❍♦✇ t♦ s♦❧✈❡ ❛♥② ♦r❞✐♥❛r② s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❆♣♣✳❊✿ Pr♦❣r❛♠♠✐♥❣ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s Logistic growth: alpha=0.2, R=1, dt=0.1 1.0 0.9 0.8 ❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥ ✶ , ✷ 0.7 u ′ ( t ) = α u ( t )( ✶ − R − ✶ u ( t )) 0.6 u 0.5 ❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r② ✶ u ( ✵ ) = U ✵ 0.4 ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s ✷ 0.3 0.2 0.1 0 5 10 15 20 25 30 35 40 45 t ❆✉❣ ✷✶✱ ✷✵✶✻ ❲❡ s❤❛❧❧ ✇r✐t❡ ❛♥ ❖❉❊ ✐♥ ❛ ❣❡♥❡r✐❝ ❢♦r♠✿ u ′ = f ( u , t ) ❊①❛♠♣❧❡s ♦♥ s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❖❉❊s✮ ❖✉r ♠❡t❤♦❞s ❛♥❞ s♦❢t✇❛r❡ s❤♦✉❧❞ ❜❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ❛♥② ❖❉❊ ❚❡r♠✐♥♦❧♦❣②✿ ❚❤❡r❡❢♦r❡ ✇❡ ♥❡❡❞ ❛♥ ❛❜str❛❝t ♥♦t❛t✐♦♥ ❢♦r ❛♥ ❛r❜✐tr❛r② ❖❉❊ ❙❝❛❧❛r ❖❉❊ ✿ ❛ s✐♥❣❧❡ ❖❉❊✱ ♦♥❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ u ′ ( t ) = f ( u ( t ) , t ) ❱❡❝t♦r ❖❉❊ ♦r s②st❡♠s ♦❢ ❖❉❊s ✿ s❡✈❡r❛❧ ❖❉❊s✱ s❡✈❡r❛❧ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ❚❤❡ t❤r❡❡ ❖❉❊s ♦♥ t❤❡ ❧❛st s❧✐❞❡ ❝♦rr❡s♣♦♥❞ t♦ ❊①❛♠♣❧❡s✿ f ( u , t ) = α u , ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ � � ✶ − u u ′ = α u ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ f ( u , t ) = α u , ❧♦❣✐st✐❝ ❣r♦✇t❤ R � � ✶ − u u ′ = α u ❧♦❣✐st✐❝ ❣r♦✇t❤ f ( u , t ) = − b | u | u + g , ❜♦❞② ✐♥ ✢✉✐❞ R u ′ + b | u | u = g ❢❛❧❧✐♥❣ ❜♦❞② ✐♥ ✢✉✐❞ ❖✉r t❛s❦✿ ✇r✐t❡ ❢✉♥❝t✐♦♥s ❛♥❞ ❝❧❛ss❡s t❤❛t t❛❦❡ f ❛s ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡ u ❛s ♦✉t♣✉t ❲❤❛t ✐s t❤❡ f ( u , t ) ❄ ❙✉❝❤ ❛❜str❛❝t f ❢✉♥❝t✐♦♥s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ ♠❛t❤❡♠❛t✐❝s Pr♦❜❧❡♠✿ ❲❡ ❝❛♥ ♠❛❦❡ ❣❡♥❡r✐❝ s♦❢t✇❛r❡ ❢♦r✿ ●✐✈❡♥ ❛♥ ❖❉❊✱ ◆✉♠❡r✐❝❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ f ′ ( x ) √ uu ′ − α ( t ) u ✸ / ✷ ( ✶ − u � b ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥✿ a f ( x ) dx R ( t )) = ✵ , ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✿ f ( x ) = ✵ ✇❤❛t ✐s t❤❡ f ( u , t ) ❄ ❆♣♣❧✐❝❛t✐♦♥s✿ ❙♦❧✉t✐♦♥✿ dx x a s✐♥ ( wx ) ✿ f ( x ) = x a s✐♥ ( wx ) ❚❤❡ t❛r❣❡t ❢♦r♠ ✐s u ′ = f ( u , t ) ✱ s♦ ✇❡ ♥❡❡❞ t♦ ✐s♦❧❛t❡ u ′ ♦♥ t❤❡ d ✶ ✷ � ✶ − ✶ ( x ✷ t❛♥❤ − ✶ x − ( ✶ + x ✷ ) − ✶ ) dx ✿ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ f ( x ) = x ✷ t❛♥❤ − ✶ x − ( ✶ + x ✷ ) − ✶ ✱ a = − ✶✱ b = ✶ u u ′ = α ( t ) u ( ✶ − R ( t )) ✸ ❙♦❧✈❡ x ✹ s✐♥ x = t❛♥ x ✿ f ( x ) = x ✹ s✐♥ x − t❛♥ x � �� � f ( u , t )
❲❡ ✉s❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ ❞❡r✐✈❛t✐✈❡s t♦ ❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ✐❞❡❛ t✉r♥ ❛♥ ❖❉❊ ✐♥t♦ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ u ′ = f ( u , t ) ❆ss✉♠❡ ✇❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ u ❛t ❞✐s❝r❡t❡ t✐♠❡ ♣♦✐♥ts t ✵ , t ✶ , . . . , t k ✳ ❆t t k ✇❡ ❤❛✈❡ t❤❡ ❖❉❊ u ′ ( t k ) = f ( u ( t k ) , t k ) ❆♣♣r♦①✐♠❛t❡ u ′ ( t k ) ❜② ❛ ❢♦r✇❛r❞ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡✱ u ′ ( t k ) ≈ u ( t k + ✶ ) − u ( t k ) ∆ t ■♥s❡rt ✐♥ t❤❡ ❖❉❊ ❛t t = t k ✿ u ( t k + ✶ ) − u ( t k ) = f ( u ( t k ) , t k ) ∆ t ❚❡r♠s ✇✐t❤ u ( t k ) ❛r❡ ❦♥♦✇♥✱ ❛♥❞ t❤✐s ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✭❞✐✛❡r❡♥❝❡✮ ❡q✉❛t✐♦♥ ❢♦r u ( t k + ✶ ) ❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ✐❞❡❛ ❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ♠❛t❤❡♠❛t✐❝s ❙♦❧✈✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ u ( t k + ✶ ) u ( t k + ✶ ) = u ( t k ) + ∆ tf ( u ( t k ) , t k ) ❚❤✐s ✐s ❛ ✈❡r② s✐♠♣❧❡ ❢♦r♠✉❧❛ t❤❛t ✇❡ ❝❛♥ ✉s❡ r❡♣❡❛t❡❞❧② ❢♦r u ( t ✶ ) ✱ u ( t ✷ ) ✱ u ( t ✸ ) ❛♥❞ s♦ ❢♦rt❤✳ ❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♥♦t❛t✐♦♥✿ ▲❡t u k ❞❡♥♦t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ u ( t ) ❛t t = t k ✳ u k + ✶ = u k + ∆ tf ( u k , t k ) ❚❤✐s ✐s ❛♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✇❡ ❝❛♥ s♦❧✈❡✦ ■❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ▲❡t✬s ❛♣♣❧② t❤❡ ♠❡t❤♦❞✦ Pr♦❜❧❡♠✿ ❚❤❡ ✇♦r❧❞✬s s✐♠♣❧❡st ❖❉❊ u ′ = u , t ∈ ( ✵ , T ] u ( t ) ❙♦❧✈❡ ❢♦r u ❛t t = t k = k ∆ t ✱ k = ✵ , ✶ , ✷ , . . . , t n ✱ t ✵ = ✵✱ t n = T ❋♦r✇❛r❞ ❊✉❧❡r ♠❡t❤♦❞✿ u k + ✶ = u k + ∆ t f ( u k , t k ) ❙♦❧✉t✐♦♥ ❜② ❤❛♥❞✿ ❲❤❛t ✐s f ❄ f ( u , t ) = u forward u k + ✶ = u k + ∆ tf ( u k , t k ) = u k + ∆ tu k = ( ✶ + ∆ t ) u k t n − 1 t n t n +1 ❋✐rst st❡♣✿ u ✶ = ( ✶ + ∆ t ) u ✵ ❜✉t ✇❤❛t ✐s u ✵ ❄
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