Calcule a temperatura u(x, t) de uma barra semi-infinita
phi(x) = { 1, 0 < x < 1; 0, x > 1 }
onde .

Passo 1

Beleza, vamos resolver esse exercício juntos! A gente precisa encontrar a temperatura de uma barra semi-infinita seguindo a equação do calor. O que isso significa? Basicamente, queremos saber como a temperatura varia ao longo da barra ao longo do tempo. Para isso, vamos usar a Equação da Difusão do Calor, que é dada por:
Aqui, é a difusividade térmica da barra. Além disso, temos condições iniciais e de contorno que vão nos ajudar a resolver a equação. Vamos precisar entender conceitos como séries de Fourier e transformadas de Fourier para resolver isso passo a passo.

Passo 2

Primeiro, vamos entender as condições que temos. A condição de contorno é , o que significa que na posição , a temperatura é sempre zero, independentemente do tempo. A condição inicial é , onde é uma função definida por:
Isso indica que inicialmente, a temperatura é 1 na região e 0 além disso.

Passo 3

Para resolver a equação do calor com essas condições, vamos usar a transformada de Fourier. A ideia é transformar a equação diferencial em uma equação algébrica mais fácil de resolver. Primeiro, aplicamos a transformada de Fourier na variável espacial . A transformada de Fourier de em relação a é:
Aplicando isso na equação do calor, obtemos uma equação para .

Passo 4

Depois de aplicar a transformada de Fourier, a equação do calor se transforma em:
Essa é uma equação diferencial simples que podemos resolver. A solução geral dessa equação é:
Aqui, é a transformada de Fourier da condição inicial .

Passo 5

Agora, precisamos encontrar . Isso significa calcular a transformada de Fourier da função :
Portanto, temos:

Passo 6

Agora, precisamos aplicar a transformada inversa de Fourier para obter de volta no domínio espacial. A transformada inversa é dada por:
Substituindo :

Passo 7

Vamos simplificar a integral dividindo-a em duas partes:
Essa integral pode ser resolvida usando propriedades da transformada de Fourier e reconhecendo que ela está relacionada com a função de erro ( ).

Passo 8

Finalmente, ao resolver a integral, chegamos à solução para :
A função é a função erro, que aparece frequentemente em soluções de equações de difusão.

Resposta

A temperatura da barra no ponto e no tempo é dada por:

Exercícios relacionados

<html><head></head><body><div>Mostre que o raio de convergência da série de Maclaurin da função <span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.566ex;” xmlns=“
http://www.w3.org/2000/svg” width=“9.473ex” height=“2.262ex” role=“img” focusable=“false” viewBox=“0 -750 4187 1000”><title>f(x) = e^x</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D453” d=“M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z”></path></g><g data-mml-node=“mo” transform=“translate(550,0)”><path data-c=“28” d=“M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z”></path></g><g data-mml-node=“mi” transform=“translate(939,0)”><path data-c=“1D465” d=“M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g><g data-mml-node=“mo” transform=“translate(1511,0)”><path data-c=“29” d=“M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z”></path></g><g data-mml-node=“mo” transform=“translate(2177.8,0)”><path data-c=“3D” d=“M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z”></path></g><g data-mml-node=“msup” transform=“translate(3233.6,0)”><g data-mml-node=“mi”><path data-c=“1D452” d=“M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z”></path></g><g data-mml-node=“TeXAtom” transform=“translate(499,363) scale(0.707)” data-mjx-texclass=“ORD”><g data-mml-node=“mi”><path data-c=“1D465” d=“M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g></g></g></g></g></svg></mjx-container></span> é <span class="math-inline ">
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Suponha que convirja quando x = -4 e divirja quando x = 6. O que pode ser dito sobre a convergência ou divergência das séries a seguir?
a.
b.
c.
d.
<html><head></head><body><div>Considere a seguinte equação diferencial:</div>
<div><span class=“math-block “>
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<html><head></head><body><div>Encontrar a solução geral das seguintes equações de segunda ordem e determine se as soluções são fundamenntais.<br>
(a) <span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.464ex;” xmlns=“
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