Documentos

1

Curso 2006-2007

Tema 1. Introducci´n a las EDP. o

Asignatura: M´todos Matem´ticos de la Ingenier´ Qu´ e a ?a ?mica Profesores: Emanuele Schiavi y Ana Isabel Mu˜ oz. n Apuntes elaborados por: C. Conde (UPM), E. Schiavi (URJC) y A.I. Mu˜ oz n (URJC).

´ Indice

1

´ Indice

1.

Generalidades. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Notaci´n yConceptos fundamentales. Ejemplos . . . . . . . . o 1.1.1.

4 5

Una aplicaci´n de la teor´ de campos . . . . . . . . 13 o ?a

2.

Ecuaciones cuasilineales de primer orden . . . . . . . . . . . . . . . . 21 2.1. Teor´ matem´tica general . . . . . . . . . . . . . . . . . . . . 22 ?a a 2.1.1. 2.1.2. 2.1.3. 2.1.4. 2.1.5. 2.1.6. 2.1.7. 2.1.8. 2.1.9. 2.1.10. 2.1.11. 2.1.12. 2.2. Interpretaci´ngeom´trica . . . . . . . . . . . . . . . 25 o e El proceso de resoluci´n . . . . . . . . . . . . . . . . 28 o Interpretaci´n geom´trica del problema del ?ujo poo e tencial: deducci´n del m´todo de las trayectorias . . 32 o e El problema de Cauchy para ecuaciones cuasilineales 36

El caso param´trico . . . . . . . . . . . . . . . . . . 41 e
? ?

Parametrizaci´n mediante la longitud de arco . . . 44 oUn ejemplo de formaci´n de ondas de choque en o una ecuaci´n cuasilineal. Ecuaci´n de B¨rgers . . . . 45 o o u
? ?

Discontinuidades y soluciones d´biles . . . . . . . . 47 e

Propagaci´n de discontinuidades en EDP de primer o orden . . . . . . . . . . . . . . . . . . . . . . . . . . 49
? ? ?

Valores iniciales discontinuos . . . . . . . . . . . . 50 Derivadas iniciales discontinuas . .. . . . . . . . . 51 Las ecuaciones de Pfa? . . . . . . . . . . . . . . . . 51

Introducci´n progresiva de las EDP hiperb´licas de primer orden 54 o o 2.2.1. 2.2.2. El caso lineal homog´neo con coe?cientes constantes e 55

El caso lineal no homog´neo con coe?cientes constantes 56 e

2.3.

Sistemas hiperb´licos con coe?cientes constantes . . . . . . . . 58 o

2

´ Indice

2.4. 2.5.2.6.

Ecuaciones hiperb´licas lineales homog´neas con coe?cientes o e variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
? ?

Sistemas hiperb´licos con coe?cientes variables . . . . . . . . 64 o

Problemas de valor inicial y de contorno para una EDP de primer orden con coe?cientes constantes . . . . . . . . . . . . 65 2.6.1. 2.6.2.
?

Problemas de valor inicial y decontorno para sistemas de primer orden con coe?cientes constantes . . 67
?

Problemas peri´dicos . . . . . . . . . . . . . . . . . 69 o 70

2.7. 3.

?

Sistemas de leyes de conservaci´n: la notaci´n de operadores o o

Ecuaciones lineales de segundo orden . . . . . . . . . . . . . . . . . . 73 3.1. 3.2. 3.3. Ecuaciones de segundo orden: curvas caracter´ ?sticas y clasi?caci´n . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 76 o Problemas de contorno . . . . . . . . . . . . . . . . . . . . . . 82 Condiciones iniciales y de contorno . . . . . . . . . . . . . . . 83

4.

Ecuaciones el´ ?pticas. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.1. Ecuaciones de Laplace y de Poisson . . . . . . . . . . . . . . . 88 4.1.1. 4.1.2. 4.1.3. 4.1.4. 4.2. 4.3.Laplaciana bidimensional en coordenadas polares . . 90 Laplaciana tri-dimensional en coordenadas cil´ ?ndricas 91 Laplaciana tri-dimensional en coordenadas esf´ricas . 91 e Laplaciana N-dimensional en coordenadas radiales . 92

Algunos fen´menos f´ o ?sico-t´cnicos que modelizan . . . . . . . . 92 e
?

F´rmulas de Green . . . . . . . . . . . . . . . . . . . . . . . 93 o Principio del m´ximo parafunciones arm´nicas . . . 96 a o Unicidad para la ecuaci´n de Laplace . . . . . . . . . 96 o

4.3.1. 4.3.2. 5.

Ecuaciones parab´licas. . . . . . . . . . . . . . . . . . . . . . . . . . . 97 o 5.1. 5.2. La ecuaci´n de difusi´n y algunas variantes . . . . . . . . . . . 97 o o
?

Algunos fen´menos f´ o ?sico-t´cnicos que modelizan . . . . . . . 101 e

6.

Anexo . . . . . . . . . . . ….

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