐t❤ s✉✐t❛❜❧② s❝❛❧❡❞ ▲❡❣❡♥❞r❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❛①✐♠❛❧ ❞❡❣r❡❡ p ✳ � ✷ � ϕ ✶ ( t ) t ✷ P m − ✷ t − ✶ , m = ✷ , . . . , ♠❛① ( ✷ , p ) , t ✶ � ✷ t − t i − ✷ � ϕ i ( t ) P m − ✶ , m = ✵ , . . . , p , i = ✷ , . . . , N − ✶ , t i − t i − ✷ � � t − t N − ✷ ϕ N ( t ) P m ✷ − ✶ , m = ✵ , . . . , p . t N − ✶ − t N − ✷

❇❛s✐s ♦❢ t❤❡ P❯▼ s♣❛❝❡ 1 m=0 m=1 m=2 m=3 0.5 0 −0.5 0 0.4 0.8 1.2 1.6 2 ❋✐❣✉r❡✿ ❇❛s✐s ❢✉♥❝t✐♦♥s ❢♦r t ✵ = ✵✱ t ✶ = ✵ . ✽ ❛♥❞ t ✷ = ✷✳

❉✐s❝r❡t❡ Pr♦❜❧❡♠ ❘❡❝❛❧❧ t❤❛t ✇❡ ✇❛♥t t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠✿ ❋✐♥❞ α j i , i = ✶ , . . . , N , j = ✶ , . . . , M s✉❝❤ t❤❛t N M � � A k , i j , l α j i = g k l , ∀ ✶ ≤ k ≤ N ∀ ✶ ≤ l ≤ M , i = ✶ j = ✶ ✇❤❡r❡ � � ϕ j ( y ) ϕ l ( x ) A k , i j , l := ✹ π � x − y � ψ k , i ( � x − y � ) d Γ y d Γ x Γ Γ ❛♥❞ � T � T � ˙ g k ψ k , i ( r ) = b i ( t − r ) b k ( t ) dt , l = g ( x , t ) ϕ l ( x ) b k ( t ) d Γ x dt . ˙ Γ ✵ ✵

❙♣❛rs✐t② P❛tt❡r♥ ♦❢ ❆ ❋✐❣✉r❡✿ ❙♣❛rs✐t② ♣❛tt❡r♥ ♦❢ ❆ ❢♦r Γ = [ ✵ , ✷ ] ✇✐t❤ M = ✽✵ ❛♥❞ T = ✸ ✇✐t❤ N = ✸✵✳