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❯♥✐✈❡r③✐t❡t ✉ ❚✉③❧✐

Pr✐r♦❞♥♦ ✲ ♠❛t❡♠❛t✐↔❦✐ ❢❛❦✉❧t❡t

❖❞s❥❡❦✿ ▼❛t❡♠❛t✐❦❛

Pr❡❞♠❡t✿ ◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛

❙❊▼■◆❆❘❙❑■ ❘❆❉

❙t✉❞❡♥t✿

❉❛✈♦r ❇❡❣❛♥♦✈✐➣

▼❡♥t♦r✿

❉r✳s❝✳❊♥❡s ❉✉✈♥❥❛❦♦✈✐➣✱ ✈❛♥r✳♣r♦❢✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✶

❙❛❞r➸❛❥

✶ ❩❛❞❛t❛❦ ✶✳ ✷

✶✳✶ ▼❡t♦❞❛ s❥❡↔✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ Pr✐♠❥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❑♦❞ ✉ ▼❛t❤❡♠❛t✐❝❛✲✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ ❑♦♠❡♥t❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✷ ❩❛❞❛t❛❦ ✷✳ ✻

✷✳✶ ◆❡✇t♦♥✲♦✈❛ ♠❡t♦❞❛ ③❛ r❥❡➨❛✈❛♥❥❡ s✐st❡♠❛ ♥❡❧✐♥❡❛r♥✐❤ ❛❧❣❡❜❛rs❦✐❤ ❥❡❞♥❛↔✐♥❛ ✻ ✷✳✷ Pr✐♠❥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❑♦❞ ✉ ▼❛t❤❡♠❛t✐❝❛✲✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ ❑♦♠❡♥t❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✸ ❩❛❞❛t❛❦ ✸✳ ✾

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✷

✶ ❩❛❞❛t❛❦ ✶✳

✶✳✶ ▼❡t♦❞❛ s❥❡↔✐❝❡

▼❡t♦❞❛ s❥❡↔✐❝❡ ❥❡ ♠♦❞✐✜❦❛❝✐❥❛ ◆❡✇t♦♥✲♦✈❡ ♠❡t♦❞❡✳ ❩❛ ♠❡t♦❞✉ s❥❡↔✐❝❡ ✈r✐❥❡❞❡ s❧❥❡❞❡➣❡ ♣r❡t♣♦st❛✈❦❡✿

• ◆❡❦❛ ♣♦st♦❥✐ x∈[a, b] t❛❦❛✈ ❞❛ ❥❡f(x) = 0✳

• ◆❡❦❛ ❥❡ ❢✉♥❦❝✐❥❛ f ♥❡♣r❡❦✐❞♥❛ ♥❛[a, b]✳

• ◆❡❦❛ ❥❡ f(a)·f(b)<0✳

❯③♠✐♠♦ s❛❞❛ ❞✈✐❥❡ t❛↔❦❡x0✐x1✐③[a, b]✱ t❛❞❛ ✐♠❛♠♦ ❞✈✐❥❡ t❛↔❦❡M0(x0, f(x0))✐M1(x1, f(x1))

♥❛ ❦r✐✈♦❥ ❢✉♥❦❝✐❥❡✳ ◆❡❦❛ ❥❡ ❦♦❞ ♥❛sx0 =b✳ ❙❛❞❛ ❛❦♦ ♣♦✈✉↔❡♠♦ s❥❡❦❛♥t✉ ❦r♦③f(x0) = f(b)

✐ ❦r♦③ f(x1)✐♠❛t ➣❡♠♦ ❦❧❛s✐↔❛♥ ♣r♦❞♦r ♣r❛✈❡ ❦r♦③ ❞✈✐❥❡ t❛↔❦❡ ✐ ❛❦♦ ✉♣♦tr✐❥❡❜✐♠♦ ❢♦r♠✉❧✉

③❛ ♣r♦❞♦r ♣r❛✈❡ ❦r♦③ ❞✈✐❥❡ t❛↔❦❡ ✐♠❛♠♦✿

s:f1(x) =f(x0) +

f(x1)−f(x0)

x1−x0

(x−x0).

❙ ♦❜③✐r♦♠ ❞❛ ❥❡ f1(x) = 0 ✐♠❛♠♦

f(x0) +

f(x1)−f(x0)

x1−x0

(x−x0) = 0

s❛❞❛ ❛❦♦ s✈❡ ♣♦♠♥♦➸✐♠♦ s❛ x1−x0 ❞❛❧❥❡ ➣❡♠♦ ✐♠❛t✐

(x1−x0)f(x0) + (f(x1)−f(x0))(x−x0) = 0

s❛ ❥♦➨ ♠❛❧♦ sr❡➒✐✈❛♥❥❛ ♥❛ ❦r❛❥✉ ❞♦❜✐❥❛♠♦

x= x0f(x1)−x1f(x0)

f(x1)−f(x0)

♣r✐ ↔❡♠✉ ❥❡ f(x0)6=f(x1)✳ ❆❦♦ ♥❛♣r❛✈✐♠♦ ♥✐③ ♦✈❛❦✈✐❤ ❥❡❞♥❛❦♦st✐

x2 =

x0f(x1)−x1f(x0)

f(x1)−f(x0)

x3 =

x1f(x2)−x2f(x1)

f(x2)−f(x1)

✐ t❛❦♦ ♥❛st❛✈✐♠♦✱ ♥❛ ❦r❛❥✉ ➣❡♠♦ ❞♦❜✐t✐

xn+1 =

xn₋1f(xn)−xnf(xn₋1)

f(xn)−f(xn−1)

♣r✐ ↔❡♠✉ ❥❡ f(xn−1)6=f(xn)✳ ❩❛ ♦✈❛❥ ♥✐③ ✈r✐❥❡❞✐

lim

n→∞xn =ξ

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✸

❙❧✐❦❛ ✶✿ ▼❡t♦❞❛ s❥❡↔✐❝❡

❑❛♦ ➨t♦ s♠♦ r❡❦❧✐ ♠❡t♦❞❛ s❥❡↔✐❝❡ ❥❡ ♠♦❞✐✜❦❛❝✐❥❛ ◆❡✇t♦♥✲♦✈❡ ♠❡t♦❞❡✳ ❙❛❞❛ ➣❡♠♦ ♣♦❦❛③❛t✐ ✐ ③❜♦❣ ↔❡❣❛✳

❩♥❛♠♦ ❞❛ ❥❡ r❡❦✉r③✐✈♥❛ ❢♦r♠✉❧❛ ③❛ ◆❡✇t♦♥✲♦✈✉ ♠❡t♦❞✉

xn+1 =xn−

f(xn)

f′_(xn). ✭✶✮

❙❛❞❛ ❛❦♦ ✉③♠❡♠♦ ❞❛ ❥❡

f′_{(xn) = lim}

xn₋1→xn

f(xn−f(xn−1))

xn−xn−1

✐♠❛♠♦ ❞❛ ❥❡

f′_(xn)≈ f(xn)−f(xn−1)

xn−xn−1

.

❆❦♦ ♣♦s❧❥❡❞♥❥✉ ❛♣r♦❦s✐♠❛❝✐❥✉ f′_(xn)✉❜❛❝✐♠♦ ✉ ✭✶✮ ❞♦❜✐t ➣❡♠♦

xn+1 =xn−

f(xn) f(xn)−f(xn

−1)

xn−xn₋1

✐ s❛❞❛ ❦❛❞❛ t♦ ♠❛❧♦ sr❡❞✐♠♦ ❞♦❜✐❥❛♠♦

xn+1 =

xn₋1f(xn)−xnf(xn₋1)

f(xn)−f(xn−1)

➨t♦ ❥❡ ✉st✈❛r✐ ❢♦r♠✉❧❛ ③❛ ♠❡t♦❞✉ s❥❡↔✐❝❡ ✐ ♥❛r❛✈♥♦ ❞❛ ♠♦r❛ ✈r✐❥❡❞✐t✐ f(xn)6=f(xn₋1)✳

✶✳✷ Pr✐♠❥❡r

▼❡t♦❞♦♠ s❥❡↔✐❝❡ s❛ t❛↔♥♦➨➣✉ ✈❡➣♦♠ ♦❞ ε= 10−4 _{r✐❥❡➨✐t✐ ❥❡❞♥❛↔✐♥✉ xe}x_{−1 = 0.}

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✺

✶✳✹ ❑♦♠❡♥t❛r

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✽

✷✳✹ ❑♦♠❡♥t❛r

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✾

✸ ❩❛❞❛t❛❦ ✸✳

✸✳✶ ▼❡t♦❞❛ ❘✉♥❣❡✲❑✉tt❛ ③❛ r❥❡➨❛✈❛♥❥❡ ❞✐❢❡r❡♥❝✐❥❛❧♥✐❤ ❥❡❞♥❛↔✐♥❛

P♦s♠❛tr❛❥♠♦ ❈❛✉❝❤②✲❡✈ ♣r♦❜❧❡♠

y′ _{=f(x, y), y(x}

0) =y0.

❏❡❞♥❛ ♦❞ ♥❛❥✈❛➸♥✐❥✐ ♠❡t♦❞❛ ③❛ r❥❡➨❛✈❛♥❥❡ ♦✈♦❣ ❈❛✉❝❤②✲❡✈♦❣ ♣r♦❜❧❡♠ ❥❡ ♠❡t♦❞❛ ❘✉♥❣❡✲ ❑✉tt❛ ✭❘❑✮✶_{✳ Pr❡t♣♦st❛✈✐♠♦ ❞❛ ♣♦③♥❛❥❡♠♦ ❛♣r♦❦s✐♠❛❝✐❥✉y}

n tr❛➸❡♥❡ ❢✉♥❦❝✐❥❡x7−→y(x) ✉ t❛↔❦✐ xn✳ ➎❡❧✐♠♦ ♦❞r❡❞✐t✐ (n+ 1)✲✈✉ ❛♣r♦❦s✐♠❛❝✐❥✉ yn+1 ✉ t❛↔❦✐xn+h✳ ❯ t✉ s✈r❤✉ ♥❛ ✐♥t❡r✈❛❧✉ (xn, xn+h) ✉ ♥❡❦♦❧✐❦♦ str❛t❡➨❦✐❤ t❛↔❛❦❛ ❛♣r♦❦s✐♠✐r❛t ➣❡♠♦ ✈r✐❥❡❞♥♦st ❢✉♥❦❝✐❥❡

x7−→f(x, y(x))✱ t❡ ♣♦♠♦➣✉ ♥❥✐❤ ➨t♦ ❜♦❧❥❡ ❛♣r♦❦s✐♠✐r❛t✐ r❛③❧✐❦✉yn+1−yn✳

◆❛❥❥❡❞♥♦st❛✈♥✐❥✐ ♣r✐♠❥❡r ✐③ ❢❛♠✐❧✐❥❡ ❘❑ ♠❡t♦❞❛ ❥❡ t③✈ ❍❡✉♥✲♦✈❛ ♠❡t♦❞❛

yn+1 =yn+ 1

2(k1 +k2) ❣❞❥❡ ❥❡ k1 =hf(xn, yn)✱ ❛ k2 =hf(xn+h, yn+k1)✳

Pr✐♠❥❡❞❜❛ ✸✳✶

❑❛❦♦ ❥❡

y(xn+1)−y(xn) =

Z xn+h

xn

dy dxdx=

Z xn+h

xn

f(x, y(x))dx

❛❦♦ ♣r❡t♣♦st❛✈✐♠♦ ❞❛ ❥❡fs❛♠♦ ❢✉♥❦❝✐❥❛ ♦❞x✱ ❍❡✉♥♦✈❛ ♠❡t♦❞❛ ♦❞❣♦✈❛r❛ tr❛♣❡③♥♦♠

♣r❛✈✐❧✉✱ ❦♦❥❛ ✐♠❛ ♣♦❣r❡➨❦✉ r❡❞❛ ✈❡❧✐↔✐♥❡O(h2_{)✳ Pr✐♠✐❥❡t✐♠♦ t❛❦♦➒❡r ❞❛ ❥❡ ③❛ s✈❛❦✉}

❛♣r♦❦s✐♠❛❝✐❥✉ yn ♣♦r❡❜♥♦ ❞✈❛ ♣✉t❛ ✐③r❛↔✉♥❛✈❛t✐ ✈r✐❥❡❞♥♦st ❢✉♥❦❝✐❥❡f✳

❑❧❛s✐↔♥❛ ❘❑ ♠❡t♦❞❛ ❞❡✜♥✐r❛♥❛ ❥❡ s❛

yn+1 =yn+ 1

6(k1+ 2k2 + 2k3+k4)

❣❞❥❡ s✉ k1 = hf(xn, yn)✱ k2 = hf(xn + h₂, yn + k₂1)✱ k3 = hf(xn + h₂, yn + k₂2) ✐ k4 =

hf(xn+h, yn+k3)✳

Pr✐♠❥❡❞❜❛ ✸✳✷

❆❦♦ ♣r❡t♣♦st❛✈✐♠♦ ❞❛ ❥❡f s❛♠♦ ❢✉♥❦❝✐❥❛ ♦❞ x✱ ♦♥❞❛ ✐③ ♣r✐♠❥❡❞❜❡ ✸✳✶ ♠♦➸❡♠♦ ♣♦✲

❦❛③❛t✐ ❞❛ ❘❑✲♠❡t♦❞❛ ♦❞❣♦✈❛r❛ ❙✐♠♣s♦♥✲♦✈♦❥ ❢♦r♠✉❧✐✱ ✉③ ③❛♠❥❡♥✉h7−→ h

2✳ ❙❥❡t✐♠♦

s❡ ❞❛ ❙✐♠♣s♦♥♦✈❛ ❢♦r♠✉❧❛ ✐♠❛ ♣♦❣r❡➨❦✉ r❡❞❛ ✈❡❧✐↔✐♥❡O(h2_{)✱ ➨t♦ s❡ ♣r❡♥♦s✐ ✐ ♥❛ ❘❑}

♠❡t♦❞✉ ✐ ✉ ♦♣➣❡♠ s❧✉↔❛❥✉ ✕ ❦❛❞❛ ❥❡ f ❢✉♥❦❝✐❥❛ ♦❞ x ✐ ♦❞ y✳ Pr✐♠✐❥❡t✐♠♦ t❛❦♦➒❡r

❞❛ ❥❡ ❦♦❞ ❘❑ ♠❡t♦❞❡ ③❛ s✈❛❦✉ ❛♣r♦❦s✐♠❛❝✐❥✉yn ♣♦tr❡❜♥♦ ↔❡t✐r✐ ♣✉t❛ ✐③r❛↔✉♥❛✈❛t✐ ✈r✐❥❡❞♥♦st ❢✉♥❦❝✐❥❡f✳

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✶✵

✸✳✷ Pr✐♠❥❡r

▼❡t♦❞♦♠ ❘✉♥❣❡ ✲ ❑✉tt❛ r✐❥❡➨✐t✐ ❈❛✉❝❤②✲❡✈ ♣r♦❜❧❡♠

y′ _=x2_−ex

+y, y(0) = 1

③❛ x∈[0,0.3]✐ ♦❝✐❥❡♥✐t✐ ❣r❡➨❦✉✳

❘❥❡➨❡♥❥❡✿ Pr✐♠✐❥❡t✐♠♦ ❞❛ ♥❛♠ ❥❡ x ∈ [0,1]✳ ❯③♠✐♠♦ s❛❞❛ ❞❛ ♥❛♠ ❥❡ h = 0.15 ✐ ♥❡❦❛ ❥❡ ∆y= 1

6(k1+ 2k2+ 2k3+k4)✳ ❋♦r♠✐r❛❥♠♦ t❛❜❡❧✉

i x y k =hf(x, y) ∆y

✵ ✵ ✶ ✵ ✵

✵✳✵✼✺ ✶ ✲✵✳✵✸✸✷✸ ✲✵✳✵✻✻✹✻

✵✳✵✼✺ ✵✳✾✻✻✼✼ ✲✵✳✵✶✺✽✷ ✲✵✳✵✸✶✻✺

✵✳✶✺ ✵✳✾✽✹✶✽ ✲✵✳✵✷✸✷✼ ✲✵✳✵✷✸✷✼

✲✵✳✵✷✵✷✸

✶ ✵✳✶✺ ✵✳✾✼✾✼✼ ✲✵✳✵✷✸✾✸ ✲✵✳✵✷✸✾✸

✵✳✷✷✺ ✵✳✾✻✼✽✶ ✲✵✳✵✸✺✵✽ ✲✵✳✵✼✵✶✼

✵✳✷✷✺ ✵✳✾✻✷✷✸ ✲✵✳✵✸✺✾✷ ✲✵✳✵✼✶✽✹

✵✳✸ ✵✳✾✹✸✽✺ ✲✵✳✵✹✼✹✵ ✲✵✳✵✹✼✹✵

✲✵✳✵✸✺✺✻

❙❛❞❛ ✐③r❛↔✉♥❛❥♠♦ y2✿

y2 =y1+ ∆y= 0.97977−0.03556 = 0.94421.

❖st❛❧♦ ♥❛♠ ❥❡ ❥♦➨ ❞❛ ♣r♦❝✐❥❡♥✐♠♦ ❣r❡➨❦✉✳ ❚♦ ➣❡♠♦ ✉↔✐♥✐t✐ t❛❦♦ ➨t♦ ➣❡♠♦ s✈❡ ✐st♦ ✉r❛❞✐t✐✱ ❛❧✐ s❛♠♦ ③❛ ❦♦r❛❦ 2h✱ ♦❞♥♦s♥♦ t♦ ❜✐ ❦♦❞ ♥❛s s❛❞❛ ❜✐❧♦ h = 0.3 ✐ ♦♥❞❛ ➣❡♠♦ ❦♦r✐st✐t✐ ❘✉♥❣❡✲♦✈✉ ♦❝❥❡♥✉ ❣r❡➨❦❡✳ ❉❛❦❧❡✱ ✐③r❛↔✉♥❛❥♠♦ k1✱ k2✱ k3✱ k4✿

k1 = 0

k2 =−0.04180

k3 =−0.04807

k4 =−0.09238.

❙❛❞❛ ✐③r❛↔✉♥❛❥♠♦

y1 = 1 +

1

6(0−0.04180−0.04807−0.09238) = 0.96963 ❖❝❥❡♥✉ ❣r❡➨❦❡ ➣❡♠♦ ✐③r❛↔✉♥❛t✐ ♥❛ s❧❥❡❞❡➣✐ ♥❛↔✐♥

|yh −y2h|

15 =

|0.94421−0.96963|

15 = 0.0017<0.005 = 1 210

−2_.

◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛ ❇❡❣❛♥♦✈✐➣ ❉❛✈♦r ✶✷

▲✐t❡r❛t✉r❛

❬✶❪ ❘✉❞♦❧❢ ❙❝✐t♦✈s❦✐✱ ◆✉♠❡r✐↔❦❛ ♠❛t❡♠❛t✐❦❛✱ ❞r✉❣♦ ✐③❞❛♥❥❡✱ ❙✈❡✉↔✐❧✐➨t❡ ❏✳❏✳ ❙tr♦s✲ s♠❛②❡r❛ ✉ ❖s✐❥❡❦✉✱ ❖❞❥❡❧ ③❛ ♠❛t❡♠❛t✐❦✉✱ ❖s✐❥❡❦✱ ✷✵✵✹✳