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❆♣♣✳❊✿ Pr♦❣r❛♠♠✐♥❣ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥✶,✷

❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r②✶ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s✷

❆✉❣ ✷✶✱ ✷✵✶✻

✶ ❍♦✇ t♦ s♦❧✈❡ ❛♥② ♦r❞✐♥❛r② s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✷ ❙②st❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭✈❡❝t♦r ❖❉❊✮

❍♦✇ t♦ s♦❧✈❡ ❛♥② ♦r❞✐♥❛r② s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥

u′(t) = αu(t)(✶ − R−✶u(t)) u(✵) = U✵

5 10 15 20 25 30 35 40 45 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 u

Logistic growth: alpha=0.2, R=1, dt=0.1

❊①❛♠♣❧❡s ♦♥ s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❖❉❊s✮

❚❡r♠✐♥♦❧♦❣②✿ ❙❝❛❧❛r ❖❉❊✿ ❛ s✐♥❣❧❡ ❖❉❊✱ ♦♥❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ❱❡❝t♦r ❖❉❊ ♦r s②st❡♠s ♦❢ ❖❉❊s✿ s❡✈❡r❛❧ ❖❉❊s✱ s❡✈❡r❛❧ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ❊①❛♠♣❧❡s✿ u′ = αu ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ u′ = αu

• ✶ − u

R

• ❧♦❣✐st✐❝ ❣r♦✇t❤

u′ + b|u|u = g ❢❛❧❧✐♥❣ ❜♦❞② ✐♥ ✢✉✐❞

❲❡ s❤❛❧❧ ✇r✐t❡ ❛♥ ❖❉❊ ✐♥ ❛ ❣❡♥❡r✐❝ ❢♦r♠✿ u′ = f (u, t)

❖✉r ♠❡t❤♦❞s ❛♥❞ s♦❢t✇❛r❡ s❤♦✉❧❞ ❜❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ❛♥② ❖❉❊ ❚❤❡r❡❢♦r❡ ✇❡ ♥❡❡❞ ❛♥ ❛❜str❛❝t ♥♦t❛t✐♦♥ ❢♦r ❛♥ ❛r❜✐tr❛r② ❖❉❊ u′(t) = f (u(t), t) ❚❤❡ t❤r❡❡ ❖❉❊s ♦♥ t❤❡ ❧❛st s❧✐❞❡ ❝♦rr❡s♣♦♥❞ t♦ f (u, t) = αu, ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ f (u, t) = αu

• ✶ − u

R

• ,

❧♦❣✐st✐❝ ❣r♦✇t❤ f (u, t) = −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)❄

Pr♦❜❧❡♠✿

• ✐✈❡♥ ❛♥ ❖❉❊✱

√uu′ − α(t)u✸/✷(✶ − u R(t)) = ✵, ✇❤❛t ✐s t❤❡ f (u, t)❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ t❛r❣❡t ❢♦r♠ ✐s u′ = f (u, t)✱ s♦ ✇❡ ♥❡❡❞ t♦ ✐s♦❧❛t❡ u′ ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ u′ = α(t)u(✶ − u R(t))

• f (u,t)

❲❤❛t ✐s t❤❡ f (u, t)❄

Pr♦❜❧❡♠✿

• ✐✈❡♥ ❛♥ ❖❉❊✱

√uu′ − α(t)u✸/✷(✶ − u R(t)) = ✵, ✇❤❛t ✐s t❤❡ f (u, t)❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ t❛r❣❡t ❢♦r♠ ✐s u′ = f (u, t)✱ s♦ ✇❡ ♥❡❡❞ t♦ ✐s♦❧❛t❡ u′ ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ u′ = α(t)u(✶ − u R(t))

• f (u,t)

❲❤❛t ✐s t❤❡ f (u, t)❄

Pr♦❜❧❡♠✿

• ✐✈❡♥ ❛♥ ❖❉❊✱

√uu′ − α(t)u✸/✷(✶ − u R(t)) = ✵, ✇❤❛t ✐s t❤❡ f (u, t)❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ t❛r❣❡t ❢♦r♠ ✐s u′ = f (u, t)✱ s♦ ✇❡ ♥❡❡❞ t♦ ✐s♦❧❛t❡ u′ ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ u′ = α(t)u(✶ − u R(t))

• f (u,t)

❙✉❝❤ ❛❜str❛❝t f ❢✉♥❝t✐♦♥s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ ♠❛t❤❡♠❛t✐❝s

❲❡ ❝❛♥ ♠❛❦❡ ❣❡♥❡r✐❝ s♦❢t✇❛r❡ ❢♦r✿ ◆✉♠❡r✐❝❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ f ′(x) ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥✿ b

a f (x)dx

◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✿ f (x) = ✵ ❆♣♣❧✐❝❛t✐♦♥s✿

✶

d dx xa s✐♥(wx)✿ f (x) = xa s✐♥(wx)

✷ ✶

−✶(x✷ t❛♥❤−✶ x − (✶ + x✷)−✶)dx✿

f (x) = x✷ t❛♥❤−✶ x − (✶ + x✷)−✶✱ a = −✶✱ b = ✶

✸ ❙♦❧✈❡ x✹ s✐♥ x = t❛♥ x✿ f (x) = x✹ s✐♥ x − t❛♥ x

❲❡ ✉s❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ ❞❡r✐✈❛t✐✈❡s t♦ t✉r♥ ❛♥ ❖❉❊ ✐♥t♦ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥

u′ = f (u, t) ❆ss✉♠❡ ✇❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ u ❛t ❞✐s❝r❡t❡ t✐♠❡ ♣♦✐♥ts t✵, t✶, . . . , tk✳ ❆t tk ✇❡ ❤❛✈❡ t❤❡ ❖❉❊ u′(tk) = f (u(tk), tk) ❆♣♣r♦①✐♠❛t❡ u′(tk) ❜② ❛ ❢♦r✇❛r❞ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡✱ u′(tk) ≈ u(tk+✶) − u(tk) ∆t ■♥s❡rt ✐♥ t❤❡ ❖❉❊ ❛t t = tk✿ u(tk+✶) − u(tk) ∆t = f (u(tk), tk) ❚❡r♠s ✇✐t❤ u(tk) ❛r❡ ❦♥♦✇♥✱ ❛♥❞ t❤✐s ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✭❞✐✛❡r❡♥❝❡✮ ❡q✉❛t✐♦♥ ❢♦r u(tk+✶)

❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ✐❞❡❛

❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ✐❞❡❛

❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ♠❛t❤❡♠❛t✐❝s

❙♦❧✈✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ u(tk+✶) u(tk+✶) = u(tk) + ∆tf (u(tk), tk) ❚❤✐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 uk ❞❡♥♦t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ u(t) ❛t t = tk✳ uk+✶ = uk + ∆tf (uk, tk) ❚❤✐s ✐s ❛♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✇❡ ❝❛♥ s♦❧✈❡✦

■❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡

forward u(t) tn tn−1 tn +1

▲❡t✬s ❛♣♣❧② t❤❡ ♠❡t❤♦❞✦

Pr♦❜❧❡♠✿ ❚❤❡ ✇♦r❧❞✬s s✐♠♣❧❡st ❖❉❊ u′ = u, t ∈ (✵, T] ❙♦❧✈❡ ❢♦r u ❛t t = tk = k∆t✱ k = ✵, ✶, ✷, . . . , tn✱ t✵ = ✵✱ tn = T ❋♦r✇❛r❞ ❊✉❧❡r ♠❡t❤♦❞✿ uk+✶ = uk + ∆t f (uk, tk) ❙♦❧✉t✐♦♥ ❜② ❤❛♥❞✿ ❲❤❛t ✐s f ❄ f (u, t) = u uk+✶ = uk + ∆tf (uk, tk) = uk + ∆tuk = (✶ + ∆t)uk ❋✐rst st❡♣✿ u✶ = (✶ + ∆t)u✵ ❜✉t ✇❤❛t ✐s u✵❄

❆♥ ❖❉❊ ♥❡❡❞s ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿ u(✵) = U✵

◆✉♠❡r✐❝s✿ ❆♥② ❖❉❊ u′ = f (u, t) ♠✉st ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u(✵) = U✵✱ ✇✐t❤ ❦♥♦✇♥ U✵✱ ♦t❤❡r✇✐s❡ ✇❡ ❝❛♥♥♦t st❛rt t❤❡ ♠❡t❤♦❞✦ ▼❛t❤❡♠❛t✐❝s✿ ■♥ ♠❛t❤❡♠❛t✐❝s✿ u(✵) = U✵ ♠✉st ❜❡ s♣❡❝✐✜❡❞ t♦ ❣❡t ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✳ ❊①❛♠♣❧❡✿ u′ = u ❙♦❧✉t✐♦♥✿ u = Cet ❢♦r ❛♥② ❝♦♥st❛♥t C✳ ❙❛② u(✵) = U✵✿ u = U✵et✳

❲❡ ❝♦♥t✐♥✉❡ s♦❧✉t✐♦♥ ❜② ❤❛♥❞

❙❛② U✵ = ✷✿ u✶ = (✶ + ∆t)u✵ = (✶ + ∆t)U✵ = (✶ + ∆t)✷ u✷ = (✶ + ∆t)u✶ = (✶ + ∆t)(✶ + ∆t)✷ = ✷(✶ + ∆t)✷ u✸ = (✶ + ∆t)u✷ = (✶ + ∆t)✷(✶ + ∆t)✷ = ✷(✶ + ∆t)✸ u✹ = (✶ + ∆t)u✸ = (✶ + ∆t)✷(✶ + ∆t)✸ = ✷(✶ + ∆t)✹ u✺ = (✶ + ∆t)u✹ = (✶ + ∆t)✷(✶ + ∆t)✹ = ✷(✶ + ∆t)✺ ✳ ✳ ✳ = ✳ ✳ ✳ uk = ✷(✶ + ∆t)k ❆❝t✉❛❧❧②✱ ✇❡ ❢♦✉♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r uk✦ ◆♦ ♥❡❡❞ t♦ ♣r♦❣r❛♠✳✳✳

❍♦✇ ❛❝❝✉r❛t❡ ✐s ♦✉r ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞❄

❊①❛❝t s♦❧✉t✐♦♥✿ u(t) = ✷et✱ u(tk) = ✷ek∆t = ✷(e∆t)k ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✿ uk = ✷(✶ + ∆t)k ❲❤❡♥ ❣♦✐♥❣ ❢r♦♠ tk t♦ tk+✶✱ t❤❡ s♦❧✉t✐♦♥ ✐s ❛♠♣❧✐✜❡❞ ❜② ❛ ❢❛❝t♦r✿ ❊①❛❝t✿ u(tk+✶) = e∆tu(tk) ◆✉♠❡r✐❝❛❧✿ uk+✶ = (✶ + ∆t)uk ❯s✐♥❣ ❚❛②❧♦r s❡r✐❡s ❢♦r ex ✇❡ s❡❡ t❤❛t e∆t−(✶+∆t) = ✶+∆t+∆t✷ ✷ +frac∆t✸✻+· · ·−(✶+∆t) = frac∆t✸✻+O(∆ ✹ ❚❤✐s ❡rr♦r ❛♣♣r♦❛❝❤❡s ✵ ❛s ∆t → ✵✳

❲❤❛t ❛❜♦✉t t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ u′ = f (u, t)❄

• ✐✈❡♥ ❛♥② U✵✿

u✶ = u✵ + ∆tf (u✵, t✵) u✷ = u✶ + ∆tf (u✶, t✶) u✸ = u✷ + ∆tf (u✷, t✷) u✹ = u✸ + ∆tf (u✸, t✸) ✳ ✳ ✳ ◆♦ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ✐♥ t❤✐s ❝❛s❡✳✳✳ ❘✉❧❡ ♦❢ t❤✉♠❜✿ ❲❤❡♥ ❤❛♥❞ ❝❛❧❝✉❧❛t✐♦♥s ❣❡t ❜♦r✐♥❣✱ ❧❡t✬s ♣r♦❣r❛♠✦

❲❡ st❛rt ✇✐t❤ ❛ s♣❡❝✐❛❧✐③❡❞ ♣r♦❣r❛♠ ❢♦r u′ = u✱ u(✵) = U✵

❆❧❣♦r✐t❤♠✿

• ✐✈❡♥ ∆t ✭❞t✮ ❛♥❞ n

❈r❡❛t❡ ❛rr❛②s t ❛♥❞ ✉ ♦❢ ❧❡♥❣t❤ n + ✶ ❙❡t ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿ ✉❬✵❪ ❂ U✵✱ t❬✵❪❂✵ ❋♦r k = ✵, ✶, ✷, . . . , n − ✶✿

t❬❦✰✶❪ ❂ t❬❦❪ ✰ ❞t ✉❬❦✰✶❪ ❂ ✭✶ ✰ ❞t✮✯✉❬❦❪

❲❡ st❛rt ✇✐t❤ ❛ s♣❡❝✐❛❧✐③❡❞ ♣r♦❣r❛♠ ❢♦r u′ = u✱ u(✵) = U✵

Pr♦❣r❛♠✿

✐♠♣♦rt ♥✉♠♣② ❛s ♥♣ ✐♠♣♦rt s②s ❞t ❂ ❢❧♦❛t✭s②s✳❛r❣✈❬✶❪✮ ❯✵ ❂ ✶ ❚ ❂ ✹ ♥ ❂ ✐♥t✭❚✴❞t✮ t ❂ ♥♣✳③❡r♦s✭♥✰✶✮ ✉ ❂ ♥♣✳③❡r♦s✭♥✰✶✮ t❬✵❪ ❂ ✵ ✉❬✵❪ ❂ ❯✵ ❢♦r ❦ ✐♥ r❛♥❣❡✭♥✮✿ t❬❦✰✶❪ ❂ t❬❦❪ ✰ ❞t ✉❬❦✰✶❪ ❂ ✭✶ ✰ ❞t✮✯✉❬❦❪ ★ ♣❧♦t ✉ ❛❣❛✐♥st t

❚❤❡ s♦❧✉t✐♦♥ ✐❢ ✇❡ ♣❧♦t u ❛❣❛✐♥st t

∆t = ✵.✹ ❛♥❞ ∆t = ✵.✷✿

10 20 30 40 50 60 0.5 1 1.5 2 2.5 3 3.5 4 u t Solution of the ODE u’=u, u(0)=1 numerical exact 10 20 30 40 50 60 0.5 1 1.5 2 2.5 3 3.5 4 u t Solution of the ODE u’=u, u(0)=1 numerical exact

❚❤❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❖❉❊ u′ = f (u, t)

❆❧❣♦r✐t❤♠✿

• ✐✈❡♥ ∆t ✭❞t✮ ❛♥❞ n

❈r❡❛t❡ ❛rr❛②s t ❛♥❞ ✉ ♦❢ ❧❡♥❣t❤ n + ✶ ❈r❡❛t❡ ❛rr❛② ✉ t♦ ❤♦❧❞ uk ❛♥❞ ❙❡t ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿ ✉❬✵❪ ❂ U✵✱ t❬✵❪❂✵ ❋♦r k = ✵, ✶, ✷, . . . , n − ✶✿

✉❬❦✰✶❪ ❂ ✉❬❦❪ ✰ ❞t✯❢✭✉❬❦❪✱ t❬❦❪✮ ✭t❤❡ ♦♥❧② ❝❤❛♥❣❡✦✮ t❬❦✰✶❪ ❂ t❬❦❪ ✰ ❞t

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❛❧❣♦r✐t❤♠ ❢♦r u′ = f (u, t)

• ❡♥❡r❛❧ ❢✉♥❝t✐♦♥✿

❞❡❢ ❋♦r✇❛r❞❊✉❧❡r✭❢✱ ❯✵✱ ❚✱ ♥✮✿ ✧✧✧❙♦❧✈❡ ✉✬❂❢✭✉✱t✮✱ ✉✭✵✮❂❯✵✱ ✇✐t❤ ♥ st❡♣s ✉♥t✐❧ t❂❚✳✧✧✧ ✐♠♣♦rt ♥✉♠♣② ❛s ♥♣ t ❂ ♥♣✳③❡r♦s✭♥✰✶✮ ✉ ❂ ♥♣✳③❡r♦s✭♥✰✶✮ ★ ✉❬❦❪ ✐s t❤❡ s♦❧✉t✐♦♥ ❛t t✐♠❡ t❬❦❪ ✉❬✵❪ ❂ ❯✵ t❬✵❪ ❂ ✵ ❞t ❂ ❚✴❢❧♦❛t✭♥✮ ❢♦r ❦ ✐♥ r❛♥❣❡✭♥✮✿ t❬❦✰✶❪ ❂ t❬❦❪ ✰ ❞t ✉❬❦✰✶❪ ❂ ✉❬❦❪ ✰ ❞t✯❢✭✉❬❦❪✱ t❬❦❪✮ r❡t✉r♥ ✉✱ t

▼❛❣✐❝✿ ❚❤✐s s✐♠♣❧❡ ❢✉♥❝t✐♦♥ ❝❛♥ s♦❧✈❡ ❛♥② ❖❉❊ ✭✦✮

❊①❛♠♣❧❡ ♦♥ ✉s✐♥❣ t❤❡ ❢✉♥❝t✐♦♥

▼❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠✿ ❙♦❧✈❡ u′ = u✱ u(✵) = ✶✱ ❢♦r t ∈ [✵, ✹]✱ ✇✐t❤ ∆t = ✵.✹ ❊①❛❝t s♦❧✉t✐♦♥✿ u(t) = et✳ ❇❛s✐❝ ❝♦❞❡✿

❞❡❢ ❢✭✉✱ t✮✿ r❡t✉r♥ ✉ ❯✵ ❂ ✶ ❚ ❂ ✸ ♥ ❂ ✸✵ ✉✱ t ❂ ❋♦r✇❛r❞❊✉❧❡r✭❢✱ ❯✵✱ ❚✱ ♥✮

❈♦♠♣❛r❡ ❡①❛❝t ❛♥❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✿

❢r♦♠ s❝✐t♦♦❧s✳st❞ ✐♠♣♦rt ♣❧♦t✱ ❡①♣ ✉❴❡①❛❝t ❂ ❡①♣✭t✮ ♣❧♦t✭t✱ ✉✱ ✬r✲✬✱ t✱ ✉❴❡①❛❝t✱ ✬❜✲✬✱ ①❧❛❜❡❧❂✬t✬✱ ②❧❛❜❡❧❂✬✉✬✱ ❧❡❣❡♥❞❂✭✬♥✉♠❡r✐❝❛❧✬✱ ✬❡①❛❝t✬✮✱ t✐t❧❡❂✧❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❖❉❊ ✉✬❂✉✱ ✉✭✵✮❂✶✧✮

◆♦✇ ②♦✉ ❝❛♥ s♦❧✈❡ ❛♥② ❖❉❊✦

❘❡❝✐♣❡✿ ■❞❡♥t✐❢② f (u, t) ✐♥ ②♦✉r ❖❉❊ ▼❛❦❡ s✉r❡ ②♦✉ ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ U✵ ■♠♣❧❡♠❡♥t t❤❡ f (u, t) ❢♦r♠✉❧❛ ✐♥ ❛ P②t❤♦♥ ❢✉♥❝t✐♦♥ ❢✭✉✱ t✮ ❈❤♦♦s❡ ∆t ♦r ♥♦ ♦❢ st❡♣s n ❈❛❧❧ ✉✱ t ❂ ❋♦r✇❛r❞❊✉❧❡r✭❢✱ ❯✵✱ ❚✱ ♥✮ ♣❧♦t✭t✱ ✉✮ ❲❛r♥✐♥❣✿ ❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ♠❡t❤♦❞ ♠❛② ❣✐✈❡ ✈❡r② ✐♥❛❝❝✉r❛t❡ s♦❧✉t✐♦♥s ✐❢ ∆t ✐s ♥♦t s✉✣❝✐❡♥t❧② s♠❛❧❧✳ ❋♦r s♦♠❡ ♣r♦❜❧❡♠s ✭❧✐❦❡ u′′ + u = ✵✮ ♦t❤❡r ♠❡t❤♦❞s s❤♦✉❧❞ ❜❡ ✉s❡❞✳

▲❡t ✉s ♠❛❦❡ ❛ ❝❧❛ss ✐♥st❡❛❞ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❢♦r s♦❧✈✐♥❣ ❖❉❊s

❯s❛❣❡ ♦❢ t❤❡ ❝❧❛ss✿

♠❡t❤♦❞ ❂ ❋♦r✇❛r❞❊✉❧❡r✭❢✱ ❞t✮ ♠❡t❤♦❞✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭❯✵✱ t✵✮ ✉✱ t ❂ ♠❡t❤♦❞✳s♦❧✈❡✭❚✮ ♣❧♦t✭t✱ ✉✮

❍♦✇❄ ❙t♦r❡ f ✱ ∆t✱ ❛♥❞ t❤❡ s❡q✉❡♥❝❡s uk✱ tk ❛s ❛ttr✐❜✉t❡s ❙♣❧✐t t❤❡ st❡♣s ✐♥ t❤❡ ❋♦r✇❛r❞❊✉❧❡r ❢✉♥❝t✐♦♥ ✐♥t♦ ❢♦✉r ♠❡t❤♦❞s✿

t❤❡ ❝♦♥str✉❝t♦r ✭❴❴✐♥✐t❴❴✮ s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥ ❢♦r u(✵) = U✵ s♦❧✈❡ ❢♦r r✉♥♥✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ t✐♠❡ st❡♣♣✐♥❣ ❛❞✈❛♥❝❡ ❢♦r ✐s♦❧❛t✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ ✉♣❞❛t✐♥❣ ❢♦r♠✉❧❛ ✭♥❡✇ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❥✉st ♥❡❡❞ ❛ ❞✐✛❡r❡♥t ❛❞✈❛♥❝❡ ♠❡t❤♦❞✱ t❤❡ r❡st ✐s t❤❡ s❛♠❡✮

❚❤❡ ❝♦❞❡ ❢♦r ❛ ❝❧❛ss ❢♦r s♦❧✈✐♥❣ ❖❉❊s ✭♣❛rt ✶✮

✐♠♣♦rt ♥✉♠♣② ❛s ♥♣ ❝❧❛ss ❋♦r✇❛r❞❊✉❧❡r❴✈✶✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❢✱ ❞t✮✿ s❡❧❢✳❢✱ s❡❧❢✳❞t ❂ ❢✱ ❞t ❞❡❢ s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭s❡❧❢✱ ❯✵✮✿ s❡❧❢✳❯✵ ❂ ❢❧♦❛t✭❯✵✮

❚❤❡ ❝♦❞❡ ❢♦r ❛ ❝❧❛ss ❢♦r s♦❧✈✐♥❣ ❖❉❊s ✭♣❛rt ✷✮

❝❧❛ss ❋♦r✇❛r❞❊✉❧❡r❴✈✶✿ ✳✳✳ ❞❡❢ s♦❧✈❡✭s❡❧❢✱ ❚✮✿ ✧✧✧❈♦♠♣✉t❡ s♦❧✉t✐♦♥ ❢♦r ✵ ❁❂ t ❁❂ ❚✳✧✧✧ ♥ ❂ ✐♥t✭r♦✉♥❞✭❚✴s❡❧❢✳❞t✮✮ ★ ♥♦ ♦❢ ✐♥t❡r✈❛❧s s❡❧❢✳✉ ❂ ♥♣✳③❡r♦s✭♥✰✶✮ s❡❧❢✳t ❂ ♥♣✳③❡r♦s✭♥✰✶✮ s❡❧❢✳✉❬✵❪ ❂ ❢❧♦❛t✭s❡❧❢✳❯✵✮ s❡❧❢✳t❬✵❪ ❂ ❢❧♦❛t✭✵✮ ❢♦r ❦ ✐♥ r❛♥❣❡✭s❡❧❢✳♥✮✿ s❡❧❢✳❦ ❂ ❦ s❡❧❢✳t❬❦✰✶❪ ❂ s❡❧❢✳t❬❦❪ ✰ s❡❧❢✳❞t s❡❧❢✳✉❬❦✰✶❪ ❂ s❡❧❢✳❛❞✈❛♥❝❡✭✮ r❡t✉r♥ s❡❧❢✳✉✱ s❡❧❢✳t ❞❡❢ ❛❞✈❛♥❝❡✭s❡❧❢✮✿ ✧✧✧❆❞✈❛♥❝❡ t❤❡ s♦❧✉t✐♦♥ ♦♥❡ t✐♠❡ st❡♣✳✧✧✧ ★ ❈r❡❛t❡ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s t♦ ❣❡t r✐❞ ♦❢ ✧s❡❧❢✳✧ ✐♥ ★ t❤❡ ♥✉♠❡r✐❝❛❧ ❢♦r♠✉❧❛ ✉✱ ❞t✱ ❢✱ ❦✱ t ❂ s❡❧❢✳✉✱ s❡❧❢✳❞t✱ s❡❧❢✳❢✱ s❡❧❢✳❦✱ s❡❧❢✳t ✉♥❡✇ ❂ ✉❬❦❪ ✰ ❞t✯❢✭✉❬❦❪✱ t❬❦❪✮ r❡t✉r♥ ✉♥❡✇

❆❧t❡r♥❛t✐✈❡ ❝❧❛ss ❝♦❞❡ ❢♦r s♦❧✈✐♥❣ ❖❉❊s ✭♣❛rt ✶✮

★ ■❞❡❛✿ ❞r♦♣ ❞t ✐♥ t❤❡ ❝♦♥str✉❝t♦r✳ ★ ▲❡t t❤❡ ✉s❡r ♣r♦✈✐❞❡ ❛❧❧ t✐♠❡ ♣♦✐♥ts ✭✐♥ s♦❧✈❡✮✳ ❝❧❛ss ❋♦r✇❛r❞❊✉❧❡r✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❢✮✿ ★ t❡st t❤❛t ❢ ✐s ❛ ❢✉♥❝t✐♦♥ ✐❢ ♥♦t ❝❛❧❧❛❜❧❡✭❢✮✿ r❛✐s❡ ❚②♣❡❊rr♦r✭✬❢ ✐s ✪s✱ ♥♦t ❛ ❢✉♥❝t✐♦♥✬ ✪ t②♣❡✭❢✮✮ s❡❧❢✳❢ ❂ ❢ ❞❡❢ s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭s❡❧❢✱ ❯✵✮✿ s❡❧❢✳❯✵ ❂ ❢❧♦❛t✭❯✵✮ ❞❡❢ s♦❧✈❡✭s❡❧❢✱ t✐♠❡❴♣♦✐♥ts✮✿ ✳✳✳

❆❧t❡r♥❛t✐✈❡ ❝❧❛ss ❝♦❞❡ ❢♦r s♦❧✈✐♥❣ ❖❉❊s ✭♣❛rt ✷✮

❝❧❛ss ❋♦r✇❛r❞❊✉❧❡r✿ ✳✳✳ ❞❡❢ s♦❧✈❡✭s❡❧❢✱ t✐♠❡❴♣♦✐♥ts✮✿ ✧✧✧❈♦♠♣✉t❡ ✉ ❢♦r t ✈❛❧✉❡s ✐♥ t✐♠❡❴♣♦✐♥ts ❧✐st✳✧✧✧ s❡❧❢✳t ❂ ♥♣✳❛s❛rr❛②✭t✐♠❡❴♣♦✐♥ts✮ s❡❧❢✳✉ ❂ ♥♣✳③❡r♦s✭❧❡♥✭t✐♠❡❴♣♦✐♥ts✮✮ s❡❧❢✳✉❬✵❪ ❂ s❡❧❢✳❯✵ ❢♦r ❦ ✐♥ r❛♥❣❡✭❧❡♥✭s❡❧❢✳t✮✲✶✮✿ s❡❧❢✳❦ ❂ ❦ s❡❧❢✳✉❬❦✰✶❪ ❂ s❡❧❢✳❛❞✈❛♥❝❡✭✮ r❡t✉r♥ s❡❧❢✳✉✱ s❡❧❢✳t ❞❡❢ ❛❞✈❛♥❝❡✭s❡❧❢✮✿ ✧✧✧❆❞✈❛♥❝❡ t❤❡ s♦❧✉t✐♦♥ ♦♥❡ t✐♠❡ st❡♣✳✧✧✧ ✉✱ ❢✱ ❦✱ t ❂ s❡❧❢✳✉✱ s❡❧❢✳❢✱ s❡❧❢✳❦✱ s❡❧❢✳t ❞t ❂ t❬❦✰✶❪ ✲ t❬❦❪ ✉♥❡✇ ❂ ✉❬❦❪ ✰ ❞t✯❢✭✉❬❦❪✱ t❬❦❪✮ r❡t✉r♥ ✉♥❡✇

❱❡r✐❢②✐♥❣ t❤❡ ❝❧❛ss ✐♠♣❧❡♠❡♥t❛t✐♦♥❀ ♠❛t❤❡♠❛t✐❝s

▼❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠✿ ■♠♣♦rt❛♥t r❡s✉❧t✿ t❤❡ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ✭❛♥❞ ♠♦st ♦t❤❡rs✮ ✇✐❧❧ ❡①❛❝t❧② r❡♣r♦❞✉❝❡ u ✐❢ ✐t ✐s ❧✐♥❡❛r ✐♥ t ✭✦✮✿ u(t) = at + b = ✵.✷t + ✸ h(t) = u(t) u′(t) = ✵.✷ + (u − h(t))✹, u(✵) = ✸, t ∈ [✵, ✸] ❚❤✐s u s❤♦✉❧❞ ❜❡ r❡♣r♦❞✉❝❡❞ t♦ ♠❛❝❤✐♥❡ ♣r❡❝✐s✐♦♥ ❢♦r ❛♥② ∆t✳

❱❡r✐❢②✐♥❣ t❤❡ ❝❧❛ss ✐♠♣❧❡♠❡♥t❛t✐♦♥❀ ✐♠♣❧❡♠❡♥t❛t✐♦♥

❈♦❞❡✿

❞❡❢ t❡st❴❋♦r✇❛r❞❊✉❧❡r❴❛❣❛✐♥st❴❧✐♥❡❛r❴s♦❧✉t✐♦♥✭✮✿ ❞❡❢ ❢✭✉✱ t✮✿ r❡t✉r♥ ✵✳✷ ✰ ✭✉ ✲ ❤✭t✮✮✯✯✹ ❞❡❢ ❤✭t✮✿ r❡t✉r♥ ✵✳✷✯t ✰ ✸ s♦❧✈❡r ❂ ❋♦r✇❛r❞❊✉❧❡r✭❢✮ s♦❧✈❡r✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭❯✵❂✸✮ ❞t ❂ ✵✳✹❀ ❚ ❂ ✸❀ ♥ ❂ ✐♥t✭r♦✉♥❞✭❢❧♦❛t✭❚✮✴❞t✮✮ t✐♠❡❴♣♦✐♥ts ❂ ♥♣✳❧✐♥s♣❛❝❡✭✵✱ ❚✱ ♥✰✶✮ ✉✱ t ❂ s♦❧✈❡r✳s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts✮ ✉❴❡①❛❝t ❂ ❤✭t✮ ❞✐❢❢ ❂ ♥♣✳❛❜s✭✉❴❡①❛❝t ✲ ✉✮✳♠❛①✭✮ t♦❧ ❂ ✶❊✲✶✹ s✉❝❝❡ss ❂ ❞✐❢❢ ❁ t♦❧ ❛ss❡rt s✉❝❝❡ss

❯s✐♥❣ ❛ ❝❧❛ss t♦ ❤♦❧❞ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ f (u, t)

▼❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠✿ u′(t) = αu(t)

• ✶ − u(t)

R

• ,

u(✵) = U✵, t ∈ [✵, ✹✵] ❈❧❛ss ❢♦r r✐❣❤t✲❤❛♥❞ s✐❞❡ f (u, t)✿

❝❧❛ss ▲♦❣✐st✐❝✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❛❧♣❤❛✱ ❘✱ ❯✵✮✿ s❡❧❢✳❛❧♣❤❛✱ s❡❧❢✳❘✱ s❡❧❢✳❯✵ ❂ ❛❧♣❤❛✱ ❢❧♦❛t✭❘✮✱ ❯✵ ❞❡❢ ❴❴❝❛❧❧❴❴✭s❡❧❢✱ ✉✱ t✮✿ ★ ❢✭✉✱t✮ r❡t✉r♥ s❡❧❢✳❛❧♣❤❛✯✉✯✭✶ ✲ ✉✴s❡❧❢✳❘✮

▼❛✐♥ ♣r♦❣r❛♠✿

♣r♦❜❧❡♠ ❂ ▲♦❣✐st✐❝✭✵✳✷✱ ✶✱ ✵✳✶✮ t✐♠❡❴♣♦✐♥ts ❂ ♥♣✳❧✐♥s♣❛❝❡✭✵✱ ✹✵✱ ✹✵✶✮ ♠❡t❤♦❞ ❂ ❋♦r✇❛r❞❊✉❧❡r✭♣r♦❜❧❡♠✮ ♠❡t❤♦❞✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭♣r♦❜❧❡♠✳❯✵✮ ✉✱ t ❂ ♠❡t❤♦❞✳s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts✮

❋✐❣✉r❡ ♦❢ t❤❡ s♦❧✉t✐♦♥

5 10 15 20 25 30 35 40 45 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 u

Logistic growth: alpha=0.2, R=1, dt=0.1

◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❢♦r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

❋♦r✇❛r❞ ❊✉❧❡r ♠❡t❤♦❞✿ uk+✶ = uk + ∆t f (uk, tk) ✹t❤✲♦r❞❡r ❘✉♥❣❡✲❑✉tt❛ ♠❡t❤♦❞✿ uk+✶ = uk + ✶ ✻ (K✶ + ✷K✷ + ✷K✸ + K✹) K✶ = ∆t f (uk, tk) K✷ = ∆t f (uk + ✶ ✷K✶, tk + ✶ ✷∆t) K✸ = ∆t f (uk + ✶ ✷K✷, tk + ✶ ✷∆t) K✹ = ∆t f (uk + K✸, tk + ∆t) ❆♥❞ ❧♦ts ♦❢ ♦t❤❡r ♠❡t❤♦❞s✦ ❍♦✇ t♦ ♣r♦❣r❛♠ ❛ ✇✐❞❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠❡t❤♦❞s❄ ❯s❡ ♦❜❥❡❝t✲♦r✐❡♥t❡❞ ♣r♦❣r❛♠♠✐♥❣✦

❆ s✉♣❡r❝❧❛ss ❢♦r ❖❉❊ ♠❡t❤♦❞s

❈♦♠♠♦♥ t❛s❦s ❢♦r ❖❉❊ s♦❧✈❡rs✿ ❙t♦r❡ t❤❡ s♦❧✉t✐♦♥ uk ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t✐♠❡ ❧❡✈❡❧s tk✱ k = ✵, ✶, ✷, . . . , n ❙t♦r❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ f (u, t) ❙❡t ❛♥❞ st♦r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❘✉♥ t❤❡ ❧♦♦♣ ♦✈❡r ❛❧❧ t✐♠❡ st❡♣s Pr✐♥❝✐♣❧❡s✿ ❈♦♠♠♦♥ ❞❛t❛ ❛♥❞ ❢✉♥❝t✐♦♥❛❧✐t② ❛r❡ ♣❧❛❝❡❞ ✐♥ s✉♣❡r❝❧❛ss ❖❉❊❙♦❧✈❡r ■s♦❧❛t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ✉♣❞❛t✐♥❣ ❢♦r♠✉❧❛ ✐♥ ❛ ♠❡t❤♦❞ ❛❞✈❛♥❝❡ ❙✉❜❝❧❛ss❡s✱ ❡✳❣✳✱ ❋♦r✇❛r❞❊✉❧❡r✱ ❥✉st ✐♠♣❧❡♠❡♥t t❤❡ s♣❡❝✐✜❝ ♥✉♠❡r✐❝❛❧ ❢♦r♠✉❧❛ ✐♥ ❛❞✈❛♥❝❡

❆ s✉♣❡r❝❧❛ss ❢♦r ❖❉❊ ♠❡t❤♦❞s

❈♦♠♠♦♥ t❛s❦s ❢♦r ❖❉❊ s♦❧✈❡rs✿ ❙t♦r❡ t❤❡ s♦❧✉t✐♦♥ uk ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t✐♠❡ ❧❡✈❡❧s tk✱ k = ✵, ✶, ✷, . . . , n ❙t♦r❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ f (u, t) ❙❡t ❛♥❞ st♦r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❘✉♥ t❤❡ ❧♦♦♣ ♦✈❡r ❛❧❧ t✐♠❡ st❡♣s Pr✐♥❝✐♣❧❡s✿ ❈♦♠♠♦♥ ❞❛t❛ ❛♥❞ ❢✉♥❝t✐♦♥❛❧✐t② ❛r❡ ♣❧❛❝❡❞ ✐♥ s✉♣❡r❝❧❛ss ❖❉❊❙♦❧✈❡r ■s♦❧❛t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ✉♣❞❛t✐♥❣ ❢♦r♠✉❧❛ ✐♥ ❛ ♠❡t❤♦❞ ❛❞✈❛♥❝❡ ❙✉❜❝❧❛ss❡s✱ ❡✳❣✳✱ ❋♦r✇❛r❞❊✉❧❡r✱ ❥✉st ✐♠♣❧❡♠❡♥t t❤❡ s♣❡❝✐✜❝ ♥✉♠❡r✐❝❛❧ ❢♦r♠✉❧❛ ✐♥ ❛❞✈❛♥❝❡

❆ s✉♣❡r❝❧❛ss ❢♦r ❖❉❊ ♠❡t❤♦❞s

❈♦♠♠♦♥ t❛s❦s ❢♦r ❖❉❊ s♦❧✈❡rs✿ ❙t♦r❡ t❤❡ s♦❧✉t✐♦♥ uk ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t✐♠❡ ❧❡✈❡❧s tk✱ k = ✵, ✶, ✷, . . . , n ❙t♦r❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥ f (u, t) ❙❡t ❛♥❞ st♦r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❘✉♥ t❤❡ ❧♦♦♣ ♦✈❡r ❛❧❧ t✐♠❡ st❡♣s Pr✐♥❝✐♣❧❡s✿ ❈♦♠♠♦♥ ❞❛t❛ ❛♥❞ ❢✉♥❝t✐♦♥❛❧✐t② ❛r❡ ♣❧❛❝❡❞ ✐♥ s✉♣❡r❝❧❛ss ❖❉❊❙♦❧✈❡r ■s♦❧❛t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ✉♣❞❛t✐♥❣ ❢♦r♠✉❧❛ ✐♥ ❛ ♠❡t❤♦❞ ❛❞✈❛♥❝❡ ❙✉❜❝❧❛ss❡s✱ ❡✳❣✳✱ ❋♦r✇❛r❞❊✉❧❡r✱ ❥✉st ✐♠♣❧❡♠❡♥t t❤❡ s♣❡❝✐✜❝ ♥✉♠❡r✐❝❛❧ ❢♦r♠✉❧❛ ✐♥ ❛❞✈❛♥❝❡

❚❤❡ s✉♣❡r❝❧❛ss ❝♦❞❡

❝❧❛ss ❖❉❊❙♦❧✈❡r✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❢✮✿ s❡❧❢✳❢ ❂ ❢ ❞❡❢ ❛❞✈❛♥❝❡✭s❡❧❢✮✿ ✧✧✧❆❞✈❛♥❝❡ s♦❧✉t✐♦♥ ♦♥❡ t✐♠❡ st❡♣✳✧✧✧ r❛✐s❡ ◆♦t■♠♣❧❡♠❡♥t❡❞❊rr♦r ★ ✐♠♣❧❡♠❡♥t ✐♥ s✉❜❝❧❛ss ❞❡❢ s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭s❡❧❢✱ ❯✵✮✿ s❡❧❢✳❯✵ ❂ ❢❧♦❛t✭❯✵✮ ❞❡❢ s♦❧✈❡✭s❡❧❢✱ t✐♠❡❴♣♦✐♥ts✮✿ s❡❧❢✳t ❂ ♥♣✳❛s❛rr❛②✭t✐♠❡❴♣♦✐♥ts✮ s❡❧❢✳✉ ❂ ♥♣✳③❡r♦s✭❧❡♥✭s❡❧❢✳t✮✮ ★ ❆ss✉♠❡ t❤❛t s❡❧❢✳t❬✵❪ ❝♦rr❡s♣♦♥❞s t♦ s❡❧❢✳❯✵ s❡❧❢✳✉❬✵❪ ❂ s❡❧❢✳❯✵ ★ ❚✐♠❡ ❧♦♦♣ ❢♦r ❦ ✐♥ r❛♥❣❡✭♥✲✶✮✿ s❡❧❢✳❦ ❂ ❦ s❡❧❢✳✉❬❦✰✶❪ ❂ s❡❧❢✳❛❞✈❛♥❝❡✭✮ r❡t✉r♥ s❡❧❢✳✉✱ s❡❧❢✳t ❞❡❢ ❛❞✈❛♥❝❡✭s❡❧❢✮✿ r❛✐s❡ ◆♦t■♠♣❧❡♠t❡❞❊rr♦r ★ t♦ ❜❡ ✐♠♣❧✳ ✐♥ s✉❜❝❧❛ss❡s

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ♠❡t❤♦❞

❙✉❜❝❧❛ss ❝♦❞❡✿

❝❧❛ss ❋♦r✇❛r❞❊✉❧❡r✭❖❉❊❙♦❧✈❡r✮✿ ❞❡❢ ❛❞✈❛♥❝❡✭s❡❧❢✮✿ ✉✱ ❢✱ ❦✱ t ❂ s❡❧❢✳✉✱ s❡❧❢✳❢✱ s❡❧❢✳❦✱ s❡❧❢✳t ❞t ❂ t❬❦✰✶❪ ✲ t❬❦❪ ✉♥❡✇ ❂ ✉❬❦❪ ✰ ❞t✯❢✭✉❬❦❪✱ t✮ r❡t✉r♥ ✉♥❡✇

❆♣♣❧✐❝❛t✐♦♥ ❝♦❞❡ ❢♦r u′ − u = ✵✱ u(✵) = ✶✱ t ∈ [✵, ✸]✱ ∆t = ✵.✶✿

❢r♦♠ ❖❉❊❙♦❧✈❡r ✐♠♣♦rt ❋♦r✇❛r❞❊✉❧❡r ❞❡❢ t❡st✶✭✉✱ t✮✿ r❡t✉r♥ ✉ ♠❡t❤♦❞ ❂ ❋♦r✇❛r❞❊✉❧❡r✭t❡st✶✮ ♠❡t❤♦❞✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭❯✵❂✶✮ ✉✱ t ❂ ♠❡t❤♦❞✳s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts❂♥♣✳❧✐♥s♣❛❝❡✭✵✱ ✸✱ ✸✶✮✮ ♣❧♦t✭t✱ ✉✮

❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ❘✉♥❣❡✲❑✉tt❛ ♠❡t❤♦❞

❙✉❜❝❧❛ss ❝♦❞❡✿

❝❧❛ss ❘✉♥❣❡❑✉tt❛✹✭❖❉❊❙♦❧✈❡r✮✿ ❞❡❢ ❛❞✈❛♥❝❡✭s❡❧❢✮✿ ✉✱ ❢✱ ❦✱ t ❂ s❡❧❢✳✉✱ s❡❧❢✳❢✱ s❡❧❢✳❦✱ s❡❧❢✳t ❞t ❂ t❬❦✰✶❪ ✲ t❬❦❪ ❞t✷ ❂ ❞t✴✷✳✵ ❑✶ ❂ ❞t✯❢✭✉❬❦❪✱ t✮ ❑✷ ❂ ❞t✯❢✭✉❬❦❪ ✰ ✵✳✺✯❑✶✱ t ✰ ❞t✷✮ ❑✸ ❂ ❞t✯❢✭✉❬❦❪ ✰ ✵✳✺✯❑✷✱ t ✰ ❞t✷✮ ❑✹ ❂ ❞t✯❢✭✉❬❦❪ ✰ ❑✸✱ t ✰ ❞t✮ ✉♥❡✇ ❂ ✉❬❦❪ ✰ ✭✶✴✻✳✵✮✯✭❑✶ ✰ ✷✯❑✷ ✰ ✷✯❑✸ ✰ ❑✹✮ r❡t✉r♥ ✉♥❡✇

❆♣♣❧✐❝❛t✐♦♥ ❝♦❞❡ ✭s❛♠❡ ❛s ❢♦r ❋♦r✇❛r❞❊✉❧❡r✮✿

❢r♦♠ ❖❉❊❙♦❧✈❡r ✐♠♣♦rt ❘✉♥❣❡❑✉tt❛✹ ❞❡❢ t❡st✶✭✉✱ t✮✿ r❡t✉r♥ ✉ ♠❡t❤♦❞ ❂ ❘✉♥❣❡❑✉tt❛✹✭t❡st✶✮ ♠❡t❤♦❞✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭❯✵❂✶✮ ✉✱ t ❂ ♠❡t❤♦❞✳s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts❂♥♣✳❧✐♥s♣❛❝❡✭✵✱ ✸✱ ✸✶✮✮ ♣❧♦t✭t✱ ✉✮

❚❤❡ ✉s❡r s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❝❤❡❝❦ ✐♥t❡r♠❡❞✐❛t❡ s♦❧✉t✐♦♥s ❛♥❞ t❡r♠✐♥❛t❡ t❤❡ t✐♠❡ st❡♣♣✐♥❣

❙♦♠❡t✐♠❡s ❛ ♣r♦♣❡rt② ♦❢ t❤❡ s♦❧✉t✐♦♥ ❞❡t❡r♠✐♥❡s ✇❤❡♥ t♦ st♦♣ t❤❡ s♦❧✉t✐♦♥ ♣r♦❝❡ss✿ ❡✳❣✳✱ ✇❤❡♥ u < ✶✵−✼ ≈ ✵✳ ❊①t❡♥s✐♦♥✿ s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts✱ t❡r♠✐♥❛t❡✮ t❡r♠✐♥❛t❡✭✉✱ t✱ st❡♣❴♥♦✮ ✐s ❝❛❧❧❡❞ ❛t ❡✈❡r② t✐♠❡ st❡♣✱ ✐s ✉s❡r✲❞❡✜♥❡❞✱ ❛♥❞ r❡t✉r♥s ❚r✉❡ ✇❤❡♥ t❤❡ t✐♠❡ st❡♣♣✐♥❣ s❤♦✉❧❞ ❜❡ t❡r♠✐♥❛t❡❞ ▲❛st ❝♦♠♣✉t❡❞ s♦❧✉t✐♦♥ ✐s ✉❬st❡♣❴♥♦❪ ❛t t✐♠❡ t❬st❡♣❴♥♦❪

❞❡❢ t❡r♠✐♥❛t❡✭✉✱ t✱ st❡♣❴♥♦✮✿ ❡♣s ❂ ✶✳✵❊✲✻ ★ s♠❛❧❧ ♥✉♠❜❡r r❡t✉r♥ ❛❜s✭✉❬st❡♣❴♥♦✱✵❪✮ ❁ ❡♣s ★ ❝❧♦s❡ ❡♥♦✉❣❤ t♦ ③❡r♦❄

✶ ❍♦✇ t♦ s♦❧✈❡ ❛♥② ♦r❞✐♥❛r② s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ✷ ❙②st❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭✈❡❝t♦r ❖❉❊✮

❙②st❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭✈❡❝t♦r ❖❉❊✮

u′ = v v′ = −u u(✵) = ✶ v(✵) = ✵

2 4 6 8 10 12 14 1.0 0.5 0.0 0.5 1.0

u v

❊①❛♠♣❧❡ ♦♥ ❛ s②st❡♠ ♦❢ ❖❉❊s ✭✈❡❝t♦r ❖❉❊✮

❚✇♦ ❖❉❊s ✇✐t❤ t✇♦ ✉♥❦♥♦✇♥s u(t) ❛♥❞ v(t)✿ u′(t) = v(t) v′(t) = −u(t) ❊❛❝❤ ✉♥❦♥♦✇♥ ♠✉st ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✱ s❛② u(✵) = ✵, v(✵) = ✶ ■♥ t❤✐s ❝❛s❡✱ ♦♥❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ t♦ ❜❡ u(t) = s✐♥(t), v(t) = ❝♦s(t) ❙②st❡♠s ♦❢ ❖❉❊s ❛♣♣❡❛r ❢r❡q✉❡♥t❧② ✐♥ ♣❤②s✐❝s✱ ❜✐♦❧♦❣②✱ ✜♥❛♥❝❡✱ ✳✳✳

❚❤❡ ❖❉❊ s②st❡♠ t❤❛t ✐s t❤❡ ✜♥❛❧ ♣r♦❥❡❝t ✐♥ t❤❡ ❝♦✉rs❡

▼♦❞❡❧ ❢♦r s♣r❡❛❞✐♥❣ ♦❢ ❛ ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✿ S′ = −βSI I ′ = βSI − νR R′ = νI ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ S(✵) = S✵ I(✵) = I✵ R(✵) = ✵

❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦♥ ❛ s②st❡♠ ♦❢ ❖❉❊s ✭✈❡❝t♦r ❖❉❊✮

❙❡❝♦♥❞✲♦r❞❡r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ❢♦r ❛ s♣r✐♥❣✲♠❛ss s②st❡♠ ✭❢r♦♠ ◆❡✇t♦♥✬s s❡❝♦♥❞ ❧❛✇✮✿ mu′′ + βu′ + ku = ✵, u(✵) = U✵, u′(✵) = ✵ ❲❡ ❝❛♥ r❡✇r✐t❡ t❤✐s ❛s ❛ s②st❡♠ ♦❢ t✇♦ ✜rst✲♦r❞❡r ❡q✉❛t✐♦♥s✱ ❜② ✐♥tr♦❞✉❝✐♥❣ t✇♦ ♥❡✇ ✉♥❦♥♦✇♥s u(✵)(t) ≡ u(t), u(✶)(t) ≡ u′(t) ❚❤❡ ✜rst✲♦r❞❡r s②st❡♠ ✐s t❤❡♥ d dt u(✵)(t) = u(✶)(t) d dt u(✶)(t) = −m−✶βu(✶) − m−✶ku(✵) ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ u(✵)(✵) = U✵✱ u(✶)(✵) = ✵

▼❛❦✐♥❣ ❛ ✢❡①✐❜❧❡ t♦♦❧❜♦① ❢♦r s♦❧✈✐♥❣ ❖❉❊s

❋♦r s❝❛❧❛r ❖❉❊s ✇❡ ❝♦✉❧❞ ♠❛❦❡ ♦♥❡ ❣❡♥❡r❛❧ ❝❧❛ss ❤✐❡r❛r❝❤② t♦ s♦❧✈❡ ✏❛❧❧✑ ♣r♦❜❧❡♠s ✇✐t❤ ❛ r❛♥❣❡ ♦❢ ♠❡t❤♦❞s ❈❛♥ ✇❡ ❡❛s✐❧② ❡①t❡♥❞ ❝❧❛ss ❤✐❡r❛r❝❤② t♦ s②st❡♠s ♦❢ ❖❉❊s❄ ❨❡s✦ ❚❤❡ ❡①❛♠♣❧❡ ❤❡r❡ ❝❛♥ ❡❛s✐❧② ❜❡ ❡①t❡♥❞❡❞ t♦ ♣r♦❢❡ss✐♦♥❛❧ ❝♦❞❡ ✭❖❞❡s♣②✮

❱❡❝t♦r ♥♦t❛t✐♦♥ ❢♦r s②st❡♠s ♦❢ ❖❉❊s✿ ✉♥❦♥♦✇♥s ❛♥❞ ❡q✉❛t✐♦♥s

• ❡♥❡r❛❧ s♦❢t✇❛r❡ ❢♦r ❛♥② ✈❡❝t♦r✴s❝❛❧❛r ❖❉❊ ❞❡♠❛♥❞s ❛ ❣❡♥❡r❛❧

♠❛t❤❡♠❛t✐❝❛❧ ♥♦t❛t✐♦♥✳ ❲❡ ✐♥tr♦❞✉❝❡ n ✉♥❦♥♦✇♥s u(✵)(t), u(✶)(t), . . . , u(n−✶)(t) ✐♥ ❛ s②st❡♠ ♦❢ n ❖❉❊s✿ d dt u(✵) = f (✵)(u(✵), u(✶), . . . , u(n−✶), t) d dt u(✶) = f (✶)(u(✵), u(✶), . . . , u(n−✶), t) ✳ ✳ ✳ = ✳ ✳ ✳ d dt u(n−✶) = f (n−✶)(u(✵), u(✶), . . . , u(n−✶), t)

❱❡❝t♦r ♥♦t❛t✐♦♥ ❢♦r s②st❡♠s ♦❢ ❖❉❊s✿ ✈❡❝t♦rs

❲❡ ❝❛♥ ❝♦❧❧❡❝t t❤❡ u(i)(t) ❢✉♥❝t✐♦♥s ❛♥❞ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢✉♥❝t✐♦♥s f (i) ✐♥ ✈❡❝t♦rs✿ u = (u(✵), u(✶), . . . , u(n−✶)) f = (f (✵), f (✶), . . . , f (n−✶)) ❚❤❡ ✜rst✲♦r❞❡r s②st❡♠ ❝❛♥ t❤❡♥ ❜❡ ✇r✐tt❡♥ u′ = f (u, t), u(✵) = U✵ ✇❤❡r❡ u ❛♥❞ f ❛r❡ ✈❡❝t♦rs ❛♥❞ U✵ ✐s ❛ ✈❡❝t♦r ♦❢ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❚❤❡ ♠❛❣✐❝ ♦❢ t❤✐s ♥♦t❛t✐♦♥✿ ❖❜s❡r✈❡ t❤❛t t❤❡ ♥♦t❛t✐♦♥ ♠❛❦❡s ❛ s❝❛❧❛r ❖❉❊ ❛♥❞ ❛ s②st❡♠ ❧♦♦❦ t❤❡ s❛♠❡✱ ❛♥❞ ✇❡ ❝❛♥ ❡❛s✐❧② ♠❛❦❡ P②t❤♦♥ ❝♦❞❡ t❤❛t ❝❛♥ ❤❛♥❞❧❡ ❜♦t❤ ❝❛s❡s ✇✐t❤✐♥ t❤❡ s❛♠❡ ❧✐♥❡s ♦❢ ❝♦❞❡ ✭✦✮

❍♦✇ t♦ ♠❛❦❡ ❝❧❛ss ❖❉❊❙♦❧✈❡r ✇♦r❦ ❢♦r s②st❡♠s ♦❢ ❖❉❊s

❘❡❝❛❧❧✿ ❖❉❊❙♦❧✈❡r ✇❛s ✇r✐tt❡♥ ❢♦r ❛ s❝❛❧❛r ❖❉❊ ◆♦✇ ✇❡ ✇❛♥t ✐t t♦ ✇♦r❦ ❢♦r ❛ s②st❡♠ u′ = f ✱ u(✵) = U✵✱ ✇❤❡r❡ u✱ f ❛♥❞ U✵ ❛r❡ ✈❡❝t♦rs ✭❛rr❛②s✮ ❲❤❛t ❛r❡ t❤❡ ♣r♦❜❧❡♠s❄ ❋♦r✇❛r❞ ❊✉❧❡r ❛♣♣❧✐❡❞ t♦ ❛ s②st❡♠✿ uk+✶

• ✈❡❝t♦r

= uk

• ✈❡❝t♦r

+∆t f (uk, tk)

• ✈❡❝t♦r

■♥ P②t❤♦♥ ❝♦❞❡✿

✉♥❡✇ ❂ ✉❬❦❪ ✰ ❞t✯❢✭✉❬❦❪✱ t✮

✇❤❡r❡ ✉ ✐s ❛ t✇♦✲❞✐♠✳ ❛rr❛② ✭✉❬❦❪ ✐s ❛ r♦✇✮ ❢ ✐s ❛ ❢✉♥❝t✐♦♥ r❡t✉r♥✐♥❣ ❛♥ ❛rr❛② ✭❛❧❧ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡s f (✵), . . . , f (n−✶)✮

❚❤❡ ❛❞❥✉st❡❞ s✉♣❡r❝❧❛ss ❝♦❞❡ ✭♣❛rt ✶✮

❚♦ ♠❛❦❡ ❖❉❊❙♦❧✈❡r ✇♦r❦ ❢♦r s②st❡♠s✿ ❊♥s✉r❡ t❤❛t ❢✭✉✱t✮ r❡t✉r♥s ❛♥ ❛rr❛②✳ ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❜❡ ❛ ❣❡♥❡r❛❧ ❛❞❥✉st♠❡♥t ✐♥ t❤❡ s✉♣❡r❝❧❛ss✦ ■♥s♣❡❝t U✵ t♦ s❡❡ ✐❢ ✐t ✐s ❛ ♥✉♠❜❡r ♦r ❧✐st✴t✉♣❧❡ ❛♥❞ ♠❛❦❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✉ ✶✲❞✐♠ ♦r ✷✲❞✐♠ ❛rr❛②

❝❧❛ss ❖❉❊❙♦❧✈❡r✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❢✮✿ ★ ❲r❛♣ ✉s❡r✬s ❢ ✐♥ ❛ ♥❡✇ ❢✉♥❝t✐♦♥ t❤❛t ❛❧✇❛②s ★ ❝♦♥✈❡rts ❧✐st✴t✉♣❧❡ t♦ ❛rr❛② ✭♦r ❧❡t ❛rr❛② ❜❡ ❛rr❛②✮ s❡❧❢✳❢ ❂ ❧❛♠❜❞❛ ✉✱ t✿ ♥♣✳❛s❛rr❛②✭❢✭✉✱ t✮✱ ❢❧♦❛t✮ ❞❡❢ s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭s❡❧❢✱ ❯✵✮✿ ✐❢ ✐s✐♥st❛♥❝❡✭❯✵✱ ✭❢❧♦❛t✱✐♥t✮✮✿ ★ s❝❛❧❛r ❖❉❊ s❡❧❢✳♥❡q ❂ ✶ ★ ♥♦ ♦❢ ❡q✉❛t✐♦♥s ❯✵ ❂ ❢❧♦❛t✭❯✵✮ ❡❧s❡✿ ★ s②st❡♠ ♦❢ ❖❉❊s ❯✵ ❂ ♥♣✳❛s❛rr❛②✭❯✵✮ s❡❧❢✳♥❡q ❂ ❯✵✳s✐③❡ ★ ♥♦ ♦❢ ❡q✉❛t✐♦♥s s❡❧❢✳❯✵ ❂ ❯✵

❚❤❡ s✉♣❡r❝❧❛ss ❝♦❞❡ ✭♣❛rt ✷✮

❝❧❛ss ❖❉❊❙♦❧✈❡r✿ ✳✳✳ ❞❡❢ s♦❧✈❡✭s❡❧❢✱ t✐♠❡❴♣♦✐♥ts✱ t❡r♠✐♥❛t❡❂◆♦♥❡✮✿ ✐❢ t❡r♠✐♥❛t❡ ✐s ◆♦♥❡✿ t❡r♠✐♥❛t❡ ❂ ❧❛♠❜❞❛ ✉✱ t✱ st❡♣❴♥♦✿ ❋❛❧s❡ s❡❧❢✳t ❂ ♥♣✳❛s❛rr❛②✭t✐♠❡❴♣♦✐♥ts✮ ♥ ❂ s❡❧❢✳t✳s✐③❡ ✐❢ s❡❧❢✳♥❡q ❂❂ ✶✿ ★ s❝❛❧❛r ❖❉❊s s❡❧❢✳✉ ❂ ♥♣✳③❡r♦s✭♥✮ ❡❧s❡✿ ★ s②st❡♠s ♦❢ ❖❉❊s s❡❧❢✳✉ ❂ ♥♣✳③❡r♦s✭✭♥✱s❡❧❢✳♥❡q✮✮ ★ ❆ss✉♠❡ t❤❛t s❡❧❢✳t❬✵❪ ❝♦rr❡s♣♦♥❞s t♦ s❡❧❢✳❯✵ s❡❧❢✳✉❬✵❪ ❂ s❡❧❢✳❯✵ ★ ❚✐♠❡ ❧♦♦♣ ❢♦r ❦ ✐♥ r❛♥❣❡✭♥✲✶✮✿ s❡❧❢✳❦ ❂ ❦ s❡❧❢✳✉❬❦✰✶❪ ❂ s❡❧❢✳❛❞✈❛♥❝❡✭✮ ✐❢ t❡r♠✐♥❛t❡✭s❡❧❢✳✉✱ s❡❧❢✳t✱ s❡❧❢✳❦✰✶✮✿ ❜r❡❛❦ ★ t❡r♠✐♥❛t❡ ❧♦♦♣ ♦✈❡r ❦ r❡t✉r♥ s❡❧❢✳✉❬✿❦✰✷❪✱ s❡❧❢✳t❬✿❦✰✷❪

❆❧❧ s✉❜❝❧❛ss❡s ❢r♦♠ t❤❡ s❝❛❧❛r ❖❉❊ ✇♦r❦s ❢♦r s②st❡♠s ❛s ✇❡❧❧

❊①❛♠♣❧❡ ♦♥ ❤♦✇ t♦ ✉s❡ t❤❡ ❣❡♥❡r❛❧ ❝❧❛ss ❤✐❡r❛r❝❤②

❙♣r✐♥❣✲♠❛ss s②st❡♠ ❢♦r♠✉❧❛t❡❞ ❛s ❛ s②st❡♠ ♦❢ ❖❉❊s✿ mu′′ + βu′ + ku = ✵, u(✵), u′(✵) ❦♥♦✇♥ u(✵) = u, u(✶) = u′ u(t) = (u(✵)(t), u(✶)(t)) f (u, t) = (u(✶)(t), −m−✶βu(✶) − m−✶ku(✵)) u′(t) = f (u, t) ❈♦❞❡ ❞❡✜♥✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

❞❡❢ ♠②❢✭✉✱ t✮✿ ★ ✉ ✐s ❛rr❛② ✇✐t❤ t✇♦ ❝♦♠♣♦♥❡♥ts ✉❬✵❪ ❛♥❞ ✉❬✶❪✿ r❡t✉r♥ ❬✉❬✶❪✱ ✲❜❡t❛✯✉❬✶❪✴♠ ✲ ❦✯✉❬✵❪✴♠❪

❆❧t❡r♥❛t✐✈❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ f ❢✉♥❝t✐♦♥ ✈✐❛ ❛ ❝❧❛ss

❇❡tt❡r ✭♥♦ ❣❧♦❜❛❧ ✈❛r✐❛❜❧❡s✮✿

❝❧❛ss ▼②❋✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ♠✱ ❦✱ ❜❡t❛✮✿ s❡❧❢✳♠✱ s❡❧❢✳❦✱ s❡❧❢✳❜❡t❛ ❂ ♠✱ ❦✱ ❜❡t❛ ❞❡❢ ❴❴❝❛❧❧❴❴✭s❡❧❢✱ ✉✱ t✮✿ ♠✱ ❦✱ ❜❡t❛ ❂ s❡❧❢✳♠✱ s❡❧❢✳❦✱ s❡❧❢✳❜❡t❛ r❡t✉r♥ ❬✉❬✶❪✱ ✲❜❡t❛✯✉❬✶❪✴♠ ✲ ❦✯✉❬✵❪✴♠❪

▼❛✐♥ ♣r♦❣r❛♠✿

❢r♦♠ ❖❉❊❙♦❧✈❡r ✐♠♣♦rt ❋♦r✇❛r❞❊✉❧❡r ★ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿ ❯✵ ❂ ❬✶✳✵✱ ✵❪ ❢ ❂ ▼②❋✭✶✳✵✱ ✶✳✵✱ ✵✳✵✮ ★ ✉✬✬ ✰ ✉ ❂ ✵ ❂❃ ✉✭t✮❂❝♦s✭t✮ s♦❧✈❡r ❂ ❋♦r✇❛r❞❊✉❧❡r✭❢✮ s♦❧✈❡r✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭❯✵✮ ❚ ❂ ✹✯♣✐❀ ❞t ❂ ♣✐✴✷✵❀ ♥ ❂ ✐♥t✭r♦✉♥❞✭❚✴❞t✮✮ t✐♠❡❴♣♦✐♥ts ❂ ♥♣✳❧✐♥s♣❛❝❡✭✵✱ ❚✱ ♥✰✶✮ ✉✱ t ❂ s♦❧✈❡r✳s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts✮ ★ ✉ ✐s ❛♥ ❛rr❛② ♦❢ ❬✉✵✱✉✶❪ ❛rr❛②s✱ ♣❧♦t ❛❧❧ ✉✵ ✈❛❧✉❡s✿ ✉✵❴✈❛❧✉❡s ❂ ✉❬✿✱✵❪ ✉✵❴❡①❛❝t ❂ ❝♦s✭t✮ ♣❧♦t✭t✱ ✉✵❴✈❛❧✉❡s✱ ✬r✲✬✱ t✱ ✉✵❴❡①❛❝t✱ ✬❜✲✬✮

❚❤r♦✇✐♥❣ ❛ ❜❛❧❧❀ ❖❉❊ ♠♦❞❡❧

◆❡✇t♦♥✬s ✷♥❞ ❧❛✇ ❢♦r ❛ ❜❛❧❧✬s tr❛❥❡❝t♦r② t❤r♦✉❣❤ ❛✐r ❧❡❛❞s t♦ dx dt = vx dvx dt = ✵ dy dt = vy dvy dt = −g ❆✐r r❡s✐st❛♥❝❡ ✐s ♥❡❣❧❡❝t❡❞ ❜✉t ❝❛♥ ❡❛s✐❧② ❜❡ ❛❞❞❡❞✦ ✹ ❖❉❊s ✇✐t❤ ✹ ✉♥❦♥♦✇♥s✿

t❤❡ ❜❛❧❧✬s ♣♦s✐t✐♦♥ x(t)✱ y(t) t❤❡ ✈❡❧♦❝✐t② vx(t)✱ vy(t)

❚❤r♦✇✐♥❣ ❛ ❜❛❧❧❀ ❝♦❞❡

❉❡✜♥❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿

❞❡❢ ❢✭✉✱ t✮✿ ①✱ ✈①✱ ②✱ ✈② ❂ ✉ ❣ ❂ ✾✳✽✶ r❡t✉r♥ ❬✈①✱ ✵✱ ✈②✱ ✲❣❪

▼❛✐♥ ♣r♦❣r❛♠✿

❢r♦♠ ❖❉❊❙♦❧✈❡r ✐♠♣♦rt ❋♦r✇❛r❞❊✉❧❡r ★ t❂✵✿ ♣r❡s❝r✐❜❡ ①✱ ②✱ ✈①✱ ✈② ① ❂ ② ❂ ✵ ★ st❛rt ❛t t❤❡ ♦r✐❣✐♥ ✈✵ ❂ ✺❀ t❤❡t❛ ❂ ✽✵✯♣✐✴✶✽✵ ★ ✈❡❧♦❝✐t② ♠❛❣♥✐t✉❞❡ ❛♥❞ ❛♥❣❧❡ ✈① ❂ ✈✵✯❝♦s✭t❤❡t❛✮ ✈② ❂ ✈✵✯s✐♥✭t❤❡t❛✮ ★ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿ ❯✵ ❂ ❬①✱ ✈①✱ ②✱ ✈②❪ s♦❧✈❡r❂ ❋♦r✇❛r❞❊✉❧❡r✭❢✮ s♦❧✈❡r✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭✉✵✮ t✐♠❡❴♣♦✐♥ts ❂ ♥♣✳❧✐♥s♣❛❝❡✭✵✱ ✶✳✷✱ ✶✵✶✮ ✉✱ t ❂ s♦❧✈❡r✳s♦❧✈❡✭t✐♠❡❴♣♦✐♥ts✮ ★ ✉ ✐s ❛♥ ❛rr❛② ♦❢ ❬①✱✈①✱②✱✈②❪ ❛rr❛②s✱ ♣❧♦t ② ✈s ①✿ ① ❂ ✉❬✿✱✵❪❀ ② ❂ ✉❬✿✱✷❪ ♣❧♦t✭①✱ ②✮

❚❤r♦✇✐♥❣ ❛ ❜❛❧❧❀ r❡s✉❧ts

❈♦♠♣❛r✐s♦♥ ♦❢ ❡①❛❝t ❛♥❞ ❋♦r✇❛r❞ ❊✉❧❡r s♦❧✉t✐♦♥s

• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 dt=0.01 numerical exact

▲♦❣✐st✐❝ ❣r♦✇t❤ ♠♦❞❡❧❀ ❖❉❊ ❛♥❞ ❝♦❞❡ ♦✈❡r✈✐❡✇

▼♦❞❡❧✿ u′ = αu(✶ − u/R(t)), u(✵) = U✵ R ✐s t❤❡ ♠❛①✐♠✉♠ ♣♦♣✉❧❛t✐♦♥ s✐③❡✱ ✇❤✐❝❤ ❝❛♥ ✈❛r② ✇✐t❤ ❝❤❛♥❣❡s ✐♥ t❤❡ ❡♥✈✐r♦♥♠❡♥t ♦✈❡r t✐♠❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❢❡❛t✉r❡s✿ ❈❧❛ss Pr♦❜❧❡♠ ❤♦❧❞s ✏❛❧❧ ♣❤②s✐❝s✑✿ α✱ R(t)✱ U✵✱ T ✭❡♥❞ t✐♠❡✮✱ f (u, t) ✐♥ ❖❉❊ ❈❧❛ss ❙♦❧✈❡r ❤♦❧❞s ✏❛❧❧ ♥✉♠❡r✐❝s✑✿ ∆t✱ s♦❧✉t✐♦♥ ♠❡t❤♦❞❀ s♦❧✈❡s t❤❡ ♣r♦❜❧❡♠ ❛♥❞ ♣❧♦ts t❤❡ s♦❧✉t✐♦♥ ❙♦❧✈❡ ❢♦r t ∈ [✵, T] ❜✉t t❡r♠✐♥❛t❡ ✇❤❡♥ |u − R| < t♦❧

▲♦❣✐st✐❝ ❣r♦✇t❤ ♠♦❞❡❧❀ ❝❧❛ss Pr♦❜❧❡♠ ✭f ✮

❝❧❛ss Pr♦❜❧❡♠✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ❛❧♣❤❛✱ ❘✱ ❯✵✱ ❚✮✿ s❡❧❢✳❛❧♣❤❛✱ s❡❧❢✳❘✱ s❡❧❢✳❯✵✱ s❡❧❢✳❚ ❂ ❛❧♣❤❛✱ ❘✱ ❯✵✱ ❚ ❞❡❢ ❴❴❝❛❧❧❴❴✭s❡❧❢✱ ✉✱ t✮✿ ✧✧✧❘❡t✉r♥ ❢✭✉✱ t✮✳✧✧✧ r❡t✉r♥ s❡❧❢✳❛❧♣❤❛✯✉✯✭✶ ✲ ✉✴s❡❧❢✳❘✭t✮✮ ❞❡❢ t❡r♠✐♥❛t❡✭s❡❧❢✱ ✉✱ t✱ st❡♣❴♥♦✮✿ ✧✧✧❚❡r♠✐♥❛t❡ ✇❤❡♥ ✉ ✐s ❝❧♦s❡ t♦ ❘✳✧✧✧ t♦❧ ❂ s❡❧❢✳❘✯✵✳✵✶ r❡t✉r♥ ❛❜s✭✉❬st❡♣❴♥♦❪ ✲ s❡❧❢✳❘✮ ❁ t♦❧ ♣r♦❜❧❡♠ ❂ Pr♦❜❧❡♠✭❛❧♣❤❛❂✵✳✶✱ ❘❂✺✵✵✱ ❯✵❂✷✱ ❚❂✶✸✵✮

▲♦❣✐st✐❝ ❣r♦✇t❤ ♠♦❞❡❧❀ ❝❧❛ss ❙♦❧✈❡r

❝❧❛ss ❙♦❧✈❡r✿ ❞❡❢ ❴❴✐♥✐t❴❴✭s❡❧❢✱ ♣r♦❜❧❡♠✱ ❞t✱ ♠❡t❤♦❞❂❖❉❊❙♦❧✈❡r✳❋♦r✇❛r❞❊✉❧❡r✮✿ s❡❧❢✳♣r♦❜❧❡♠✱ s❡❧❢✳❞t ❂ ♣r♦❜❧❡♠✱ ❞t s❡❧❢✳♠❡t❤♦❞ ❂ ♠❡t❤♦❞ ❞❡❢ s♦❧✈❡✭s❡❧❢✮✿ s♦❧✈❡r ❂ s❡❧❢✳♠❡t❤♦❞✭s❡❧❢✳♣r♦❜❧❡♠✮ s♦❧✈❡r✳s❡t❴✐♥✐t✐❛❧❴❝♦♥❞✐t✐♦♥✭s❡❧❢✳♣r♦❜❧❡♠✳❯✵✮ ♥ ❂ ✐♥t✭r♦✉♥❞✭s❡❧❢✳♣r♦❜❧❡♠✳❚✴s❡❧❢✳❞t✮✮ t❴♣♦✐♥ts ❂ ♥♣✳❧✐♥s♣❛❝❡✭✵✱ s❡❧❢✳♣r♦❜❧❡♠✳❚✱ ♥✰✶✮ s❡❧❢✳✉✱ s❡❧❢✳t ❂ s♦❧✈❡r✳s♦❧✈❡✭t❴♣♦✐♥ts✱ s❡❧❢✳♣r♦❜❧❡♠✳t❡r♠✐♥❛t❡✮ ❞❡❢ ♣❧♦t✭s❡❧❢✮✿ ♣❧♦t✭s❡❧❢✳t✱ s❡❧❢✳✉✮ ♣r♦❜❧❡♠ ❂ Pr♦❜❧❡♠✭❛❧♣❤❛❂✵✳✶✱ ❯✵❂✷✱ ❚❂✶✸✵✱ ❘❂❧❛♠❜❞❛ t✿ ✺✵✵ ✐❢ t ❁ ✻✵ ❡❧s❡ ✶✵✵✮ s♦❧✈❡r ❂ ❙♦❧✈❡r✭♣r♦❜❧❡♠✱ ❞t❂✶✳✮ s♦❧✈❡r✳s♦❧✈❡✭✮ s♦❧✈❡r✳♣❧♦t✭✮ ♣r✐♥t ✬♠❛① ✉✿✬✱ s♦❧✈❡r✳✉✳♠❛①✭✮

▲♦❣✐st✐❝ ❣r♦✇t❤ ♠♦❞❡❧❀ r❡s✉❧ts

50 100 150 200 250 300 350 20 40 60 80 100 120 alpha=0.1, U0=2, dt=0.0253906