❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
❘❊❨◆❆▲❉❖ ❉✬❆▲❊❙❙❆◆❉❘❖ ◆❊❚❖
❊❙❚❯❉❖ ❉❆❙ ❙❖▲❯➬Õ❊❙ ❆◆❆▲❮❚■❈❆❙ ❉❆ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ❯◆■❉■▼❊◆❙■❖◆❆▲ ❊ ❇■❉■▼❊◆❙■❖◆❆▲
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
❊❙❚❯❉❖ ❉❆❙ ❙❖▲❯➬Õ❊❙ ❆◆❆▲❮❚■❈❆❙ ❉❆ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ❯◆■❉■▼❊◆❙■❖◆❆▲ ❊ ❇■❉■▼❊◆❙■❖◆❆▲
❘❡②♥❛❧❞♦ ❉✬❆❧❡ss❛♥❞r♦ ◆❡t♦ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ▲✉ís ❱❡♥❡③✉❡❧❛
D'Alessandro Neto, Reynaldo
Estudo das Soluções Analíticas da Equação do Calor Unidimensional e Bidimensional / Reynaldo D'Alessandro Neto. -- 2016.
72 f. : 30 cm.
Dissertação (mestrado)-Universidade Federal de São Carlos, campus Sorocaba, Sorocaba
Orientador: Antonio Luís Venezuela
Banca examinadora: Luiza Amália Pinto Cantão, Wladimir Seixas Bibliografia
1. EDP. 2. Equação do Calor. 3. Técnica da Transformada Integral Clássica. I. Orientador. II. Universidade Federal de São Carlos. III. Título.
❊❙❚❯❉❖ ❉❆❙ ❙❖▲❯➬Õ❊❙ ❆◆❆▲❮❚■❈❆❙ ❉❆ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ❯◆■❉■▼❊◆❙■❖◆❆▲ ❊ ❇■❉■▼❊◆❙■❖◆❆▲
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡s✲ tr❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛✲ ❝✐♦♥❛❧ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s✱ ❝♦♠♦ ❡①✐✲ ❣ê♥❝✐❛ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ♠❡str❡ s♦❜ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❆♥t♦♥✐♦ ▲✉ís ❱❡♥❡③✉❡❧❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❡ ❛♠✐❣♦✱ Pr♦❢✳ ❆♥t♦♥✐♦ ▲✉ís ❱❡♥❡③✉❡❧❛✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ❡s❢♦rç♦ ❡♠ ♠❡ ❛✉①✐❧✐❛r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ♠♦str❛♥❞♦✲s❡ ❞✐s♣♦st♦ ❡ ♣r❡s❡♥t❡ ❞❡s❞❡ ♦s ❡st✉❞♦s ✈✐s❛♥❞♦ ♦ ❡①❛♠❡ ❞❡ q✉❛❧✐✜❝❛çã♦✳
❆♦s ♠❡✉s ♣❛✐s✱ ❘❡②♥❛❧❞♦ ❏♦sé ❉✬❆❧❡ss❛♥❞r♦ ❡ ▼❛r✐❛ ❚❡r❡s❛ ❆♣❛r❡❝✐❞❛ ❆❧✲ ✈❡s ❉✬❆❧❡ss❛♥❞r♦✱ q✉❡ ♠❡ ❛♣♦✐❛♠ ❡ ❛❥✉❞❛♠ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ s❡♠♣r❡ ❛❝r❡❞✐t❛r❛♠ ❡♠ ♠✐♠✱ ❡ ♣♦r ✐ss♦ ❞❡✈♦ ♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡t❡r♥♦✳ ❆ ♠✐♥❤❛ ♥♦✐✈❛ ❡ ❝♦❧❡❣❛ ❞❡ ❝❧❛ss❡✱ ▼❛r✐❛♥❛ ❈❛♣❡❧✐♥ ❋❛❜r✐❝✐♦✱ q✉❡ ❛❧é♠ ❞❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦ ♠❡str❛❞♦✱ ♠❡ ❛♣♦✐❛ ❝♦♥st❛♥t❡♠❡♥t❡✱ ♥♦s ♠♦♠❡♥t♦s ❢❡❧✐③❡s ❡ ❞✐❢í❝❡✐s ❞❡ss❛ ❝❛♠✐♥❤❛❞❛✳
❘❡s✉♠♦
❆s ♣r♦♣r✐❡❞❛❞❡s tér♠✐❝❛s ❞♦s ♠❛t❡r✐❛✐s sã♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦s ♣r♦❥❡t♦s ♠❡❝â♥✐❝♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦s q✉❡ ❡♥✈♦❧✈❡♠ s✐st❡♠❛s tér♠✐❝♦s✳ ❆ s✐✲ ♠✉❧❛çã♦ ❡ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ❝❛♠♣♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♣❡❧♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r✱ ❛✉①✐❧✐❛ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ tér♠✐❝♦✱ ✐st♦ é✱ ♥♦s ❢♦r♥❡❝❡ ✐♥❢♦r♠❛çõ❡s ♣ré✈✐❛s ❞❡ ❝♦♠♦ ❛ t❡♠♣❡r❛t✉r❛ ✈❛r✐❛ ❝♦♠ ❛ ♣♦s✐çã♦ ❡ ♦ t❡♠♣♦ ❡♠ ✉♠ só❧✐❞♦✱ ❡ ❛ss✐♠✱ ♣♦❞❡r ❝❛r❛❝t❡r✐③❛r ♦ ♠❛t❡r✐❛❧ tér♠✐❝❛♠❡♥t❡ ❡ s❛❜❡r ❛s ❝♦♥❞✐çõ❡s ❛♣r♦♣r✐❛❞❛s ❛ s❡ ✐♠♣♦r ❛♦ ♦❜❥❡t♦ ❡♠ ❡st✉❞♦✳ ❊st❛ ❞✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦ ❡st✉❞❛ té❝♥✐❝❛s ❞❡ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❞❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ✉♥✐ ❡ ❜✐❞✐♠❡♥s✐♦♥❛❧✳ ▼♦❞❡❧❛✲s❡ ♦s ❢❡♥ô♠❡♥♦s ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r ❛ ♣❛rt✐r ❞❡ ♣r♦❝❡❞✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s r❡❛❧✐③❛❞♦s ❝♦♠ ❛ ▲❡✐ ❞❡ ❘❡s❢r✐❛♠❡♥t♦ ❞❡ ❋♦✉r✐❡r✳ ❆♣ós ❛ ♠♦❞❡❧❛❣❡♠✱ ❢♦✐ ❝♦♥str✉í❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣ré✲r❡q✉✐s✐t♦s q✉❡ ❡♥❣❧♦❜❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ P❛r❝✐❛❧ ✭❊❉P✮ ❡ ❛❧❣✉♥s ♠ét♦❞♦s q✉❡ ❞❛rã♦ ♦ ❡♠❜❛s❛♠❡♥t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ té❝✲ ♥✐❝❛ ❞❡s❡❥❛❞❛✳ P♦r ✜♠✱ ❢❛③✲s❡ ❛ ❛♥á❧✐s❡ ❞♦s ♠♦❞❡❧♦s ❡♥❝♦♥tr❛❞♦s ❛ ♣❛rt✐r ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ❣rá✜❝♦s ❡ t❛❜❡❧❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✳
❆❜str❛❝t
❚❤❡ ❚❤❡r♠❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♠❛t❡r✐❛❧s ❛r❡ ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ ❢♦r ♠❡❝❤❛♥✐❝❛❧ ♣r♦❥❡❝ts✱ ❡s♣❡❝✐❛❧❧② t❤♦s❡ ✐♥✈♦❧✈✐♥❣ t❤❡r♠❛❧ s②st❡♠s✳ ❚❤❡ s✐♠✉❧❛t✐♦♥ ❛♥❞ ❞❡✲ t❡r♠✐♥✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❢♦r t❤❡ t❡♠♣❡r❛t✉r❡ ✜❡❧❞ ❦♥♦✇♥ ❛s ❤❡❛t ❡q✉❛t✐♦♥✱ ❛ss✐sts ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t❤❡r♠❛❧ ❜❡❤❛✈✐♦r✱ t❤❛t ✐s ✱ ✐t ❣✐✲ ✈❡s ✉s ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ♦♥ ❤♦✇ t❤❡ t❡♠♣❡r❛t✉r❡ ✈❛r✐❡s ✇✐t❤ t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t✐♠❡ ✐♥ ❛ s♦❧✐❞✱ ❛♥❞ s♦ ♣♦✇❡r t❤❡r♠❛❧❧② ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♠❛t❡r✐❛❧ ❛♥❞ ❦♥♦✇ t❤❡ ❛♣♣r♦♣r✐❛t❡ ❝♦♥❞✐t✐♦♥s t♦ ✐♠♣♦s❡ ♦♥ t❤❡ ♦❜❥❡❝t ✉♥❞❡r st✉❞②✳ ❚❤✐s ♠❛s✲ t❡r t❤❡s✐s st✉❞✐❡s t❡❝❤♥✐q✉❡s ♦❢ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♦♥❡✲ ❛♥❞ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❤❡❛t✳ ▼♦❞❡❧s t♦ t❤❡ ♣❤❡♥♦♠❡♥❛ ♦❢ ❤❡❛t tr❛♥s❢❡r ❢r♦♠ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❝❡❞✉r❡s ♣❡r❢♦r♠❡❞ ✇✐t❤ ❋♦✉r✐❡r ❝♦♦❧✐♥❣ ❧❛✇ ✱ ❛❢t❡r ♠♦❞❡❧✐♥❣✱ ❛ s❡q✉❡♥❝❡ ✇❛s ❜✉✐❧t ♣r❡ r❡q✉✐r❡♠❡♥ts ✇❤✐❝❤ ✐♥❝❧✉❞❡s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ✭P❉❊✮ ❛♥❞ s♦♠❡ ♠❡t❤♦❞s t❤❛t ❣✐✈❡ t❤❡ ♥❡❝❡ss❛r② ❜❛s✐s ❢♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❞❡s✐r❡❞ t❡❝❤♥✐q✉❡✳ ❋✐♥❛❧❧②✱ ♠❛❦❡s t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ♠♦❞❡❧s ❢♦✉♥❞ ❢r♦♠ t❤❡ ✉s❡ ♦❢ ❣r❛♣❤✐❝s ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ t❛❜❧❡s ✳
▲✐st❛ ❞❡ ❙í♠❜♦❧♦s
❙í♠❜♦❧♦ ◗✉❛♥t✐❞❛❞❡ ❋ís✐❝❛ ❊s❝❛❧❛r ❯♥✐❞❛❞❡s
A ➪r❡❛ ❞❛ ♣❛r❡❞❡ q✉❡ s❡♣❛r❛ ❛s ♣❧❛❝❛s✴➪r❡❛ ❞❛ s❡❝çã♦ tr❛♥s✈❡rs❛❧ ♠
C ❈❛♣❛❝✐❞❛❞❡ ❚ér♠✐❝❛ J/K
d ❉✐stâ♥❝✐❛ ❞❛s P❧❛❝❛s ♠
E ❊♥❡r❣✐❛ ❏
α ❈♦♥st❛♥t❡ ❞❡ ❉✐❢✉s✐✈✐❞❛❞❡ ❚ér♠✐❝❛ m2/s
▲x ❈♦♠♣r✐♠❡♥t♦ ❞❛ ❇❛rr❛✴P❧❛❝❛ ♥❛ ❈♦♦r❞❡♥❛❞❛ x ♠
▲y ❈♦♠♣r✐♠❡♥t♦ ❞❛ P❧❛❝❛ ♥❛ ❈♦♦r❞❡♥❛❞❛ y ♠
m ▼❛ss❛ ❦❣
Q ◗✉❛♥t✐❞❛❞❡ ❞❡ ❈❛❧♦r ❲
t ❚❡♠♣♦ s
❚1 ❚❡♠♣❡r❛t✉r❛ ❞❛ P❧❛❝❛ P1 ◦❈
❚2 ❚❡♠♣❡r❛t✉r❛ ❞❛ P❧❛❝❛ P2 ◦❈
f(x) ❚❡♠♣❡r❛t✉r❛ ■♥✐❝✐❛❧ ❞❛ ❇❛rr❛ ◦❈
f(x, y) ❚❡♠♣❡r❛t✉r❛ ■♥✐❝✐❛❧ ❞❛ P❧❛❝❛ ◦❈
˜
T ❚❡♠♣❡r❛t✉r❛ ❋✐♥❛❧ ◦❈
W ❚r❛❜❛❧❤♦ ❏
❙í♠❜♦❧♦s ●r❡❣♦s
γ ◗✉❛♥t✐❞❛❞❡ ❞❡ ❈❛❧♦r ❝❛❧
λn ❆✉t♦✈❛❧♦r❡s
L ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
ρ ❉❡♥s✐❞❛❞❡ kg/m3
Ψn ❆✉t♦❢✉♥çõ❡s ❆ss♦❝✐❛❞❛s ❛♦sλn
˜
▲✐st❛ ❞❡ ❋✐❣✉r❛s
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✺
✶✳✶ ▼♦t✐✈❛çõ❡s ❡ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷ ▼♦❞❡❧❛❣❡♠ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ✈✐❛ ▲❡✐ ❞❡ ❘❡s❢r✐❛♠❡♥t♦ ❞❡
❋♦✉r✐❡r ✷✵
✷✳✶ ▲❡✐ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❋❧✉①♦ ❞❡ ❈❛❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸ ❘❡s♦❧✉çã♦ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ✷✽ ✸✳✶ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ P❛r❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ❘❡s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❞❡ ❊❉Ps ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷✳✶ ▼ét♦❞♦ ❞❛ ❙❡♣❛r❛çã♦ ❞❡ ❱❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷✳✷ ❘❡s♦❧✉çã♦ ✈✐❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✷✳✸ ❚é❝♥✐❝❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥t❡❣r❛❧ ❈❧áss✐❝❛ ✲ ❚❚■❈ ✳ ✳ ✳ ✹✸ ✹ ❘❡s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ P❛r❝✐❛❧ ❞♦ ❈❛❧♦r ❯♥✐❞✐♠❡♥✲
s✐♦♥❛❧ ❡ ❇✐❞✐♠❡♥s✐♦♥❛❧ ✹✺
✹✳✶ ❊q✉❛çã♦ ❯♥✐❞✐♠❡♥s✐♦♥❛❧ ❞♦ ❈❛❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✶✳✶ ❘❡s♦❧✉çã♦ ✈✐❛ ❚❚■❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✷ ❊q✉❛çã♦ ❇✐❞✐♠❡♥s✐♦♥❛❧ ❞♦ ❈❛❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷✳✶ ❘❡s♦❧✉çã♦ ✈✐❛ ❙❡♣❛r❛çã♦ ❞❡ ❱❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✺ ❆♥á❧✐s❡ ❞♦s ❘❡s✉❧t❛❞♦s ✺✾
✺✳✶ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❯♥✐❞✐♠❡♥s✐♦♥❛❧ ✕ ❙♦❧✉çã♦ ✈✐❛ ❚❚■❈ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❇✐❞✐♠❡♥s✐♦♥❛❧ ✕ ❙♦❧✉çã♦ ✈✐❛ ❙❡♣❛r❛çã♦ ❱❛r✐á✈❡✐s ✻✸
✶
⑤
■♥tr♦❞✉çã♦
✶✳✶ ▼♦t✐✈❛çõ❡s ❡ ❖❜❥❡t✐✈♦s
❈♦♠ ♦ ❛✈❛♥ç♦ t❡❝♥♦❧ó❣✐❝♦✱ ♦s ❡st✉❞♦s q✉❡ ❡♥✈♦❧✈❡♠ ❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r ❣❛♥❤❛♠ ✉♠ ❣r❛♥❞❡ ❞❡st❛q✉❡✱ ❥á q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ♣r♦❝❡ss♦s ✐♥❞✉str✐❛✐s ❡ ❞❡ ♣r♦❥❡t♦s ❞❡ ✉s✐♥❛s ♥✉❝❧❡❛r❡s ❡ tér♠✐❝❛s ✉t✐❧✐③❛♠ ❡q✉✐♣❛♠❡♥t♦s ❞❡ tr♦❝❛ ❞❡ ❝❛❧♦r ❝♦♠♦ ❣❡r❛❞♦r❡s ❞❡ ✈❛♣♦r✱ ❢♦r♥♦s✱ ♠♦t♦r❡s ❞❡ ❝❛❧♦r✱ ❝♦♥❞❡♥s❛❞♦r❡s ❡ ♦✉tr♦s✳ ❖ ♠❡s♠♦ ❛❝♦♥t❡❝❡ ♥❛ ár❡❛ ❞❡ ♣r♦❞✉çã♦ ❞❡ ❡♥❡r❣✐❛✱ q✉❡ ❛t✉❛❧♠❡♥t❡ é ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ❡ ❡stá ❡♠ ♣r♦❝❡ss♦ ❞❡ ❡①♣❛♥sã♦ ❝♦♠ ♣r♦❥❡t♦s ♥♦ ❝♦♥tr♦❧❡ ❞♦ ♠❡✐♦ ❛♠❜✐❡♥t❡✳
❊①✐st❡♠ ♦✉tr♦s ♣r♦❝❡ss♦s q✉❡ ❡stã♦ ♣r❡s❡♥t❡s ❡♠ ♥♦ss♦ ❞✐❛✲❛✲❞✐❛✱ ❝♦♠♦ ♦s ❝♦♥✈❡rs♦r❡s ❝❛t❛❧ít✐❝♦s ♣r❡s❡♥t❡s ♥♦s ♠♦t♦r❡s ❞❡ ❝♦♠❜✉stã♦ ✐♥t❡r♥❛ ❞♦s ❛✉✲ t♦♠ó✈❡✐s✱ ❛s ✉♥✐❞❛❞❡s ❞❡ r❡❢r✐❣❡r❛çã♦ ❡ ❛r✲❝♦♥❞✐❝✐♦♥❛❞♦✱ ♦s ❡q✉✐♣❛♠❡♥t♦s ❡❧❡✲ trô♥✐❝♦s✱ ❛ r❡❢r✐❣❡r❛çã♦ ❞❡ ♠♦t♦r❡s ❡❧étr✐❝♦s✱ ♦s tr❛♥s❢♦r♠❛❞♦r❡s ❡ ❣❡r❛❞♦r❡s ❡❧étr✐❝♦s✱ ❛q✉❡❝✐♠❡♥t♦ ❡ r❡❢r✐❣❡r❛çã♦ ❞❡ ♣r♦❝❡ss♦s q✉í♠✐❝♦s✱ ❛ ♠✐♥✐♠✐③❛çã♦ ❞❡ ♣❡r❞❛s ❞❡ ❝❛❧♦r ❡♠ ❝♦♥str✉çõ❡s ❡ ❛♣r✐♠♦r❛♠❡♥t♦ ❞❡ té❝♥✐❝❛s ❞❡ ✐s♦❧❛♠❡♥t♦ tér♠✐❝♦✳
❈♦♠ ❡ss❛ ✈❛st❛ ❣❛♠❛ ❞❡ ❛♣❧✐❝❛çõ❡s✱ ✈❡♠♦s q✉❡ ♦s ♣r♦❜❧❡♠❛s r❡❧❛t✐✈♦s ❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r ❛♣❛r❡❝❡♠ ❝♦♠♦ ❡♥♦r♠❡s ❞❡s❛✜♦s ❛ s❡ r❡s♦❧✈❡r✳ ❆ss✐♠✱ ♠❛t❡♠át✐❝♦s✱ ❢ís✐❝♦s ❡ ❡♥❣❡♥❤❡✐r♦s ❡stã♦ ❝♦♥st❛♥t❡♠❡♥t❡ ❝♦♥❢r♦♥t❛♥❞♦ ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ♠❛①✐♠✐③❛r ❡✴♦✉ ♠✐♥✐♠✐③❛r t❛①❛s ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r✳
❙❡❣✉♥❞♦ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ ❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r ♦❝♦rr❡ ♣♦r ❝♦♥❞✉çã♦✱ ❝♦♥✈❡❝çã♦ ❡ r❛❞✐❛çã♦✱ ♠❛s ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ♣♦r ❝♦♠❜✐♥❛çã♦ ❞❛s ♠❡s♠❛s✳
✶✳✶✳ ▼❖❚■❱❆➬Õ❊❙ ❊ ❖❇❏❊❚■❱❖❙ ✶✻
❆ ♠❛✐♦r✐❛ ❞♦s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r sã♦ tr❛✲ t❛❞♦s ❛ ♣❛rt✐r ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ✭❊❉P✮✱ ❡ ❝♦♠ ✐ss♦✱ ❛ ♣r♦❝✉r❛ ♣♦r s♦❧✉çõ❡s ❡①❛t❛s ❛✉♠❡♥t♦✉ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡✳ ❆ ♠❛✐♦r✐❛ ❞❡ss❛s ❡q✉❛çõ❡s sã♦ ❞❡ ❣r❛♥❞❡ ❞✐✜❝✉❧❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❡ ♣r❡❝✐s❛♠ s❡r r❡s♦❧✈✐❞❛s ❡♠ ✉♠ ❝✉rt♦ ❡s♣❛ç♦ ❞❡ t❡♠♣♦✳ P❛r❛ ✐ss♦✱ ❛s té❝♥✐❝❛s ♥✉♠ér✐❝❛s ❡stã♦ s❡ s♦❜r❡ss❛✐♥❞♦ ❡♠ r❡❧❛çã♦ ❛s té❝♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s ❡ ❛♥❛❧ít✐❝❛s✱ ♣♦✐s ❛❝❡❧❡r❛ ♦ ♣r♦❝❡ss♦ ❞❡ r❡s♦❧✉✲ çã♦ ❡ ❣❛♥❤❛♠ ♥♦ ❢❛t♦r t❡♠♣♦✱ ❡ ♥♦ ✜♥❛♥❝❡✐r♦✱ ❥á q✉❡ ❛ ❝❛❞❛ ♥♦✈♦ ❡①♣❡r✐♠❡♥t♦ ❣❛st♦s ❛❞✐❝✐♦♥❛✐s ❛♣❛r❡❝❡♠✳
❖ ♣r♦❜❧❡♠❛ ❞♦s ♠ét♦❞♦s ♣✉r❛♠❡♥t❡ ♥✉♠ér✐❝♦s é ❛ ❞❡♠♦r❛ ❞♦ ♣r♦❝❡ss❛✲ ♠❡♥t♦✱ ❡❧❡✈❛♥❞♦ ♦s ❝✉st♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ♣♦✐s ♣❛r❛ s❡ t❡r ✉♠❛ ót✐♠❛ ♣r❡❝✐sã♦ é ♥❡❝❡ssár✐♦ ✉♠❛ ♠❛❧❤❛ ❝♦♠ ✉♠ ♥ú♠❡r♦ ♠❛✐♦r ❞❡ ♣♦♥t♦s✱ ♦ q✉❡ ❛❝❛❜❛ ✐♥✈✐❛✲ ❜✐❧✐③❛♥❞♦ ❛s s♦❧✉çõ❡s✳
◆❡ss❡ ❝♦♥t❡①t♦✱ ❡st❛ ❞✐ss❡rt❛çã♦ ♣r♦❝✉r❛ ♠♦str❛r ✉♠❛ té❝♥✐❝❛ ❛♥❛❧ít✐❝❛✱ q✉❡ ❛❜r❛♥❣❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♣r♦❝❡❞✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❡ ❝♦♥❝❡✐t♦s ❢ís✐❝♦s ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✳ ❈♦♠ ✐ss♦✱ t❡♠♦s ✉♠❛ ❛✈❛❧✐❛çã♦ ❝♦♠ ♠❛✐♦r ♣r❡❝✐sã♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s tér♠✐❝❛s ❞♦s ♠❛t❡r✐❛s q✉❡ s❡ tr❛❜❛❧❤❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ r❡s✉❧t❛❞♦ ❛♥❛❧ít✐❝♦ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐✲ ❛✐s✱ ❢♦r♥❡❝❡ ♠❡❧❤♦r s✉❜sí❞✐♦ ♣❛r❛ ❛♥á❧✐s❡s tér♠✐❝❛s ❞❡ ♠❛t❡r✐❛✐s ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r✳
❖ ♦❜❥❡t✐✈♦ ❣❡r❛❧ ❞❡ss❛ ❞✐ss❡rt❛çã♦ é r❡s♦❧✈❡r ❛ ❊❉P q✉❡ ♠♦❞❡❧❛ ♦s ♣r♦❝❡ss♦s ❞❡ tr❛♥s♣♦rt❡ ❞❡ ❝❛❧♦r ✉♥✐ ❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❡♠ ❣❡♦♠❡tr✐❛ r❡t❛♥❣✉❧❛r ♣♦r ♠❡✐♦ ❞❛ té❝♥✐❝❛ ❞❛ tr❛♥s❢♦r♠❛❞❛ ✐♥t❡❣r❛❧ ❝❧áss✐❝❛ ❡ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
P❛r❛ ❛t✐♥❣✐r ♦ ♦❜❥❡t✐✈♦ ♣r♦♣♦st♦✱ ♦ tr❛❜❛❧❤♦ ❢♦✐ ❡str✉t✉r❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ t❡♠ ❝♦♠♦ ♣r♦♣♦st❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ♥❛s ❝♦♦r❞❡♥❛❞❛s r❡t❛♥❣✉❧❛r❡s ♥❛s ✈❡rsõ❡s ✉♥✐ ❡ ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ❛ ♣❛rt✐r ❞❛ ❝♦♥❤❡❝✐❞❛ ▲❡✐ ❞❡ ❋♦✉r✐❡r✳
✶✳✷✳ ❘❊❱■❙➹❖ ❇■❇▲■❖●❘➪❋■❈❆ ✶✼
▲❛♣❧❛❝❡✱ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❡ ❛ té❝♥✐❝❛ ❞❛ tr❛♥s❢♦r♠❛❞❛ ✐♥t❡❣r❛❧ ❝❧áss✐❝❛✳ ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❚é❝♥✐❝❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥t❡✲ ❣r❛❧ ❈❧áss✐❝❛ ✭❚❚■❈✮✱ ❡ ♣♦r ♠❡✐♦ ❞❡❧❛ ❛ r❡s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❯♥✐✲ ❞✐♠❡♥s✐♦♥❛❧✳ ❆ r❡s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r ❇✐❞✐♠❡♥s✐♦♥❛❧ s❡rá r❡❛❧✐③❛❞❛ ♣♦r ♠❡✐♦ ❞❛ té❝♥✐❝❛ ❙❡♣❛r❛çã♦ ❞❡ ❱❛r✐á✈❡✐s✳
❆ ✈❛❧✐❞❛çã♦ s❡rá ❢❡✐t❛ ❛ ♣❛rt✐r ❞❡ r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❞❛ ❧✐t❡r❛t✉r❛✳ ❖ q✉❛rt♦ ❝❛♣ít✉❧♦ é ❞❡st✐♥❛❞♦ ❛ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s✱ ♣❛r❛ ✐ss♦ s❡rá ❝♦♥str✉í❞♦✱ ❛ ♣❛rt✐r ❞❛s s♦❧✉çõ❡s ♦❜t✐❞❛s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳
✶✳✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛
❊ss❛ s❡çã♦ ♣r♦❝✉r❛ ❢❛③❡r ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❜✐❜❧✐♦❣rá✜❝♦ ❞♦s ❛✉t♦r❡s q✉❡ ❡st✉❞❛r❛♠ ❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ♦❜r❛s q✉❡ t❡♥❤❛♠ ❝♦♠♦ ❢♦❝♦ té❝♥✐❝❛s ❛♥❛❧ít✐❝❛s✱ ♥✉♠ér✐❝❛s ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥❞✉çã♦ ❡ ❛ ❞❡s❝r✐çã♦ ❞❛ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r ❡♠ ♠❛t❡r✐❛✐s✳
❉♦✉❣❧❛s ❏r ❡ ❘❛❝❤❢♦r❞ ❏r ✭✶✾✺✻✮ ✉t✐❧✐③❛r❛♠ ♦ ❝❤❛♠❛❞♦ ♠ét♦❞♦ ❞❛s ❞✐❢❡✲ r❡♥ç❛s ✜♥✐t❛s ❛♣❧✐❝❛❞♦ ❛♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r tr❛♥s✐❡♥t❡ ♣❛r❛ ❞✉❛s ❡ três ❞✐♠❡♥sõ❡s✳ ❋r❛♥❦❡❧✱ ❱✐❝❦ ❡ Ö③✐s✐❦ ✭✶✾✽✻✮✱ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❣❡r❛❧ ♣❛r❛ ❛ ❝♦♥❞✉çã♦ ❤✐♣❡r❜ó❧✐❝❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥❞♦ ❢♦r s✉❜♠❡t✐❞♦ ❛ ✉♠ ✢✉①♦ ❞❡ ❝❛❧♦r ❡♠ ♠❡✐♦ ❝♦♠♣♦st♦ ❡ t❛♠❜é♠ ♥♦s ❛♣r❡s❡♥t♦✉ ❛ s♦❧✉çã♦ ♣❛r❛ s✐st❡♠❛s tr✐❞✐♠❡♥s✐♦♥❛✐s ❞❡ ❝♦♦r❞❡♥❛❞❛s ♦rt♦❣♦♥❛✐s✱ ♥❡ss❡ ❝❛s♦ ❛ té❝♥✐❝❛ ❞❛ tr❛♥s❢♦r♠❛❞❛ ✐♥t❡❣r❛❧ ❣❡♥❡r❛❧✐③❛❞❛ ❢♦✐ ✉t✐❧✐③❛❞❛ ♣❛r❛ s❡ ♦❜t❡r ❛ s♦❧✉çã♦✳
❆♥t❛❦✐ ✭✶✾✾✻✮ ❛♣❧✐❝♦✉ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ s❡ ♦❜t❡r ❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r ❤✐♣❡r❜ó❧✐❝❛ ❡♠ ♣❧❛❝❛s s❡♠✐✲✐♥✜♥✐t❛s ❡♠ ❝♦rr❡♥t❡s ❝♦♥✈❡❝t✐✈❛s✳ ▼✐❦❤❛✐❧♦✈ ❡ Ö③✐s✐❦ ✭✶✾✽✹✮ ✉t✐❧✐③❛r❛♠ ♦ s♦❢t✇❛r❡ ▼❛t❤❡✲ ♠❛t✐❝❛ ♣❛r❛ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❞❡ ❝❛❧♦r ❤✐♣❡r❜ó❧✐❝❛✱ ❛♣r❡s❡♥t❛♥❞♦ ❞❡❢❛s❛❣❡♠ ❞❡ ❢❛s❡ ❡ ❛♠♣❧✐t✉❞❡ ❞❡ ♦s❝✐❧❛çõ❡s ❞❡ t❡♠♣❡r❛t✉r❛✳
✶✳✷✳ ❘❊❱■❙➹❖ ❇■❇▲■❖●❘➪❋■❈❆ ✶✽
❞❛ tr❛♥s❢♦r♠❛❞❛ ✐♥t❡❣r❛❧ ✜♥✐t❛✳
▼♦♥t❡ ✭✶✾✾✾✮ ❛♥❛❧✐s♦✉ ❛ r❡s♣♦st❛ tr❛♥s✐❡♥t❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ ✉♠ ❝♦♥✲ ❞✉t♦r ❝♦♠♣♦st♦ ❞❡ ♠✉❧t✐❝❛♠❛❞❛s ❡♠ ✈❛r✐❛çõ❡s ❜r✉s❝❛s ❞❡ t❡♠♣❡r❛t✉r❛✳ ❆ s♦❧✉çã♦ ❢♦✐ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞♦ ♠ét♦❞♦ ❞❛ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ♣❛r❛ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞♦ ❝❛❧♦r✳ ❘❡❣✐s✱ ❈♦tt❛ ❡ ❚❛♥ ✭✷✵✵✵✮ ❞❡s❝r❡✈❡r❛♠ ❛ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r tr❛♥s✐❡♥t❡ ❡♠ ❜❛rr❛s ❞❡ ❝♦♠❜✉stí✈❡❧ ♥✉❝❧❡❛r ♣♦r ✉♠ ♠ét♦❞♦ ❞❡ ♣❛râ♠❡✲ tr♦ ❛❣r✉♣❛❞♦ ♠❡❧❤♦r❛❞♦✱ ❝♦♠ ✐ss♦ ♦ ❛✉t♦r ❛❧❝❛♥ç♦✉ ♠❡❧❤♦r✐❛s s✐❣♥✐✜❝❛t✐✈❛s ♥❛ ❢♦r♠✉❧❛çã♦ ❝❧áss✐❝❛ ❞❡ ♣❛râ♠❡tr♦s ❝♦♥❝❡♥tr❛❞♦s✳
❙❛❞❛t ✭✷✵✵✹✮ ❢❡③ ❛ ❛♥á❧✐s❡ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r ✉♥✐❞✐✲ ♠❡♥s✐♦♥❛❧ ❡ tr❛♥s✐❡♥t❡ ✉t✐❧✐③❛♥❞♦ ♦ ❝❤❛♠❛❞♦ ♠ét♦❞♦ ❞❡ ♣❡rt✉r❜❛çã♦✱ ❝♦♠ ✐ss♦ ❡❧❡ ♠♦str♦✉ q✉❡ ♠♦❞❡❧♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ ♦ ❝❡♥tr♦✱ s✉♣❡r❢í❝✐❡ ❡ t❡♠✲ ♣❡r❛t✉r❛ ♠é❞✐❛ ♣♦❞❡♠ s❡r ❞❡s❡♥✈♦❧✈✐❞♦s ♥❛ ❣❡♦♠❡tr✐❛ ❝✐❧í♥❞r✐❝❛✱ ❡s❢ér✐❝❛ ❡ ❞❡ ♣❧❛❝❛✳ ❙✉✱ ❚❛♥ ❡ ❙✉ ✭✷✵✵✾✮ tr❛❜❛❧❤❛r❛♠ ♥❛ ♠❡❧❤♦r✐❛ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ♣❛râ✲ ♠❡tr♦s ❝♦♥❝❡♥tr❛❞♦s ❞❡ ♠♦❞❡❧♦s ❛ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r tr❛♥s✐❡♥t❡ ❡♠ ✉♠❛ ♣❧❛❝❛✱ ❝♦♠ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ♥ã♦ s❡♥❞♦ tr❛t❛❞❛ ❝♦♠ ❝♦♥st❛♥t❡✱ ❝♦♠ ✐ss♦ ❡❧❛ ❞❡♣❡♥❞❡r✐❛ ❞❛ t❡♠♣❡r❛t✉r❛✳ ❖ ❛✉t♦r ❝♦♠♣❛r♦✉ ♦s s❡✉s r❡s✉❧t❛❞♦s ❝♦♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r✳
❈♦ss❛❧✐ ✭✷✵✵✽✮ ♠♦str♦✉ s♦❧✉çõ❡s ❛♥❛❧ít✐❝❛s ♣❛r❛ ✉♠ ❝✐❧✐♥❞r♦ ❤♦♠♦❣ê✲ ♥❡♦ ❞❡s❡♥✈♦❧✈✐❞❛s ❛tr❛✈és ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ♣❛r❛ ❝♦♥❞✉çã♦ ❞❡ ❝❛❧♦r ♣❡r✐ó❞✐❝❛ ❢♦r♥❡❝❡♥❞♦ ✉♠ ❝❛♠♣♦ ❞❡ t❡♠♣❡r❛t✉r❛ ✢✉t✉❛♥t❡✳
✶✳✷✳ ❘❊❱■❙➹❖ ❇■❇▲■❖●❘➪❋■❈❆ ✶✾
♠ér✐❝♦ ❢♦✐ ♦❜t✐❞♦ ♣❡❧♦ ❝♦♥❤❡❝✐❞♦ ♠ét♦❞♦ ❞❡ ●❡❛r✳ ❈♦♠♦ ❢♦r♠❛ ❞❡ ✈❛❧✐❞❛çã♦ ♦ ❛✉t♦r ✉t✐❧✐③♦✉ ❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❡ ❝♦♠ ♦s ♣r♦❞✉③✐❞♦s ❛ ♣❛rt✐r ❞❛ ✉t✐❧✐③❛çã♦ ❞♦ ♠ét♦❞♦ ❞❛ tr❛♥s❢♦r♠❛ ❞❡ ▲❛♣❧❛❝❡ ❡ ♦ ♠ét♦❞♦ ❞❡ ●❡❛r ❡♠ ❝♦r♣♦s ❝♦♠ ✈♦❧✉♠❡s ✜♥✐t♦s✳
✷
⑤
▼♦❞❡❧❛❣❡♠ ❞❛ ❊q✉❛çã♦ ❞♦ ❈❛✲
❧♦r ✈✐❛ ▲❡✐ ❞❡ ❘❡s❢r✐❛♠❡♥t♦ ❞❡
❋♦✉r✐❡r
❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❛ ♠♦❞❡❧❛❣❡♠ ❞❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ❛ ♣❛rt✐r ❞❛ ▲❡✐ ❞❡ ❋♦✉r✐❡r✳ P❛r❛ ✐ss♦✱ ✉t✐❧✐③❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❛ ❢ís✐❝❛ ❞❡st❡ ♣r♦❜❧❡♠❛ ❡ ❛ t❡♦r✐❛ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳
✷✳✶ ▲❡✐ ❞❡ ❋♦✉r✐❡r
❆ ❧❡✐ ❢ís✐❝❛ q✉❡ s❡r✈✐rá ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♦ ♥♦ss♦ ❡st✉❞♦ ❞❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r é ❛ ▲❡✐ ❞❡ ❋♦✉r✐❡r✳ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ ❡①♣❧✐❝❛♠ q✉❡ ❡ss❡ ♠♦❞❡❧♦ r❡❧❛❝✐♦♥❛ ❛ t❡♠♣❡r❛t✉r❛ ❝♦♠ ♦ ♠♦✈✐♠❡♥t♦ ❞❛s ♣❛rtí❝✉❧❛s ❡♠ ✉♠ ❝♦♥❞✉t♦r ❞❡ ❝❛❧♦r✳ ❊ ❝♦♠ ♦ r❡❝❡❜✐♠❡♥t♦ ❞❡ ❝❛❧♦r✱ ❛s ♠♦❧é❝✉❧❛s ✈✐❜r❛♠ ♠❛✐s ✐♥t❡♥s❛♠❡♥t❡ ❡ ❛ss✐♠ ❛ ❡♥❡r❣✐❛ é ♣❛ss❛❞❛ ♣❛r❛ ♦✉tr❛ ♣❛rtí❝✉❧❛✱ ❝♦♠ ✐ss♦ t❡♠♦s ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❝❛❧♦r✳
❆ ▲❡✐ ❞❡ ❋♦✉r✐❡r✱ q✉❡ r❡❣❡ ❡ss❡ ♣r♦❝❡ss♦ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ ❝❛❧♦r✱ ❢♦✐ ❞❡t❡r♠✐♥❛❞❛ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❏❡❛♥✲❇❛♣t✐st❡ ❋♦✉✲ r✐❡r ✭✶✼✻✽✲✶✽✸✵✮✳ ❊ss❛ ❧❡✐ ♥♦s ❞✐③ q✉❡ s❡ ❞✉❛s ♣❧❛❝❛s P1 ❡ P2 sã♦ ♠❛♥t✐❞❛s ❛
t❡♠♣❡r❛t✉r❛ ❝♦♥st❛♥t❡✱ ❞✐s♣♦st❛s ♣❛r❛❧❡❧❛♠❡♥t❡ ❡ s❡♣❛r❛❞❛s ♣♦r ✉♠❛ ♣❛r❡❞❡ ❛ ✉♠❛ ❞✐stâ♥❝✐❛d ✉♠❛ ❞❛ ♦✉tr❛✱ ❛ss✐♠ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✶✿
✷✳✷✳ ❋▲❯❳❖ ❉❊ ❈❆▲❖❘ ✷✶
❋✐❣✉r❛ ✶✿ ❙✉♣❡r❢í❝✐❡ ❞❡ ❈♦♥tr♦❧❡ ✲ ❋❧✉①♦ ❞❡ ❈❛❧♦r
❋♦♥t❡✿ ❊❧❛❜♦r❛❞♦ ♣❡❧♦ ❛✉t♦r
❍❛✈❡rá ♣❛ss❛❣❡♠ ❞❡ ❝❛❧♦r ❞❛ ♣❧❛❝❛ ♠❛✐s q✉❡♥t❡ ♣❛r❛ ❛ ♠❛✐s ❢r✐❛✱ ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦r✱ ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ tr❛♥s❢❡r✐❞❛ ❞❡ ✉♠❛ ♣❛r❛ ♦✉tr❛ é ❞❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
Q= KA˜ (T2−T1)
d ✭✷✳✶✮
❙❡❣✉♥❞♦ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ ♦ ♣r♦❝❡ss♦ ❛❝✐♠❛ s❡ ❛♣❧✐❝❛ t❛♥t♦ ❛ tr❛♥s♠✐ssã♦ ❞❡ ❝❛❧♦r q✉❛♥❞♦ ❛s ♣❧❛❝❛s ❡stã♦ s❡♣❛r❛❞❛s ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ✉♠❛ ❞❛ ♦✉tr❛✱ q✉❛♥t♦ ♣❛r❛ ♠❛t❡r✐❛✐s ❡♠ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ t❡♠♣❡r❛t✉r❛ ❛♣❛r❡❝❡ ♥❛s s✉❛s ❡①tr❡♠✐❞❛❞❡s✳ ◆❡ss❡s ❝❛s♦s✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ❞✐stâ♥❝✐❛d❝♦♠♦ ❛ ❡s♣❡ss✉r❛✱
s❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ t❡♠♣❡r❛t✉r❛ é ♦❜s❡r✈❛❞❛ ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞♦ ♠❛t❡r✐❛❧✱ ♦✉ ❛ ❞✐stâ♥❝✐❛ ❞♦s ♣♦♥t♦s ❡♠ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ t❡♠♣❡r❛t✉r❛ ♦❝♦rr❡✱A❝♦♠♦ ❛ ár❡❛
❞❛ s❡❝çã♦ tr❛♥s✈❡rs❛❧ ❞♦ ♠❛t❡r✐❛❧ ❡ T2✱ T1 ❛s t❡♠♣❡r❛t✉r❛s ❞❛s s✉❛s ❡①tr❡♠✐✲
❞❛❞❡s✱ ♦♥❞❡K˜ r❡♣r❡s❡♥t❛ ❛ ❈♦♥❞✉t✐✈✐❞❛❞❡ ❚ér♠✐❝❛✱ ♥❛s ✉♥✐❞❛❞❡s ❲✴♠❑✳
✷✳✷ ❋❧✉①♦ ❞❡ ❈❛❧♦r
✷✳✷✳ ❋▲❯❳❖ ❉❊ ❈❆▲❖❘ ✷✷
q✉❡ ❛ t❡♠♣❡r❛t✉r❛ ✐♥❞❡♣❡♥❞❡ ❞❛s ❝♦♦r❞❡♥❛❞❛sy ❡z✳ ❈♦♠ ✐ss♦ ❡♥❝♦♥tr❛r❡♠♦s
♦ ✢✉①♦ ❞❡ ❝❛❧♦r ❡♠ ✉♠❛ ❜❛rr❛ ❞❡ ✉♠ ♠❛t❡r✐❛❧ ❝♦♥❞✉t♦r✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛♣❡♥❛s ✉♠❛ ❞✐♠❡♥sã♦ ♥♦s ❝á❧❝✉❧♦s✳
❚♦♠❛♠♦s ❡♥tã♦✱ ❞✉❛s s❡❝çõ❡s ❞❛ ❜❛rr❛ ❧♦❝❛❧✐③❛❞❛ ❡♠ x ❡ ❡♠ x+d✱
r❡♣r❡s❡♥t❛♥❞♦ ✉♠❛ ♠❡❞✐çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❡♠ ✉♠ ♦✉tr♦ ♣♦♥t♦ ❞❛ ❜❛rr❛ ❞✐❢❡✲ r❡♥t❡ ❞♦ ♣♦♥t♦x✐♥✐❝✐❛❧✳ ❆♦ s❡ ❛♣❧✐❝❛r ❛ ❧❡✐ ❞❡ ❋♦✉r✐❡r✱ ♥♦s ❞❡♣❛r❛♠♦✲♥♦s ❝♦♠
✉♠ ♣r♦❜❧❡♠❛✿ ❆s t❡♠♣❡r❛t✉r❛s ❞❡ss❛s s❡❝çõ❡s ✈❛r✐❛♠ ❝♦♠ ♦ t❡♠♣♦✱ ❞❡ ❢♦r♠❛ q✉❡ ♥ã♦ sã♦ ❝♦♥st❛♥t❡s ❝♦♠♦ r❡q✉❡r ❛ ❧❡✐✳ P❛r❛ r❡s♦❧✈❡r♠♦s ❡ss❡ ♣r♦❜❧❡♠❛✱ ❞❡✜♥♠♦s ♦ ✢✉①♦ ❞❡ ❝❛❧♦r✱ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ✜①❛♠♦s ♦ t❡♠♣♦ t✱ t❡♠♦sT1❂ u✭x, t✮ ❡T2 =u(x+d, t),✳ P♦r ✜♠✱ ❛♣❧✐❝❛♠♦s ♦ ❧✐♠✐t❡ ❝♦♠ d t❡♥❞❡♥❞♦ ❛ ③❡r♦✱
❝♦♠♦ s❡ s❡❣✉❡✿
q(x, t) = lim
d→0
KA(T2−T1)
d = limd→0
KA(u(x+d, t)−u(x, t))
d
= KA.lim
d→0
u(x+d, t)−u(x, t)
d =KA∂xu(x, t)
❙❡❣✉♥❞♦ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮✱ ❞❡✜♥✐♠♦s ♦ ✢✉①♦ ❞❡ ❝❛❧♦r ♥❛ ❞✐r❡çã♦ ♣♦s✐t✐✈❛ ❞♦ ❡✐①♦x ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧q ❞❛❞❛ ♣♦r✿
q(x, t) =−KA∂xu(x, t)
❖ s✐♥❛❧ ♥❡❣❛t✐✈♦ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛ é ❥✉st✐✜❝❛❞♦ ♣♦r ❇♦②❝❡ ❡ ❉✐♣r✐♠❛ ✭✷✵✶✷✮ ♣❡❧♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦✿ ❙❡ ❛ t❡♠♣❡r❛t✉r❛ ❝r❡s❝❡r ❝♦♠x✱∂xu(x, t)s❡rá
♣♦s✐t✐✈♦✱ ♠❛s ❝♦♠♦ ♦ ❝❛❧♦r ✢✉✐ ♣❛r❛ ❛ ❡sq✉❡r❞❛✱q✭x, t✮ ❞❡✈❡rá s❡r ♥❡❣❛t✐✈♦✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ u ❞❡❝r❡s❝❡ss❡ ❝♦♠ x✱ ∂xu(x, t) s❡rá ♥❡❣❛t✐✈♦✱ ♠❛s
❝♦♠♦ ♦ ❝❛❧♦r ✢✉✐ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ q✭x, t✮ ❞❡✈❡ s❡r ♣♦s✐t✐✈♦✳ ❆ ❥✉st✐✜❝❛t✐✈❛ s❡
✷✳✸✳ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ✷✸
q✉❡r tr❛♥s❢♦r♠❛çã♦ ❝✉❥♦ ❡❢❡✐t♦ ✜♥❛❧ ♣♦ss❛ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❡❧❡✈❛çã♦ ❞❡ ✉♠ ♣❡s♦✳
✷✳✸ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r
P❛r❛ ♦❜t❡r♠♦s ❛ ❡q✉çã♦ ❞♦ ❝❛❧♦r ✉t✐❧✐③❛♠♦s ❛❧❣✉♠❛s ❝♦♥❤❡❝✐❞❛s r❡❧❛çõ❡s ❞❛ ❢ís✐❝❛✳ ■♥✐❝✐❛❧♠❡♥t❡ ❢❛③❡♠♦s ✉s♦ ❞❛ ❡q✉❛çã♦ ❞❡ tr❛♥s♣♦rt❡ ❞❡ ❡♥❡r❣✐❛ tér♠✐❝❛ q✉❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ é ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ♣r✐♠❡✐r❛ ❧❡✐ ❞❛ t❡r♠♦❞✐♥â♠✐❝❛ ❡ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦✿
dE
dT =Q−W ✭✷✳✷✮
❊st❛❜❡❧❡❝❡♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❡ ❡♥❡r❣✐❛E ♣❛r❛ ✉♠ s✐st❡♠❛ é ✐❣✉❛❧ ❛ s✉❜✲
tr❛çã♦ ❞♦s ✢✉①♦s ❞❡ ❝❛❧♦r✱Q✱ ❡ tr❛❜❛❧❤♦✱W✱ q✉❡ ❝r✉③❛♠ ❛ ❢r♦♥t❡✐r❛ ❞♦ s✐st❡♠❛✳
❉❛❞❛ ✉♠❛ ❜❛rr❛ ❝♦♠ ❝♦♠♣r✐♠❡♥t♦L❡ ✉♠ ♣♦♥t♦ ①0 t❛❧ q✉❡0< x0 < L✱
✜①❛♠♦s ①0 ❡ ①0+δ✱ s❡♥❞♦ δ ♠✉✐t♦ ♠❡♥♦r q✉❡ ❛ ✉♥✐❞❛❞❡✳ ❱❡♠♦s ❛ q✉❛♥t✐❞❛❞❡
❞❡ ❝❛❧♦r γ q✉❡ ❡♥tr❛ ♥♦ s✐st❡♠❛✱ ♥♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ t0
❡ t0+τ✱ s❡♥❞♦ τ é ♠✉✐t♦ ♠❡♥♦r q✉❡ ❛ ✉♥✐❞❛❞❡✱ ❝♦♠ ✐♥t❡r✈❛❧♦s t♦♠❛❞♦s t❛♥t♦
♣❛r❛ ♦ ♣♦♥t♦ ❞❛ ❜❛rr❛ q✉❛♥t♦ ♣❛r❛ ♦ t❡♠♣♦✳ ❆ss✐♠✱ ✉s❛♠♦s ♦ ✢✉①♦ ❞❡ ❝❛❧♦r
q(x, t)✱ ❡ ❛ ❡q✉❛çã♦ ❞❡ tr❛♥s♣♦rt❡ ❞❡ ❡♥❡r❣✐❛ tér♠✐❝❛✱ ❡ ❞❡✜♥✐♠♦s ❛ q✉❛♥t✐❞❛❞❡
❞❡ ❝❛❧♦r✴❡♥❡r❣✐❛ tér♠✐❝❛✱ ♣❡❧❛ ❡①♣r❡ssã♦✿
γ =
t0+τ
Z
t0
q(x0, t)dt−
t0+τ
Z
t0
q(x0+δ, t)dt ✭✷✳✸✮
❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ t❡♠♦s✿
γ =
Z t0+τ
t0
q(x0, t)−q(x0+δ, t)dt⇒
γ =
t0+τ
Z
t0
✷✳✸✳ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ✷✹
❊♥tã♦✿
γ =
t0+τ
Z
t0
KA.[(∂xu(x0+δ, t))−(∂xu(x0, t))]dt ✭✷✳✹✮
❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✿
∂xu(x0 +δ, t))−(∂xu(x0, t)) =
x0+τ
Z
x0
∂x2u(x, t)dx ✭✷✳✺✮
❊ ❛ss✐♠✿
γ =
t0+τ
Z
t0
x0+τ
Z
x0
K∂x2u(x, t)dxAdt ✭✷✳✻✮
❙❡❣✉♥❞♦ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦r ♥❡❝❡ssár✐❛ ♣❛r❛ ❛✉♠❡♥t❛r ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠❛ ♠❛ss❛ ✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ s✉❜stâ♥✲ ❝✐❛✱ ♣❡❧❛ r❡❧❛çã♦ ❞❡ss❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦r ∆γ, ❢♦r♥❡❝✐❞❛ ❛ ✉♠ ❝♦r♣♦✱ ❡ ♦
❝♦rr❡s♣♦♥❞❡♥t❡ ❛❝rés❝✐♠♦ ❞❛ t❡♠♣❡r❛t✉r❛∆T✿
C = ∆γ
∆T
❆ ❝❛♣❛❝✐❞❛❞❡ tér♠✐❝❛ ❞❡ ✉♠ ❝♦r♣♦ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❛ss❛ m é ❞❡✜♥✐❞❛
♣♦r ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ ❝♦♠♦ ♥✉♠❡r✐❝❛♠❡♥t❡ ✐❣✉❛❧ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦r ♣❛r❛ ❡❧❡✈❛r ❛ t❡♠♣❡r❛t✉r❛ ❡♠ ✉♠ ❣r❛✉✳ P♦r ✐ss♦✱ ♣♦❞❡ s❡r ❝❤❛♠❛❞❛ ❞❡ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞❡ ✉♠❛ s✉❜stâ♥❝✐❛✱ ❞❡♥♦t❛❞♦ ♣❡❧❛ ❧❡tr❛c✿
c=
∆γ
∆T
m =
∆γ m∆T
✷✳✸✳ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ✷✺
♣❡r❛t✉r❛✳ P♦r ✐ss♦✱ t❡♠♦s q✉❡ ♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞❡ ✉♠❛ s✉❜stâ♥❝✐❛ ❞❛❞❛ ❛ t❡♠♣❡r❛t✉r❛✿
c= △γ
m△T
❆ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦r ♣❛r❛ ❛✉♠❡♥t❛r ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠ ❚1 ♣❛r❛ ✉♠
❚2✱ s❡rá ❛ s❡❣✉✐♥t❡ ✐♥t❡❣r❛❧✿
γ =m
T2
Z
T1 cdT
❙❡❣✉♥❞♦ ■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❛❞✈✐♥❞❛ ❞❛ ♠❡❝â♥✐❝❛ ❞♦s ✢✉✐❞♦s✿
m =
x1
Z
x0
ρAdx
❆ ♣❛rt✐r ❞❛ r❡❧❛çã♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡ ❛ ♠❛ss❛ é ❝❛❧❝✉❧❛❞❛ ✉t✐❧✐③❛♥❞♦✲ s❡ ❛ ✐♥t❡❣r❛❧ ❞❛ ❞❡♥s✐❞❛❞❡✱ρ✱ ✈❡③❡s ❛ ár❡❛ ✈❛r✐❛♥❞♦ ♥♦ ✐♥t❡r✈❛❧♦ ❬①0, x0 +τ]✳
P♦❞❡♠♦s ❞❡❞✉③✐r q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦rγ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❬t1,t2]
♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✿
γ =
x0+τ
Z
x0
t2
Z
t1
cρ∂tu(x, t)AdT dx
P♦rt❛♥t♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❧♦r✱ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦✿
γ =
t0+τ
Z
t0
x0+τ
Z
x0
✷✳✸✳ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ✷✻
■❣✉❛❧❛♥❞♦ ❛s ❡①♣r❡ssõ❡s ❛❝✐♠❛✱ ♦❜t❡♠♦s✿
t0+τ
Z
t0
x0+τ
Z
x0
K∂x2u(x, t)dxAdt =
t0+τ
Z
t0
x0+τ
Z
x0
c∂tu(x, t)dxρAdt
⇒ K∂x2u(x, t) =cρ∂tu(x, t)
❈♦♠♦ ❡ss❛ ú❧t✐♠❛ r❡❧❛çã♦ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦t >0 ❡ t♦❞♦0< x0 < L❡
t♦❞♦τ >0 ❡ δ >0, tê♠✲s❡ q✉❡✿
∂tu=α∂x2u ✭✷✳✽✮
❝♦♠✿
α = K
cρ
■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt ✭✶✾✾✵✮ ♥♦s ❞❡✜♥❡ q✉❡ α é ❛ ❞✐❢✉s✐✈✐❞❛❞❡ tér♠✐❝❛✱
q✉❡ ♣♦ss✉✐ ❛s ❞✐♠❡♥sõ❡s m2/s✳ ❆ ❡q✉❛çã♦ ∂
tu = α∂x2u é ❝❤❛♠❛❞❛ ❡q✉❛çã♦
✷✳✸✳ ❊◗❯❆➬➹❖ ❉❖ ❈❆▲❖❘ ✷✼
❚❛❜❡❧❛ ✶✿ ❈♦♥st❛♥t❡ ❞❡ ❉✐❢✉s✐✈✐❞❛❞❡ ❞❡ ▼❛t❡r✐❛✐s ▼❛t❡r✐❛❧ α(10−4m2
/s)
Pr❛t❛ ✶✳✼✶
❈♦❜r❡ ✶✳✶✹
❆❧✉♠í♥✐♦ ✵✳✽✻ ❋❡rr♦ ❋✉♥❞✐❞♦ ✵✳✶✷ ●r❛♥✐t♦ ✵✳✵✶✶
❋♦♥t❡✿✭■♥❝r♦♣❡r❛ ❡ ❉❡✇✐tt✱✶✾✾✵✮
❆ ❊q✉❛çã♦ ✭✷✳✽✮ ❡♥✈♦❧✈❡ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❝❛❧♦r ❧✐♥❡❛r✳ ❆ss✐♠✱ ❛❧é♠ ❞♦ t❡♠♣♦ t✱ ❝♦♥s✐❞❡r❛♠♦s ❛♣❡♥❛s ✉♠❛ ❞✐♠❡♥sã♦✱ x✱ ♥♦s ❝á❧❝✉❧♦s✳ ◆♦ ❝❛s♦
❡s♣❡❝✐❛❧ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ❝❛❧♦r ❡♠ ✉♠ ♠❡✐♦ ✐s♦tró♣✐❝♦ ❡ ❤♦♠♦❣ê♥❡♦ ❡♠ ✉♠ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❇♦②❝❡ ❡ ❉✐♣r✐♠❛ ✭✷✵✶✷✮ ❝♦♥s✐❞❡r❛ ❛s três ❞✐♠❡♥sõ❡s✱
x, y, z ♥❛ ❡q✉❛çã♦✳ ❆ss✐♠✱ ✜❝❛rá ❝♦♠ ❛ s❡❣✉✐♥t❡s ❡str✉t✉r❛✿ ∂u
∂t =α
∂2u ∂x2 +
∂2u ∂y2 +
∂2u ∂z2
◆❡st❛ ❞✐ss❡rt❛çã♦ ♣r❡t❡♥❞❡♠♦s r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ❝♦♥s✐❞❡✲ r❛♥❞♦✱ ❛❧é♠ ❞♦ t❡♠♣♦✱ ✉♠❛ ❞✐♠❡♥sã♦✱ x ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❊q✉❛çã♦ ❞♦ ❈❛❧♦r
❯♥✐❞✐♠❡♥s✐♦♥❛❧✱ ❡ ❞✉❛s ❞✐♠❡♥sõ❡s✱x, y ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞♦ ❈❛❧♦r ❇✐✲
❞✐♠❡♥s✐♦♥❛❧✳ ❊ss❛ ú❧t✐♠❛✱ ♣♦ss✉✐ ❛ ♠❡s♠❛ ❡str✉t✉r❛ ❞❛ q✉❡ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❛❝✐♠❛✱ ❛♣❡♥❛s r❡t✐r❛♥❞♦ ❛ ✈❛r✐á✈❡❧z✱ ❛♣r❡s❡♥t❛♥❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❯♥✐❞✐♠❡♥s✐♦♥❛❧✿∂F
∂T =α
∂2F
∂x2 ✭✷✳✾✮
❇✐❞✐♠❡♥s✐♦♥❛❧✿∂u
∂t =α
∂2u ∂x2 +
∂2u ∂y2
✸
⑤
❘❡s♦❧✉çã♦ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡✲
r❡♥❝✐❛✐s P❛r❝✐❛✐s
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ P❛r✲ ❝✐❛❧✱ ❡ ❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ❛♥❛❧ít✐❝❛s✿ ❙❡♣❛r❛çã♦ ❞❡ ❱❛r✐á✈❡✐s❀ ❚❚■❈ ❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✳
✸✳✶ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ P❛r❝✐❛❧
❯♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦✿
F
x, y, ..., u(x, y, ...),∂u ∂x,
∂u ∂y, ...,
∂2u ∂x2,
∂2u ∂y2, ...
= 0
é ❝❤❛♠❛❞❛ ❞❡ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ P❛r❝✐❛❧ ✭❊❉P✮ q✉❡ ❡♥✈♦❧✈❡ ✈❛r✐á✈❡✐s ✐♥❞❡✲ ♣❡♥❞❡♥t❡s ✭x, y, z✮ ❡ ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡♥t❡s u✳ ❙❡❣✉♥❞♦ ■ór✐♦✱ ❛ ♦r❞❡♠ ❞❡ ✉♠❛
❊❉P é ❞❛❞❛ ♣❡❧❛ ♦r❞❡♠ ❞❡ s✉❛ ♠❛✐♦r ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ♣r❡s❡♥t❡ ♥❛ ❡q✉❛çã♦✱ ❡ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ❡s❝r✐t❛✱ ♣♦❞❡♠♦s r❡❝♦rr❡r ❛ ♠❛✐s ❞❡ ✉♠❛ ♥♦t❛çã♦ ♣❛r❛ ❞❡s✐❣♥❛r ❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❝♦♠ r❡❧❛çã♦ ❛x✿
∂xu=
∂u ∂x
❆s ❊❉Ps t❛♠❜é♠ ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s ❡♠ t❡r♠♦s ❞❛ s✉❛ ❧✐♥❡❛r✐❞❛❞❡✳ ❆ss✐♠✱ ✉♠❛ ❊❉P ❧✐♥❡❛r ❞❡✈❡rá s❡r ❧✐♥❡❛r r❡❧❛t✐✈❛♠❡♥t❡ à ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❡ às s✉❛s ❞❡r✐✈❛❞❛s✳ ◆♦ ❝❛s♦ ❞❡ ✉♠❛ ❊❉P ❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ t❡r❡♠♦s
✸✳✶✳ ❊◗❯❆➬➹❖ ❉■❋❊❘❊◆❈■❆▲ P❆❘❈■❆▲ ✷✾
♦ ❝❛s♦ ❣❡r❛❧✱ ♣❛r❛ ❞✉❛s ✈❛r✐á✈❡✐s✿
a∂xxu+b∂xyu+c∂yyu+d∂xu+e∂yu+f u=g
♦♥❞❡ u é ✉♠❛ ❢✉♥çã♦ ❞❡ x, y ❝❤❛♠❛❞❛ ❢✉♥çã♦ ✐♥❝ó❣♥✐t❛✱ ❡♠ q✉❡ a, b, c, d, e, f
sã♦ ❝♦❡✜❝✐❡♥t❡s q✉❡ ♣♦❞❡♠ s❡r ❡♠ ❢✉♥çã♦ ❞❡x, y✱ ♠❛s ♥ã♦ ❞❡u✳ ❙❡❣✉♥❞♦ ■ór✐♦
✭✷✵✶✷✮ s❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❢♦r❡♠ ❡♠ ❢✉♥çã♦ ❞❡x, y✱ ❛ ❊❉P ❞✐③✲s❡ ❞❡ ❝♦❡✜❝✐❡♥t❡s
❝♦♥st❛♥t❡s✳ ❙❡g = 0✱ ❛ ❊❉P é ❝❤❛♠❛❞❛ ❞❡ ❤♦♠♦❣ê♥❡❛✳
❊♠ ■ór✐♦ ✭✷✵✶✷✮ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛s ❊❉Ps ❧✐♥❡❛r❡s ❞❡ 2a ♦r❞❡♠ ♣♦❞❡♠
♦❝♦rr❡r ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❊❉P✳ ❯♠❛ ❊❉P ❧✐♥❡❛r é✿ ❍✐♣❡r❜ó❧✐❝❛ s❡∆ =b2−4ac >0❀ ❊❧í♣t✐❝❛ s❡∆ =b2−4ac <0 ❡ P❛r❛❜ó❧✐❝❛ s❡
∆ =b2−4ac= 0✳
❉❡ss❛ ❢♦r♠❛✱ ■ór✐♦ ✭✷✵✶✷✮ ❡①♣❧✐❝❛ ✉♠ ✐♠♣♦rt❛♥t❡ ♣r✐♥❝í♣✐♦ q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♥❛ r❡s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❞❛s ❊❉P✬s✳ P❛r❛ ✐ss♦ é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ❛ t❡r♠✐♥♦❧♦❣✐❛ ♦♣❡r❛❞♦r ✭♦✉ tr❛♥s❢♦r♠❛çã♦✮ q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ■ór✐♦ ✭✷✵✶✷✮✱ é ✉s❛❞❛ ♣❛r❛ ❡♥❢❛t✐③❛r q✉❡ ❛ ❢✉♥çã♦ L ❡stá ❞❡✜♥✐❞❛ ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❢✉♥çõ❡s✱
✐st♦ é✱L❧❡✈❛ ✉♠❛ ❢✉♥çã♦ u✭❝♦♠ ❞❡t❡r♠✐♥❛❞❛s ♣r♦♣r✐❡❞❛❞❡s✮ ❡♠ ♦✉tr❛ ❢✉♥çã♦ Lu✳ ❖ ♦♣❡r❛❞♦r L é ✉♠ ❡①❡♠♣❧♦ ❞❡ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧✳
❚❡♦r❡♠❛ ✸✳✶ ✭Pr✐♥❝í♣✐♦ ❞❛ ❙✉♣❡r♣♦s✐çã♦✮ ❙❡❥❛ ▲ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥✲ ❝✐❛❧ ♣❛r❝✐❛❧ ❞❡ ♦r❞❡♠ ❦ ❝✉❥♦s ❝♦❡✜❝✐❡♥t❡s ❡stã♦ ❞❡✜♥✐❞♦s ❡♠ ✉♠ ❞♦♠í♥✐♦ ❛❜❡rt♦
Ω⊆Rn✳ ❙✉♣♦♥❤❛ q✉❡ {um}
∞
m=1 é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ❞❡ ❝❧❛ss❡ Ck ❡♠ Ω
s❛t✐s❢❛③❡♥❞♦ ❛ ❊❉P ❧✐♥❡❛r✳ ❊♥tã♦✱ s❡ {αm}
∞
m=1 é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡s❝❛❧❛r❡s
t❛❧ q✉❡ ❛ sér✐❡
u(x) =
∞
X
m=1
αmum(x)
é ❝♦♥✈❡r❣❡♥t❡ ❡ k ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧ t❡r♠♦ ❛ t❡r♠♦ ❡♠ Ω✳ ❆ ❞❡♠♦♥str❛çã♦
q✉❡ s❡rá ❞❡s❝r✐t❛ ❛❜❛✐①♦ ❢♦✐ r❡t✐r❛❞❛ ❞❡ ■ór✐♦ ✭✷✵✶✷✮✳ ❉❡♠♦♥str❛çã♦
✸✳✶✳ ❊◗❯❆➬➹❖ ❉■❋❊❘❊◆❈■❆▲ P❆❘❈■❆▲ ✸✵
❝❛s♦s ❡♠ q✉❡ k = 1 ❡ k = 2✳ ◆❡ss❡s ❝❛s♦s✱ ♣♦r ❤✐♣ót❡s❡✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ xǫΩ✱ 1≤i, j ≤n
u(x) =
∞
X
m=1
αmum(x)
Diu(x) =
∞
X
m=1
αmDium(x)
DjDiu(x) =
∞
X
m=1
αmDjDium(x)
❡ ❡ss❛s sér✐❡s ❝♦♥✈❡r❣❡♠✳ P♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦xǫΩ✿
Lu(x) =
n
X
i,j=1
αij(x)DjDiu(x) + n
X
j=1
bj(x)Dju(x) +c(x)u(x)
=
n
X
i,j=1 αij(x)
∞
X
m=1
αmDjDium(x) + n
X
j=1 bj(x)
∞
X
m=1
αmDjum(x)
+ c(x)
∞
X
m=1
αmum(x)
= ∞ X m=1 αm " n X i,j=1
αij(x)DjDium(x) + n
X
j=1
bj(x)Djum(x) +c(x)um(x)
#
=
∞
X
m=1
αm(Lum)(x) = 0
♦ q✉❡ ❞❡♠♦♥str❛ ♦ t❡♦r❡♠❛ ❛❝✐♠❛ ♣❛r❛ ♦s ❝❛s♦s ❡♠ q✉❡k = 1 ❡ k= 2✳
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✶
❢❡♥ô♠❡♥♦s ❢ís✐❝♦s ❡st❛❝✐♦♥ár✐♦s✱ ❝♦♠♦ ❛s ❝♦♥❞✐çõ❡s ❞♦ t✐♣♦✿
αu(x) +β∂u
∂n(x) =f(x) xǫ∂Ω
♦♥❞❡α ❡β sã♦ ❝♦♥st❛♥t❡s ❞❛❞❛s✱f ✉♠❛ ❢✉♥çã♦ ❞❛❞❛ ❡♠∂Ω ❡ ∂u∂n é ❛ ❞❡r✐✈❛❞❛
❞❡u ♥❛ ❞✐r❡çã♦∂Ω✳ ◆♦ ❝❛s♦ ❞❡β = 0✱ ❛ ❝♦♥❞✐çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❈♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t❀ ♥♦ ❝❛s♦ ❞❡α= 0✱ t❡♠♦s ✉♠❛ ❈♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥✳
✸✳✷ ❘❡s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❞❡ ❊❉Ps
◆❡ss❛ s❡çã♦ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s té❝♥✐❝❛s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❛ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❞❡ ✉♠❛ ❊❉P✳
✸✳✷✳✶ ▼ét♦❞♦ ❞❛ ❙❡♣❛r❛çã♦ ❞❡ ❱❛r✐á✈❡✐s
❉❡ ❛❝♦r❞♦ ❝♦♠ ■ór✐♦ ✭✷✵✶✷✮✱ ♥♦ ❝❛s♦ ❣❡r❛❧ ❞❡ ✉♠❛ ❊❉P✱ ❝✉❥❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ é u(x, y)✱ ♦ ♠ét♦❞♦ ❞❡ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❜❛s❡✐❛✲s❡ ♥❛ ♣♦ss✐✲
❜✐❧✐❞❛❞❡ ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ u✱ r❡❧❛t✐✈❛♠❡♥t❡ às ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s x ❡ y✱
♣♦❞❡r s❡r ❡①♣r❡ss❛ ❡♠ t❡r♠♦s ❞♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ❢✉♥çõ❡s✿ X ❡ Y✱ ❞❡✜♥✐❞❛s
♣♦rX =X(x)❡ Y =Y(y)✱ ♦✉ s❡❥❛✿
u(x, y) = X(x)Y(y) ✭✸✳✶✮
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❛♠♦s ✉♠❛ ❊❉P ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ q✉❡ ❡♥✈♦❧✈❡ ❞✉❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s✿
∂u
∂x +
∂u
∂y = 2(x+y)u ✭✸✳✷✮
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✷
q✉❡✿
Y dX
dx +X
dY
dy = 2(x+y)XY
❉✐✈✐❞✐♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣♦r XY ❡ r❡♦r❣❛♥✐③❛♥❞♦✲❛✱ t❡♠♦s✿
1
X dX
dx −2x=−
1
Y dY
dy + 2y ✭✸✳✸✮
❱❡♠♦s q✉❡✱ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❊q✉❛çã♦ ✭✸✳✸✮ ❡stá ❡♠ ❢✉♥çã♦ ❞❛ ✈❛r✐á✲ ✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡x❡ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❡♠ ❢✉♥çã♦ ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡y✳■ór✐♦
✭✷✵✶✷✮ ❝♦♠❡♥t❛ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ só ♣♦❞❡rá s❡r ✈á❧✐❞❛ s❡ ❛♠❜♦s ♦s ❧❛❞♦s ❢♦✲ r❡♠ ✐❣✉❛✐s ❛ ✉♠❛ ♠❡s♠❛ ❝♦♥st❛♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡x ❡ y✳ ❈♦♠ r❡❧❛çã♦ ❛
❊q✉❛çã♦ ✭✸✳✸✮ ❝♦♥s✐❞❡r❛♠♦s ❛ ❝♦♥st❛♥t❡ ❞❡ s❡♣❛r❛çã♦✱ k✱ ❙❡❣✉❡ q✉❡✿
1
X dX
dx −2x=k ✭✸✳✹✮
❡
−1
Y dY
dy + 2y=k ✭✸✳✺✮
❘❡♦r❣❛♥✐③❛♥❞♦✿
dX
dx −X(k+ 2x) = 0 ✭✸✳✻✮
❡
dY
dy +Y(k−2y) = 0 ✭✸✳✼✮
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✸
❝♦♥s✐❞❡r❛♥❞♦p(x) =k+ 2x ❡ q(x) = 0✳ ❆ss✐♠✿
X(x) = c0
µ(x) =c0e
kx+x2
✭✸✳✽✮
s❡♥❞♦c0 ❛ ❝♦♥st❛♥t❡ ❞❡ ✐♥t❡❣r❛çã♦✳
❖ ♠ét♦❞♦ ♣❛r❛ s❡ r❡s♦❧✈❡r ❛ ❊q✉❛çã♦ ✭✸✳✼✮ é ❛♥á❧♦❣♦ ❛♦ ❛♥t❡r✐♦r✱ ❛♣❧✐✲ ❝❛♥❞♦ ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ❝❤❡❣❛♠♦s ❛ s♦❧✉çã♦✿
Y(y) =c1ey
2−ky
✭✸✳✾✮
s❡♥❞♦c1 ❛ ❝♦♥st❛♥t❡ ❞❡ ✐♥t❡❣r❛çã♦✳
❙✉❜st✐t✉✐♠♦s ❛s ❊q✉❛çõ❡s ✭✸✳✽✮ ❡ ✭✸✳✾✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✶✮✱ ♦❜t❡♠♦s✿
u(x, y) = c0c1ex
2+kx+y2−ky
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ ❝❛❧♦r ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ ❞❡s❝r❡✲ ✈❡♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠ ❝♦r♣♦✱ ❛♦ ❧♦♥❣♦ ❞❛ ❞✐r❡çã♦ x✱ ❡♠
❢✉♥çã♦ ❞♦ t❡♠♣♦t✳
∂u ∂t =α
∂2u
∂x2 06x61 t >0 ✭✸✳✶✵✮
s❡♥❞♦ u(x, t) r❡♣r❡s❡♥t❛♥❞♦ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠❛ ❝❡rt❛ ❜❛rr❛ ♠❡tá❧✐❝❛ ♥❛ ♣♦s✐çã♦x❡ ♥♦ ✐♥st❛♥t❡t❡ é ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ❞♦ ♠❡t❛❧✳ ❙❡ ♣r❡t❡♥❞❡r♠♦s
♦❜t❡r ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦❜❧❡♠❛✱ ■ór✐♦ ✭✷✵✶✷✮ ❡①♣❧✐❝❛ q✉❡ ❞❡✈❡♠♦s ❝♦♥❤❡❝❡r ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ s♦❜r❡ t ❡ ❞✉❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ s♦❜r❡ x✱
❝❤❛♠❛❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ✭♦✉ ❞❡ ❢r♦♥t❡✐r❛✮✳ ◆❡ss❡ ❝❛s♦✱ ❞❡✈❡♠♦s s❡r ✐♥❢♦r♠❛r ❛♣❡♥❛s ❞✉❛s ❝♦♥❞✐çõ❡s✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❊❉P ❡♥✈♦❧✈❡ ✉♠❛ ❞❡r✐✈❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡ ✉♠❛ ❞❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ s♦❜r❡ ❝❛❞❛ ✉♠❛ ❞❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✹
s❡ ❞✐str✐❜✉✐ ❛♦ ❧♦♥❣♦ ❞❛ ❜❛rr❛ ♥♦ ✐♥st❛♥t❡t= 0✿
u(x,0) =f(x) ✭✸✳✶✶✮
❙✉♣♦♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ❝♦♥❤❡❝✐❞❛✳ ❆s ❝♦♥❞✐çõ❡s ❢r♦♥t❡✐r❛ ❝♦rr❡s✲
♣♦♥❞❡♠ ♥♦r♠❛❧♠❡♥t❡ à t❡♠♣❡r❛t✉r❛ ❞❛ ❜❛rr❛ ❡♠ ❝❛❞❛ ❡①tr❡♠✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♣❛r❛ x= 0 ❡ x= 1✳ ❙✉♣♦♥❞♦✱ q✉❡ ❡ss❛s t❡♠♣❡r❛t✉r❛s sã♦ ❝♦♥st❛♥t❡s ❡ ✐❣✉❛✐s
❛ ③❡r♦✱ t❡♠♦s✿
u(0, t) = 0 ✭✸✳✶✷✮
❡
u(1, t) = 0 ✭✸✳✶✸✮
❆s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❛❝✐♠❛ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❤♦♠♦❣ê♥❡❛s ♦✉ ❞❡ ❉✐r✐❝❤❧❡t✳ ◆❡ss❡ ❝❛♣ít✉❧♦✱ r❡s♦❧✈❡r❡♠♦s ❛ ❡q✉❛çã♦ ❞♦ ❝❛❧♦r ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♣♦r s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ s♦❧✉çã♦ ❞❛ ♠❡s♠❛ ✉t✐❧✐③❛♥❞♦ ♦✉tr❛ té❝♥✐❝❛ ❡ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ♥ã♦ ❤♦♠♦❣ê♥❡❛s✳
■♥✐❝✐❛♠♦s ❡♥tã♦ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❚❛❧ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ ❢❡✐t♦✱ ✈❛♠♦s ♣r♦❝✉r❛r r❡♣r❡s❡♥t❛ru(x, t) ❝♦♠♦✿
u(x, t) = X(x)T(t) ✭✸✳✶✹✮
❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✉❛çã♦ ✭✸✳✶✹✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✶✵✮✱ t❡♠♦s✿
XdT
dt =αT
d2X dx2
❈♦♠ ❛ s❡♣❛r❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ♣♦❞❡♠♦s ♦❜t❡r✿
1
α
1
T dT
dt =
1
X d2X
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✺
❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❡ss❛ ✐❣✉❛❧❞❛❞❡ só s❡rá ✈á❧✐❞❛ ♣❛r❛ q✉❛❧q✉❡r t ❡ q✉❛❧q✉❡r x s❡ ❛♠❜♦s ♦s ❧❛❞♦s ❢♦r❡♠ ✐❣✉❛✐s ❛ ✉♠❛ ♠❡s♠❛
❝♦♥st❛♥t❡✳ ❆ss✐♠ t♦♠❛♠♦s✿
1
α
1
T dT
dt =−k
2 ✭✸✳✶✺✮
❡ 1
X d2X
dx2 =−k
2 ✭✸✳✶✻✮
❉❛í✿
dT
dt +αk
2T = 0 ✭✸✳✶✼✮
d2X dx2 +k
2X = 0 ✭✸✳✶✽✮
❆ ❊q✉❛çã♦ ✭✸✳✶✼✮ é ❛♥á❧♦❣❛ ❛ ❊q✉❛çã♦ ✭✸✳✻✮ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ ❯t✐✲ ❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞♦ ❢❛t♦r ✐♥t❡❣r❛♥t❡✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ s♦❧✉çã♦✿
T(t) = ce−αk2t
s❡♥❞♦c ❛ ❝♦♥st❛♥t❡ ❞❡ ✐♥t❡❣r❛çã♦✳ ❈♦♥s✐❞❡r❛♠♦sc= 1✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛✲
❧✐❞❛❞❡✳
❆ ❊❉❖✱ ❞❛❞❛ ♣❡❧❛ ❊q✉❛çã♦ ✭✸✳✶✽✮✱ é ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛ ❞❡ s❡❣✉♥❞❛ ♦r✲ ❞❡♠✳ ❙❛❜❡♠♦s q✉❡✱ ❛ s✉❛ s♦❧✉çã♦ ❣❡r❛❧ s❡rá ❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ❞✉❛s s♦❧✉çõ❡s ♣❛rt✐❝✉❧❛r❡s✱ ❝♦♥❢♦r♠❡ ♦ ♣r✐♥❝í♣✐♦ ❞❛ s✉♣❡r♣♦s✐çã♦✱ ♥❛s q✉❛✐s ❞❡✈❡✲ rã♦ s❡r ❞♦ t✐♣♦ erx✳ ❖ ♣❛râ♠❡tr♦ r é ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❛s r❛í③❡s ❞❛ ❡q✉❛çã♦
❝❛r❛❝t❡ríst✐❝❛r2+k2r= 0✳ ❆ss✐♠✿
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✻
s❡♥❞♦A ❡ B ❝♦♥st❛♥t❡s r❡❛✐s✳
❖ ♣r♦❜❧❡♠❛ ❞❛❞♦ ♣❡❧❛ ❊q✉❛çã♦ ✭✸✳✶✽✮ é ❝❤❛♠❛❞♦ ❞❡ ❙t✉r♠✲▲✐♦✉✈✐❧❧❡✳ ❙✉❜st✐t✉✐♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞❛s ❊q✉❛çõ❡s ✭✸✳✶✷✮ ❡ ✭✸✳✶✸✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✶✹✮✱ ♦❜t❡♠♦s✿
X(0)T(t) = 0
X(1)T(t) = 0
❈♦♠♦T(t)6= 0✱ ♣❛r❛ ∀t∈R, t≥0✱ t❡♠♦s q✉❡✿
X(1) = 0 ✭✸✳✷✵✮
❡
X(0) = 0 ✭✸✳✷✶✮
❆♣❧✐❝❛♥❞♦ ❛s ❊q✉❛çõ❡s ✭✸✳✷✵✮ ❡ ✭✸✳✷✶✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✶✾✮✱ t❡♠♦s✿ P❛r❛ (i) x = 0 ⇒ A = 0 ❡ ♣❛r❛ (ii) x = 1 ⇒ Bsen(k) = 0✳ ❈♦♠♦
B 6= 0✱ ❧♦❣♦ sen(k) = 0 ❡ ❝♦♠ ✐ss♦✿
k =nπ,∀n= 0,1,2, ... ✭✸✳✷✷✮
❉❛í ❛s ❢✉♥çõ❡s X ❡T sã♦ ❡s❝r✐t❛s ❝♦♠♦✿
X(x) = Bsen(nπx) ✭✸✳✷✸✮
T(t) = e−(nπα)2t
✭✸✳✷✹✮ ❙✉❜st✐t✉✐♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ♥❛ ❊q✉❛çã♦ ✭✸✳✶✹✮✱ t❡♠♦s q✉❡✿
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✼
❡ ❝❤❛♠❛♥❞♦ bn =B✱ t❡♠♦s✿
u(x, t) = bnsen(nπx)e
−α(nπ)2t
✭✸✳✷✺✮
♣❛r❛ n= 1,2, ...
P❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ s✉♣❡r♣♦s✐çã♦✿
u(x, t) =
∞
X
n=1
anun(x, t)
❈♦♠ ✐ss♦✱ ❡①✐st❡♠ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ♣♦ssí✈❡✐s✳ ❆ s♦❧✉çã♦ ❝♦♠♣❧❡t❛ ❞♦ ♣r♦❜❧❡♠❛ é ❞❛❞❛ ♣❡❧♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s✳
u(x, t) =
∞
X
n=1 bne
−α(nπ
L)
2t
sen(nπx) ✭✸✳✷✻✮
P❛r❛ ❞❡t❡r♠✐♥❛r ♦s ❝♦❡✜❝✐❡♥t❡s bn ❞❛ ❡q✉❛çã♦✱ ❡ ❛ss✐♠ ❞❡✜♥✐r♠♦s ❛
s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦❜❧❡♠❛✱ ❞❡✈❡♠♦s ❛♣❧✐❝❛r ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✿
u(x,0) =f(x)⇒
∞
X
n=1
bnsen(nπx) =f(x) ✭✸✳✷✼✮
■ór✐♦ ✭✷✵✶✷✮ ♥♦s ❡①♣❧✐❝❛ ♦s ❝♦❡✜❝✐❡♥t❡s bn s❡rã♦ ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❛
❡①♣❛♥sã♦ ❞❛ f ♥❛ sér✐❡ s❡♥♦ ❞❡ ❋♦✉r✐❡r ♣❛r❛ ♦ ✐♥t❡r✈❛❧♦ ❞❛❞♦ ♥❛ ❝♦♥❞✐çã♦
✐♥✐❝✐❛❧0< x <1✳ ❆ss✐♠ bn s❡rá ♦❜t✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
f(x) =
∞
X
n=0
bnsen(nπx)
❉❡✜♥✐♠♦sXn(x) = sen(nπx)
❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ♣❛r❛ ❛s ❢✉♥çõ❡sg ❡h é ❞❡✜♥✐❞♦ ♣♦r ❈❛❧❧✐♦❧✐✱ ❉♦♠✐♥✲
❣✉❡s ❡ ❈♦st❛ ✭✶✾✽✸✮✿
hg(x), h(x)i=
1
Z
0
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✽
❚♦♠❛♥❞♦g(x) = f(x) ❡h(x) = Xm(x)✱ t❡♠♦s✿
hf(x), Xm(x)i =
* ∞
X
n=0
bnXn(x), Xm(x)
+
=
∞
X
n=0
bnhXn(x), Xm(x)i
♠❛s✿
hXn(x), Xm(x)i=
0, m6=n
Nm, m=n
❝♦♠ ✐ss♦✿
bnhXn(x), Xm(x)i=bn
1
Z
0
Xn2(nπx)dx=bn
1
Z
0
sen2(nπx)dx= 1 2bn
▲♦❣♦✿
1
2bn =hf(x), Xm(x)i ⇒bn= 2
1
Z
0
f(x)sen2(nπx)dx
❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✉❛çã♦ ❛❝✐♠❛ ♥❛ ❊q✉❛çã♦ ✭✸✳✷✺✮✱ ♦❜t❡♠♦s✿
u(x, t) =
∞
X
0
2sen(nπx)e−α(nπ)2t
1
Z
0
f(x)sen(nπx)dx
P♦rt❛♥t♦ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡stá ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡✜♥✐❞❛✳
✸✳✷✳✷ ❘❡s♦❧✉çã♦ ✈✐❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
✸✳✷✳ ❘❊❙❖▲❯➬➹❖ ❆◆❆▲❮❚■❈❆ ❉❊ ❊❉P❙ ✸✾
❉❡✜♥✐çã♦ ✸✳✶ ✭❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✮ ❙❡❥❛F :R∗
+ 7→R✉♠❛ ❢✉♥çã♦✳
❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡F(t) é ❞❡✜♥✐❞❛ ♣♦r ❙♣✐❡❣❡❧ ✭✶✾✻✺✮✿
L {F(t)}=f(s) =
∞
Z
0 e−st
F(t)dt ✭✸✳✷✾✮
❝♦♠ s∈R✳
❯♠❛ ❢✉♥çã♦F é ✉♠❛ ❢✉♥çã♦ ❞❡ ♦r❞❡♠ ❡①♣♦♥❡♥❝✐❛❧η✱ s❡ ❡①✐st❡♠ ❝♦♥s✲
t❛♥t❡s m >0❡ η t❛✐s q✉❡ ♣❛r❛ t♦❞♦ t > N✿
e
−ηt
F(t) < M
q✉❛♥❞♦t → ∞✳
❈♦♠♣❧❡t❛♥t♦ ❛ ❞❡✜♥✐çã♦✱ ❙♣✐❡❣❡❧ ✭✶✾✻✺✮ ♥♦s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛✱ ❛tr❛✈és ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿
❚❡♦r❡♠❛ ✸✳✷ ✭❊①✐stê♥❝✐❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✮ ❙❡F é ✉♠❛ ❢✉♥✲
çã♦ s❡❝❝✐♦♥❛❧♠❡♥t❡ ❝♦♥tí♥✉❛ ♥✉♠ ✐♥t❡r✈❛❧♦ ✜♥✐t♦ 0 ≤ t ≤ N ❡ ❞❡ ♦r❞❡♠ ❡①✲
♣♦♥❡♥❝✐❛❧ η✱ ♣❛r❛ t > M✱ ❡♥tã♦ ❡①✐st❡ s✉❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ f(s)✱ ∀s∈R, s > η✳
❈♦♠ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡✜♥✐❞❛✱ ❧✐st❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ✭❙♣✐❡❣❡❧✱✶✾✻✺✱ ♣✳✹✮✿
✭✶✮ Pr♦♣r✐❡❞❛❞❡ ❞❛ ❧✐♥❡❛r✐❞❛❞❡
❙❡c1 ❡ c2 sã♦ ❝♦♥st❛♥t❡s q✉❛✐sq✉❡r✱ ❡♥q✉❛♥t♦ F1(t)❡ F2(t) sã♦ ❢✉♥çõ❡s
❝♦♠ tr❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡f1(s) ❡ f2(s)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿
L {c1F1(t) +c2F2(t)}=c1L{F1(t)}+c2L{F2(t)}=c1f1(s) +c2f2(s)
✭✷✮ Pr♦♣r✐❡❞❛❞❡ ❞❛s ❉❡r✐✈❛❞❛s
