Equa\303\247\303\265es integraisvon Dr. M. P\303\266hlmannEmail: Poehlmann001@aol.comResolvendo equa\303\247\303\265es integrais simples de Volterra usando a transformada de Laplace.
Com a ajuda da transforma\303\247\303\243o de Laplace, as equa\303\247\303\265es integrais lineares de Volterra podem ser resolvidas com relativa facilidade. As equa\303\247\303\265es integrais s\303\243o transformadas em uma equa\303\247\303\243o de imagem alg\303\251brica que pode ser facilmente resolvida usando a transformada de Laplace. O mesmo m\303\251todo tamb\303\251m pode ser usado para resolver equa\303\247\303\265es diferenciais integro lineares simples, como mostrado na planilha.
Carregando o pacote para transforma\303\247\303\243o integral - pacote "inttrans".
Exemplo 1: Come\303\247amos com uma equa\303\247\303\243o integral linear da forma: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A fun\303\247\303\243o que voc\303\252 est\303\241 procurando
"x (t)"
est\303\241 presente \303\240 esquerda e integral \303\240 direita da Equa\303\247\303\243o (1). Esta equa\303\247\303\243o integral n\303\243o pode ser rastreada at\303\251 uma equa\303\247\303\243o diferencial ordin\303\241ria, conforme mostrado a seguir: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 diferencia\303\247\303\243o da equa\303\247\303\243o integral (1) leva a uma equa\303\247\303\243o diferencial integral linear. Resolver esta equa\303\247\303\243o est\303\241, portanto, se tornando mais complicado do que f\303\241cil.
No entanto, se aplicarmos a transformada de Laplace \303\240 equa\303\247\303\243o (1), obtemos uma equa\303\247\303\243o de imagem alg\303\251brica simples. A transforma\303\247\303\243o da equa\303\247\303\243o integral (1) produz: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A equa\303\247\303\243o alg\303\251brica (3) agora pode ser facilmente decomposta na vari\303\241vel transformada
"
"laplace (x (t), t, s)"
"dissolva e voc\303\252 obt\303\251m: 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 equa\303\247\303\243o (4) agora pode ser transformada novamente no dom\303\255nio do tempo (transforma\303\247\303\243o inversa de Laplace com "invlaplace"). Isso d\303\241 a solu\303\247\303\243o para a equa\303\247\303\243o integral (1):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Queremos verificar se a solu\303\247\303\243o est\303\241 correta e, portanto, definir a fun\303\247\303\243o da solu\303\247\303\243o x (t) com a equa\303\247\303\243o (5):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 a fun\303\247\303\243o de solu\303\247\303\243o na equa\303\247\303\243o integral (1) e simplificar a representa\303\247\303\243o integral resulta em: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Os lados esquerdo e direito da equa\303\247\303\243o (7) s\303\243o id\303\252nticos. Ent\303\243o a solu\303\247\303\243o est\303\241 certa!
Exemplo 2: No exemplo a seguir, escolhemos uma equa\303\247\303\243o integral de Volterrasche da forma: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 equa\303\247\303\243o integral (8) pode ser transformada na equa\303\247\303\243o de imagem com a ajuda da transforma\303\247\303\243o de 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transformar. A resolu\303\247\303\243o ap\303\263s a transforma\303\247\303\243o de Laplace fornece:
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 transforma\303\247\303\243o inversa da equa\303\247\303\243o (10) no dom\303\255nio do tempo fornece a solu\303\247\303\243o para a equa\303\247\303\243o integral (8):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Defini\303\247\303\243o da fun\303\247\303\243o de solu\303\247\303\243o x (t):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 a corre\303\247\303\243o da solu\303\247\303\243o inserindo a fun\303\247\303\243o da solu\303\247\303\243o (12) na equa\303\247\303\243o integral e simplifica\303\247\303\243o: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Os lados direito e esquerdo da equa\303\247\303\243o (13) s\303\243o id\303\252nticos. A solu\303\247\303\243o (11) est\303\241 correta!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 3: Solu\303\247\303\243o de uma equa\303\247\303\243o diferencial linear de Volterra Integro
A equa\303\247\303\243o diferencial integro surge ao diferenciar uma equa\303\247\303\243o integral de Volterra de acordo com a vari\303\241vel de tempo. No nosso exemplo, diferenciamos a equa\303\247\303\243o integral (8) e obtemos a equa\303\247\303\243o integrodiferencial: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\303\247\303\265es integrodiferenciais lineares tamb\303\251m podem ser resolvidas pela transforma\303\247\303\243o de Laplace. O uso da transforma\303\247\303\243o direta de Laplace produz: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As equa\303\247\303\265es integrodiferenciais lineares tamb\303\251m podem ser resolvidas pela transforma\303\247\303\243o de Laplace. O uso da transforma\303\247\303\243o direta de Laplace produz: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A transforma\303\247\303\243o inversa no dom\303\255nio do tempo fornece a solu\303\247\303\243o da equa\303\247\303\243o diferencial inteira (14):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A condi\303\247\303\243o inicial ainda est\303\241 contida na solu\303\247\303\243o
"x (0)"
. No nosso caso, usamos: 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Representa\303\247\303\243o da solu\303\247\303\243o da equa\303\247\303\243o (14):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Defini\303\247\303\243o da fun\303\247\303\243o de solu\303\247\303\243o x (t):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Verifique a corre\303\247\303\243o da solu\303\247\303\243o inserindo-a na equa\303\247\303\243o integrodiferencial (14):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 4: Equa\303\247\303\243o integral de Volterra com fun\303\247\303\243o hiperb\303\263lica
Resolvemos a seguinte equa\303\247\303\243o integral linear, na qual as fun\303\247\303\265es hiperb\303\263licas ocorrem no n\303\272cleo integral: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Etapa 1: colocar a transforma\303\247\303\243o na \303\241rea da 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Etapa 2: Resolvendo a equa\303\247\303\243o da imagem ap\303\263s a transforma\303\247\303\243o de 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Etapa 3: Transforme a equa\303\247\303\243o da imagem novamente no dom\303\255nio do 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Defini\303\247\303\243o da fun\303\247\303\243o da solu\303\247\303\243o x (t)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Verificando a corre\303\247\303\243o da solu\303\247\303\243o (inserindo na equa\303\247\303\243o integral):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Representa\303\247\303\243o da solu\303\247\303\243o da equa\303\247\303\243o integral (21):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 5: Solu\303\247\303\243o de um sistema de duas equa\303\247\303\265es integrais
As transformadas de Laplace tamb\303\251m podem ser usadas para resolver sistemas simples de equa\303\247\303\265es integrais. Queremos resolver um sistema de duas equa\303\247\303\265es integrais lineares. As equa\303\247\303\265es s\303\243o:
- Equa\303\247\303\243o 1: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- Equa\303\247\303\243o 2: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Etapa 1: colocar a transforma\303\247\303\243o na \303\241rea da 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Etapa 2:Ap\303\263s a transforma\303\247\303\243o, obtemos um sistema de duas equa\303\247\303\265es alg\303\251bricas, que podem ser resolvidas com o solucionador "resolver" ap\303\263s as duas transforma\303\247\303\265es. Fazemos as substitui\303\247\303\265es por uma quest\303\243o de 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Para os transformados, obtemos as duas equa\303\247\303\265es (sistema):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Etapa 3: transforma\303\247\303\243o de volta ao dom\303\255nio do tempo
Obtemos as fun\303\247\303\265es x [1] (t) e x [2] (t) as solu\303\247\303\265es :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\303\247\303\265es funcionais da solu\303\247\303\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Verificando a exatid\303\243o das solu\303\247\303\265es, inserindo-as nas equa\303\247\303\265es integrais (27) e 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lados esquerdo e direito das equa\303\247\303\265es da amostra coincidem. As solu\303\247\303\265es est\303\243o certas!
Representa\303\247\303\243o do comportamento das solu\303\247\303\265es no 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dos valores iniciais em t = 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