Solucion Del Examn Parcial

7/23/2019 Solucion Del Examn Parcial

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Universidad

Nacional de Callao

Ciencia y Tecnología rumbo al tercer milenio

7

ECUACIONES DIFERENCIALES PARCIALES

Profesor: CASTRO VIDAL RAUL PEDRO

Integrantes:

ABANTO LÓPEZ MIJAIL ANDRÉS 132322033 ADVINCULA!EVARISTO!IVAN RIC"ARD 1323220#$ RICALDI CRISTOBAL JUNIOR ARTURO 13232200%1 SUCASAIRE!MERMA!C"RISTIAN LUCAS 13232201&# 'EDEME(ER DE LAS CASAS JOSÉ ANDRES 1323220212 (ABAR!PRINCIPE!CRISTIAN ENRI)UE 13232101&

ECUACIONESDIFERENCIALES

GRUPO:



ECUACIONES DIFERENCIALESPARCIALES

I. OBJETIOS

• R*+,n,+*- +/ai+a- na *+a+in 4i5*-*n+ia/ 6a-+ia/.• R*,/7*- na *+a+in 4i5*-*n+ia/ 6a-+ia/ +,n */ 89:,4, 4*

*6a-a+in 4* 7a-ia;/*.• R*,/7*- /a a6/i+a+i,n* 4* na *+a+in 4i5*-*n+ia/ 6a-+ia/ +,n */

89:,4, 4* *6a-a+in 4* 7a-ia;/*.

II. INTRODUCCION

M+<a 7*+* n, 6-**n:a8, an:* 6-,;/*8a 8a:*8á:i+,= 5>i+,=?>8i+,= *:+. ?* 6a-*+*n n, :*n*- ,/+in= ana/i@a8, n*:-,4a:,= n*:-a 7a-ia;/* :-a:a8, 4* -*,/7*- */ 6-,;/*8a 4* 5,-8a+,n7*n+i,na/= a> ?* 4*;*8, ;+a- ,:-, :i6, 4* ,/+i,n* n :an:,8á +,86/*a. La ,/+in 4* *+a+i,n* 4i5*-*n+ia/* 6a-+ia/*8+<a 7*+* * 6*4* i86/i+a- a -*,/7*- na *+a+in 4i5*-*n+ia/6a-+ia/ 4* *gn4, ,-4*n. P-,;/*8a :>6i+, ,n /a 6-,6aga+in 4*/

,ni4, , 4*/ +a/,-= /a */*+:-,:á:i+a= /a */*+:-,4iná8i+a= /a 4iná8i+a 4*i4,= /a */a:i+i4a4= /a 8*+áni+a +án:i+a 8+<, ,:-,. S* /a+,n,+* :a8;i9n +,8, *+a+i,n* 4i5*-*n+ia/* 6a-+ia/*.

III. DEFINICION DE ECUACION DIFERENCIAL PARCIAL!ORDEN " LINEALIDAD

Una *+a+in 4i5*-*n+ia/ 6a-+ia/ EDP * a?*//a +a in+gni:a ,n5n+i,n* 4* 4i7*-a 7a-ia;/*= +,n /a 6*+/ia-i4a4 4* ?* *n 4i+<a*+a+in g-an n, ,/, /a 6-,6ia 5n+i,n* in, :a8;i9n

4*-i7a4a. Una 7a-ia;/* in4*6*n4i*n:* 6*4*n *- :i*86, 6,i+in= ,;i*n 4, +,,-4*na4a 4* 6,i+in , na +,8;ina+in 4* 7a-ia 7a-ia;/*+,n */ :i*86,.

1. #Enton$es! %&' es &na e$&a$()n *(feren$(a+ ,ar$(a+-

UNIVERSIDAD NACIONAL DEL CALLAOPágina 1



E na *6-*in 8a:*8á:i+a ?* +,n:i*n* na , 8á 7a-ia;/*4*6*n4i*n:* 4, , 8á 7a-ia;/* 4*6*n4i*n:*. Una *+a+in4i5*-*n+ia/ *n 4*-i7a4a 6a-+ia/* 6,- *8*an@a +,n /a EDO= * na*+a+in 4,n4* na +i*-:a 5n+in in+gni:a 7i*n* 4*ni4a 6,- na-*/a+in *n:-* 4*-i7a4a 6a-+ia/* +,n -*6*+:, a /a 7a-ia;/*

in4*6*n4i*n:*.

Si u=u ( x , y , z ) , na *+a+in 4i5*-*n+ia/ *n 4*-i7a4a 6a-+ia/* *->aF

F ( x , y , z , ∂u

∂ x ,

∂u

∂ y ,

∂ u

∂ z ,

∂2u

∂ x2 ,

∂2u

∂ y2 , ∂

2u

∂ z2 ,

∂2u

∂ x∂ y ,…)=0

Ee/,+os:

a∂2u

∂ x2+

∂2u

∂ y2=0

;∂2u

∂ x2=

∂2u

∂ t 2−2

∂ u

∂ t

0. Or*en &na E$&a$()n D(feren$(a+ Par$(a+

La *+a+i,n* 4i5*-*n+ia/* 6a-+ia/* * 6*4*n +/ai+a- 4* a+*-4, a ,-4*n= /in*a/i4a4 :i6, 4* +,n4i+i,n* 4* 5-,n:*-a.

E/ ,-4*n 4* na *+a+in 4i5*-*n+ia/ 6a-+ia/ *:á 4*:*-8ina4, 6,- /a

4*-i7a4a 6a-+ia/ 4* 8a,- ,-4*n 6-**n:* *n /a *6-*in +,8, *8*:-a *n /, igi*n:* **86/,F

Ee/,+o:

a




∂2u

∂ x2+

∂2u

∂ y2=0

E na EDP 4* *gn4, ,-4*n

2 La igi*n:* EDP

u× ∂u

∂ x +

∂u

∂ y=0

E na PDE 4* 6-i8*- ,-4*n.

3. E4(sten$(a 5 &n($(*a*

An?* */ an:, 4* /a *i:*n+ia ni+i4a4 4* /a ,/+i,n* 4*/a *+a+i,n* 4i5*-*n+ia/* ,-4ina-ia :i*n* na -*6*:a 8a:i5a+:,-ia -*8i4a *n */ :*,-*8a 4* Pi+a-4!Lin4*/G5= */ 8i8, an:,6a-a /a *+a+i,n* *n 4*-i7a4a 6a-+ia/* *:á /*, 4* *:a-a:i5a+:,-ia8*n:* -**/:,. An?* *i:* n :*,-*8a g*n*-a/=*/ :*,-*8a 4* Ca+<!H,7a/*7aa= ?* a-8a ?* 6a-a na EDP= ?** ana/>:i+a *n /a 5n+in in+gni:a 4*-i7a4a= :i*n* na ni+a,/+in ana/>:i+a.

An?* *:* -*/:a4, ?* 6a-*+* *:a;/*+*- /a *i:*n+ia ni+i4a4 4*/a ,/+i,n*= a6a-*+*n **86/, 4* EDP 4* 6-i8*- ,-4*n +,+,*+i*n:* :i*n*n 4*-i7a4a 4* +a/?i*- ,-4*n an?* in *-ana/>:i+a 6*-, ?* n, :i*n*n ,/+in. In+/, i /a ,/+in 4* na EDP*i:* * ni+a= 9:a 6*4* :*n*- 6-,6i*4a4* in4**a;/*.

Un **86/, 4* +,86,-:a8i*n:, 6a:,/gi+, * /a *+*n+ia 4* 6-,;/*8a4* Ca+< 4*6*n4i*n:* 4*/ 6a-á8*:-, n 6a-a /a *+a+in 4* La6/a+*F

∂2u

∂ x2+

∂2u

∂ y2=0

C,n +,n4i+i,n* ini+ia/*


https://es.wikipedia.org/wiki/Funci%C3%B3n_anal%C3%ADtica

https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial#Orden_de_la_ecuaci.C3.B3n

https://es.wikipedia.org/w/index.php?title=Condici%C3%B3n_inicial&action=edit&redlink=1

https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial#Orden_de_la_ecuaci.C3.B3n

https://es.wikipedia.org/w/index.php?title=Condici%C3%B3n_inicial&action=edit&redlink=1

https://es.wikipedia.org/wiki/Funci%C3%B3n_anal%C3%ADtica



u ( x ,0)=0, ∂u

∂ y ( x ,0 )=

sinnx

n

D,n4* n * n *n:*-,. La 4*-i7a4a 4* +,n -*6*+:, a K y *

a6-,i8a a 0 ni5,-8*8*n:* *n x a 8*4i4a ?* n * in+-*8*n:a=

6*-, /a ,/+in *F

u ( x , y )=(sinh ny ) (sinh nx )

n2

E:a ,/+in * a6-,i8a a inni:, i nx n, * n *n:*-, 8/:i6/, 4*

6a-a +a/?i*- 7a/,- 4* . E/ 6-,;/*8a 4* Ca+< 6a-a /a *+a+in 4*La6/a+* * 4*n,8ina K8a/ 6-,6*:, , 8a/ 4*ni4,= 6*:, ?* /a,/+in n, 4*6*n4* +,n:ina8*n:* 4* /, 4a:, 4*/ 6-,;/*8a. E:,6-,;/*8a 8a/ 4*ni4, n, ,n a/8*n:* a:i5a+:,-i, 6a-a /aa6/i+a+i,n* 5>i+a.

I. CLASIFICACI6N DE LAS EDP DE SEGUNDOORDEN.

La EDP 4* *gn4, ,-4*n * +/ai+an <a;i:a/8*n:* 4*n:-, 4* +a:-,:i6, 4* EDP ?* ,n 4* in:*-9 5n4a8*n:a/= a +,n:ina+in * 4an**86/, 4* *:, +a:-, :i6,F

E$&a$()n No/2re T(,o

∇2u=0 La6/a+* E/>6:i+a

UNIVERSIDAD NACIONAL DEL CALLAOPágina



∂2u

∂ t 2=c

2∇2u

On4a "i6*-;/i+a

∂u

∂ t

=k ∇2u

Di5in Pa-a;/i+a

∇2u=ku "*/8<,/:@ E/>6:i+a

C,n 8a,- g*n*-a/i4a4= i * :i*n* na *+a+in 4* *gn4, ,-4*n 4*/:i6,F

A u xx+2 Bu xy+C u yy+ D u x+ E u y+ F =0

• * 4i+* ?* * elíptica i /a 8a:-i@ Z =[ A B

B C ] :i*n* n

4*:*-8inan:* 8a,- a 0.

• S* 4i+* ?* * parabólica i /a 8a:-i@ Z =[ A B

B C ] :i*n* n

4*:*-8inan:* iga/ a 0.

• S* 4i+* ?* * hiperbólica i /a 8a:-i@ Z =[ A B

B C ] :i*n* n

4*:*-8inan:* 8*n,- a 0.

. LA TRANSFOR7ADA DE FOURIER

1. DEFINICI6N " PROPIEDADES FUNDA7ENTALES

Una 4* /a <*--a8i*n:a 8á :i/* *n */ *:4i, 4* /a E+a+i,n* *nD*-i7a4a Pa-+ia/* EDP * /a :-an5,-8a4a 4* ,-i*-. Da4a na5n+in 5 L1 Rn 4*ni8, :-an5,-8a+in 4* ,-i*- 4*/ 8,4,igi*n:*F

UNIVERSIDAD NACIONAL DEL CALLAOPágina #



2 π ¿¿

f (ξ )=1 /¿

C,n7i*n* *Qa/a- ?* 5 *:á ;i*n 4*5ini4a 6a-a :,4, R

n

6*:, ?*e

ix·ξ

5 L1Rn. E:, * *5*+:i7a8*n:* +i*-:, a ?*F

|e−ix·ξf ( x)|=|f ( x)|∈ L

1( Rn)

D* *:a 6-,6i*4a4 * 4*4+* ?*F

‖f ‖ L∞( Rn)≤‖f ‖ L

1( Rn)

P,- ,:-a 6a-:*= */ T*,-*8a 4* /a C,n7*-g*n+ia D,8ina4a TCD 6*-8i:*7*- ?* 5 * :a8;i9n na 5n+in +,n:ina. P,- :an:,= 5 BCRn.

La :-an5,-8a4a 4* ,-i*- :i*n* 7a-ia 6-,6i*4a4* 5n4a8*n:a/* ?*/a <a+*n n, ,/a8*n:* na <*--a8i*n:a 8 :i/ *n */ +,n:*:, 4* /aEDP in, :a8;i9n *n 8+<, ,:-, á8;i:, 4* /a Ma:*8á:i+a 4* /a,:-a Ci*n+ia T*+n,/,g>a. Una 4* *//a *:á -*/a+i,na4a +,n */+,86,-:a8i*n:, 4* /a :-an5,-8a4a 4* ,-i*- *n -*/a+in a /,,6*-a4,-* 4* 4*-i7a+in. T*n*8,F

^∂ f ∂ x

(ξ )= 1

(2π )n

2∫ R

n

❑

e−ixξ ∂ f ∂ x

( x ) dx= iξ

(2 π )n

2∫ R

n

❑

e−ixξ f ( x ) dx=iξ f (ξ )

-a+ia a *:a 6-,6i*4a4= /a EDP /in*a/* +,n +,*+i*n:* +,n:an:* *-*4+*n a *+a+i,n* a/g*;-ai+a /in*a/* *n */ *6a+i, 4* ,-i*-. En*5*+:,= +,ni4*-*8, 6,- **86/, /a *+a+in 4* La6/a+*F

−∆ u=f en Rn

T,8an4, /a :-an5,-8a4a 4* ,-i*- ,;:*n*8,F

|ξ|2 u ( ξ )=f (ξ ) ξenRn…(1)

UNIVERSIDAD NACIONAL DEL CALLAOPágina %



En /a ,;:*n+in 4* /a *+a+in an:*-i,- <*8, :i/i@a4, 4* 8an*-a**n+ia/ ?*

ξ

i ¿¿¿2¿

∂2u

∂ x i

2=¿−∑

i=1

n

¿

−∆ u=−^

∑i=1

n∂

2u

∂ xi

2=−∑

i=1

n

¿

D* 1 P,4*8, ,;:*n*- 4* 8an*-a in8*4ia:a na *6-*in 6a-a /a

:-an5,-8a4a 4* ,-i*- u 4* /a ,/+in F

u (ξ )=f ( ξ )/|ξ|2

P*-,= 6a-a ,;:*n*- /a ,/+in n*+*i:a8, in7*-:i- 4* /a :-an5,-8a4a4* ,-i*-. La :-an5,-8a4a in7*-a 4* ,-i*- * 4*n* 4*/ 8,4,igi*n:*F

f ( x )= 1

(2 π )n2

∫ R

n

❑

e−ixξ

f (ξ )dξ

N*7a8*n:*= i 5 L1 Rn = f *:á ;i*n 4*ni4a * na 5n+in 4*

BCRn .

La aiana * n **86/, 6a-:i+/a-8*n:* i86,-:an:* *n */ ?* /a:-an5,-8a4a 4* ,-i*- * 5á+i/ 4* +a/+/a- *6/>+i:a8*n:*. T*n*8,

^e−t x

2

(ξ )= 1

2t

n /2

e−|ξ2|/4 t

En *5*+:,=

^e−t | x|

2

(ξ )= 1

(2π )n

2

∫ R

n

❑

e−ixξ

e−t | x|2

dx=∏ =1

n1

(2π )1

2

∫ R

n

❑

e−i x ξ e

−t | x |2

d x




Va8, a +a/+/a- /a igi*n:* in:*g-a/F

iξ

2 t

x+¿¿¿2−t ¿e¿

∫ Rn

❑

e−ixξ

e−t | x|2

dx=∫ Rn

❑

¿

D,n4* * */ +,n:,-n, I8 @ W2: 4*/ 6/an, +,86/*,. D*5,-8an4, /,+,n:,-n, /a in:*g-a/= 6,- */ :*,-*8a 4* /, -*i4,= 6*4* -*4+i-* a/

+á/+/, 4* /a 8i8a ,;-* */ ** -*a/F

∫ !

❑

e−t z

2

dz=∫−∞

∞

e−t x

2

dx=√ π / t

D* /a 4, /:i8a *+a+i,n* ,;:*n*8,F

π

t ¿¿

∫ R

❑

e−ixξ

e−t x

2

dx=e

−ξ 2

4 t ¿

En *5*+:,= 4a4a 5 Rn= i 4*ni8, /a 5n+in

ft ( x )= f (tx)

T*n*8,

f (ξ )=1

t n

n /2

f (ξ / t )

UNIVERSIDAD NACIONAL DEL CALLAOPágina $



Pa-a +,86-,;a- <a+*8, /, igi*n:*F

f t (ξ )= 1

(2π )n2

∫ R

e−ixξ

f t ( x ) dx= 1

(2 π )n2

∫ R

n

e−ixξ

f ( tx ) dx= 1

(2π )n2 t

n

∫ R

n

e−iy ξt f ( y ) dy

La i4*n:i4a4 ,;-* /a :-an5,-8a4a 4* ,-i*- 4* /a aiana * 8X:i/. En 6a-:i+/a- 6-,6,-+i,na /a ,/+in 5n4a8*n:a/ 4* /a *+a+in4*/ +a/,-.

En *5*+:,= -*+,-4*8, ?* * /a ,/+in 4*/ i:*8a

−| x|2 "/4 t

" ( x , t )=(4 πt )

−n

2

exp ¿

Y

{¿−e$t%"=0en Rn x (0,∞ )

" ( x ,0 )=& ( x ) en Rn

T,8an4, /a :-an5,-8a4a 4* ,-i*- *n /a 6-i8*-a *+a+in

YY

"t +|ξ|2ϱ=0,ξϵ R

n, t >0

P,- /, :an:,

" ( ξ , t )=e−|ξ|2t " (ξ ,0 ) …(¿∗¿)

C,n7i*n* ,;*-7a- ?* *n YY /, * :,8a /a :-an5,-8a4a 4* ,-i*-*n /a 7a-ia;/* *6a+ia/= /, ?* :-an5,-8a /a *+a+in *n 4*-i7a4a6a-+ia/* *n na *+a+in 4i5*-*n+ia/ ,-4ina-ia +,n 6a-á8*:-, .

P,- ,:-a 6a-:*= *n 7i-:4 4*/ 4a:, ini+ia/ 4* Y :*n*8,

UNIVERSIDAD NACIONAL DEL CALLAOPágina &



" ( ξ ,0 )= & (ξ)

P,- ,:-a 6a-:*

¿∗¿∗¿

& ' 1

(2π )n

2

…¿

S,n 7a-ia /a 8an*-a 4* :i+a- /a i4*n:i4a4 4* /a *+a+in an:*-i,-.Una 6-i8*-a * i86/*8*n:* a6/i+a- /a 4*ni+in 4* /a :-an5,-8a4a 4*,-i*-= :*ni*n4, *n +*n:a ?* [ * na 8*4i4a. S* ,;:i*n*F

& 0 (ξ )= 1

(2π )n2

∫ R

n

e−ixξ

& 0 ( x )dx= 1

(2π )n2

O:-a 6,i;i/i4a4 * ,;*-7a- ?* /a 5a8i/ia

( t

π )

n/2

e−| x|2 t

E na a6-,i8a+in 4* /a i4*n:i4a4 +an4, : \ ] 4* 8,4, ?*

∫ Rn

(

(t

π

)n /2

e−| x|2t ) ( x ) dx*) (0)

Pa-a :,4, ^ BC Rn. En:,n+*

& 0 (ξ )=limt * ∞

( t π )n

2^e−t | x|2

( ξ )=limt * ∞

1

(2π )n

2

e−|ξ|

2/4 t =

1

(2π )n

2

C,8;inan4, YYY YYYY ,;:*n*8,

ϱ ( ξ , t )= 1

(2π )n

2

e−|ξ|2t




Ba:a 6,- X/:i8, a6/i+a- /a :-an5,-8a4a in7*-a 4* ,-i*- 6a-a,;:*n*- = : a 6a-:i- 4* :-an5,-8a4a 4* ,-i*- 4* /a *+a+inan:*-i,-. O;:*n*8, *n:,n+*

"( x , t

)=4 πt

−n

2

exp

(−| x|2

4 t

)E:a /:i8a 5,-8/a * ,;:i*n* 4* 8an*-a 5á+i/ g-a+ia a *:a ,:-a

*+a+inF^e−t x

2

(+ )= 1

2 t

n /2

e−|+ 2|/4 t

En 7i-:4 4* __ * ,;:i*n*F

f ( + )= f (−+)

Siga8, a<,-a *:4ian4, /a 6-,6i*4a4* 5n4a8*n:a/* 4* /a:-an5,-8a4a 4* ,-i*-. La i4*n:i4a4 4* P/an+<*-*/ ga-an:i@a ?*

‖f ‖ L

2 ( Rn )=‖f ‖ L2 ( Rn )

Pa-a :,4, 5 L 1 R n ` L 2 R n.

An:* 4* 6-,;a- /a /:i8a *+a+in ,;*-7a8, ?* *:a i4*n:i4a4

ga-an:i@a ?* /a :-an5,-8a4a 4* ,-i*- 4*n* na i,8*:->a 4* L 2 Rn *n L 2 R n . A4*8á= /a :-an5,-8a4a 4* ,-i*- 6*4* *:*n4*-*6,- 4*ni4a4 a na i,8*:->a 4*ni4a *n :,4, */ *6a+i, L2 Rn.

Pa-a 6-,;a- * n*+*i:aF

∫ R

n

( x ) -( x )dx=∫ R

n

( x ) - ( x)dx

Pa-a :,4, 7= L1 Rn. C,n7i*n* *n 6-i8*- /ga- ,;*-7a- ?* a8;ain:*g-a/* *:án ;i*n 4*ni4a 6*:, ?* +,8, 7= L1 Rn =*n:,n+* 7= L] Rn = 6,- /, ?* 7 L1 Rn.

La i4*n:i4a4 an:*-i,- * 5á+i/ 4* +,86-,;a-. En *5*+:,




∫ R

n

❑

( x ) -( x )dx=∫ R

n

❑

( x ) 1

(2π )n2

∫ R

❑

e−ixy

- ( y )dy= 1

(2 π )n2

∬ R

n

❑

e−ixy

( x ) - ( y ) dxdy

)* +,in+i4* +,n /a *6-*in 4*

∫ R

n

( x ) -( x )dx

A6/i+an4, *:a i4*n:i4a4 +,n 7 * bcdd 2 an4, /a *6-*in 4*7 an:* +a/+/a4a

y

^

-(¿)e−+|ξ|2dξ=

1

(2 + )n

2

∫ R

n

❑

- ( x)e−| x|2/ 4+

dx

∫ R

n

❑

¿

O;:*n*8, ?*

Da4a L 1 R n ` L 2 R n = 4*ni8, 7 F eb f 7?* 6*-:*n*+* a L 1 R n ` CR n . T*n*8,

-=u∗=(2π )n /2 u ϵ L∞ ( Rn )… (. )

E:a * ,:-a 4* /a 6-,6i*4a4* i86,-:an:* 4* /a :-an5,-8a4a 4*,-i*-F /a :-an5,-8a4a 4* ,-i*- 4* /a +,n7,/+i,n *= 84/, na+,n:an:* 8/:i6/i+a:i7a= iga/ a/ 6-,4+:, 4* /a :-an5,-8a4a 4*

,-i*-.En *:* +a,= +,8, 7 eb= :*n*8,

(ξ )= 1

(2 π )n2

∫ R

n

e−ixξ

u (− x )dx=u (ξ)




6,- :an:, - ¿u/ (ξ)∨¿2((2 π )n/2¿

C,8, * +,n:ina :*n*8, :a8;i9n

2+¿¿

¿n/2¿2π ¿¿1¿

lim+* ∞

¿

A4*8á

2π ¿¿

¿n/2¿

∫ R

n

❑

|u|2

dξ=1

¿

L, +a/ +,n+/* /a 6-*;a 4* /a i4*n:i4a4 4* P/an+<*-*/.

C,86-,;a8, a<,-a /a 5,-8/a V ?* 6,n* *n -*/a+in /a +,n7,/+i,n /a :-an5,-8a4a 4* ,-i*-. T*n*8,

2π ¿¿

e−i+x❑ (¿u∗)( x )dx

- ( + )=^u∗ (+ )=¿

D*69 4* 4*a--,//a-= in:*g-a- 7a8, a ,;:*n*- /a igi*n:* *+a+inF




2π ¿¿

- ( + )=¿

P-,;*8, a/gna 6-,6i*4a4* 8á 4* /a :-an5,-8a4a 4* ,-i*-. Lai4*n:i4a4 4* Pa-*7a/ ga-an:i@a ?*

(¿u d+)

(¿u dx)=∫ R

n

❑

¿

∫ Rn

❑

¿

Pa-a +,86-,;a-/,= 4a4, = 7 C= a6/i+a8, /a i4*n:i4a4 4*

P/an+<*-*/ a /a 5n+in h 7 4* 8,4, ?*

‖u+∝‖2

L2( R n)=‖u+∝‖2 L2 ( Rn )

E6an4i*n4, *:a i4*n:i4a4 :*n*8,

∫ R

n

[|u|2+|0 |2+u (0 )+u( 0 )] dx=∫ R

n

[|u|2+|0 |2+´u ( 0 )+u( 0 ´ )] dξ

T,8an4, +*i7a8*n:* 1 i *n *:a i4*n:i4a4 ,;:*n*8,.C,86-,;a8, 6,- X/:i8, /a 5-8/a 4* in7*-in 4* /a :-an5,-8a4a 4*,-i*-= * 4*+i-=

(f )∨= f (

Da4, @ R n c 0 +,ni4*-a8, /a 5n+in + ( x)=eix·z−+∨ x∨2

( T*n*8,

+ ( ξ )= 1

(2π )n2∫ R

n

❑

e

−ix ( y− z)−+| x|2

1 dx= 1

(2+ )n2

e−| y− z|2/4 +

En *:* 6n:, <*8, a4, /a 5-8/aF

)* i4*n:i+a /a :-an5,-8a4a 4* ,-i*- 4* /a aiana




∫ R

❑

e−ixξ

e−tx

2

dx=e−ξ

2/4 t ( π

t )1 /2

D*4+i8, ?*

∫ R

n

❑

f ( ξ ) e−izξ−+|ξ|2

dξ= 1

(2 + )n

2

∫ R

n

❑

f ( x ) e−| x− z|

2

4+ dx (

A<,-a ;i*n= /a *6-*in 4* /a 4*-*+<a 4* *:a i4*n:i4a4 +,n7*-g* a 5@6a-a +ai :,4, @ R n = 8i*n:-a ?* /a 4* /a i@?i*-4a +,n7*-g* a 5j @. E:, +,n+/* /a 6-*;a 4* /a 5-8/a 4* in7*-in. O:-a 4* /a6-,6i*4a4* 8á +/ái+a 4* /a T-an5,-8a4a 4* ,-i*- * /a ?* *,;:i*n* *n -*/a+in a/ ,6*-a4,- 4* :-a/a+in. En *5*+:,= 4a4a na5n+in 5 L 1 R 4*ni8, na :-a/a+in 4* /a 8i8a 4* a86/i:4 a R nF

f % ( x )=f ( x+%)

La :-an5,-8a4a 4* ,-i*- 4* /a :-a/a4a4a 5a 7i*n* 4a4a 6,-F

f % ( + )=ei%(+

f (+ )

En *5*+:,= 8*4ian:* n i86/* +a8;i, 4* 7a-ia;/* * ,;:i*n*

UNIVERSIDAD NACIONAL DEL CALLAOPágina 1#



2 π ¿¿

¿n/2¿2 π ¿¿¿

n

2¿¿¿

f % ( + )=1¿

2π ¿¿

¿ n

2¿

¿ f ❑ ( y ) dy=ei%(+

f (+ )¿

f % (+ )=1¿

D* 8an*-a aná/,ga= * +,86-*;a ?*

^e

i%(+f ( + )=f −% ( + )

Pa-a /a T-an5,-8a4a In7*-a 4* ,-i*- * ,;:i*n*n 5-8/a*8*an:*F

f % ( x)=e−i% (x

f ( x)

(

ei%(+

f ¿¿¿

UNIVERSIDAD NACIONAL DEL CALLAOPágina 1%



I. LA ECUACI6N DE TRANSPORTE LINEAL

La *+a+i,n* ?* 8,4*/an 5*n8*n, 4* 6-,6aga+in 4* ,n4a 7i;-a+i,n* :>6i+a8*n:* E+a+i,n* *n D*-i7a4a 6a-+ia/* EDP 4*,-4*n 4,. Sin *8;-ag, *n :,4a *//a ;a+*n /a *+a+i,n* 4* T-an6,-:* 4* ,-4*n *n n, ?* ana/i@a8, *n *:a *++in.

E/ 8,4*/, 8á *n+i//, *F

1 ut h u x 0

C,8, * 7i, an:*-i,-8*n:* * 5á+i/ +,86-,;a- ?* u=u( x , t ) * ,/+in4* *:a *+a+in i ,/, i * +,n:an:* a /, /a-g, 4* /a />n*a+a-a+:*->:i+a.

2 h : +:*

D* *:* 8,4, 4*4+i8, ?* /a *+a+i,n* 4* 1 ,n 4* /a 5,-8aF

3 u=f ( x+t )

D,n4* 5 * */ 6*-/ 4* /a ,/+in *n */ in:an:* :0 F

u ( x ,0 )=f ( x )

C,8, * 4i, an:*-i,-8*n:* ,;:7i8, *:a 5-8/a 4* -*6-**n:a+in4* /a ,/+in 8*4ian:* /a :-an5,-8a4a 4* ,-i*-.

La ,/+in 3 4* 1 * *n:,n+* na i86/* ,n4a 6-a *n */ ?* */6*-/ 5 * :-an6,-:a a7an@a *n */ ** -*a/ a 7*/,+i4a4 +,n:an:* n,.

A/ in7*-:i- */ *n:i4, 4*/ :i*86, <a+i*n4, */ +a8;i, 4* 7a-ia;/* : a :k /a*+a+in * :-an5,-8a *nF

# ut +u x=0

Ca ,/+i,n* ,n 4* /a 5,-8aF




% u=1 ( x , t )

T-a:án4,* 4* ,n4a 7ia*-a ?* * 6-,6agan *n 4i-*++in ,6*:a +,n7*/,+i4a4 n,. V*8, 6,- :an:, ?* /a ,/+i,n* 4* /a *+a+in 4*:-an6,-:* 6*4*n +a/+/a-* 4* 8an*-a *6/>+i:a ?* *n *//a *,;*-7a n *n+i//, 5*n8*n, 4* :-an6,-:* /in*a/ in 4*5,-8a+in.

C,8, 4*+>a8, *:a *+a+in 4* :-an6,-:* 4* ,-4*n n, ;a+* *n8+<a 4* /a *+a+i,n* 49 ,n4a ?* in:*-7i*n*n *n /a :*,->a 4* /a7i;-a+i,n*. A> 6,- **86/, */ ,6*-a4,- 4* ,n4a

∂t 2∂ x2= (∂t ∂ x )( ∂t +∂ x )=(∂t +∂ x )(∂t ∂ x)

En */ +a, *n */ ?* */ :-an6,-:* * 6-,4@+a *n n 8*4i, n,<,8,g9n*,= /a *+a+in +,--*6,n4i*n:* a4,6:a /a 5,-8aF

ut +C ( x ) u x=0

En *:* +a,= n*7a8*n:* /a ,/+in * +,n:an:* a /, /a-g, 4* /a+a-a+:*->:i+a ?* ,n ,/+i,n* 4*F

$ x 2 ( t )=c ( x (t ) ) , t >0 3x (0)= x0

En *5*+:, * 5á+i/ +,86-,;a- ?* i * ,/+in 4* * :i*n* ?*:= : * in4*6*n4i*n:* 4* : 6a-a +a/?i*- +-7a +a-a+:*->:i+a.

En -*a/i4a4= :a/ +,8, ,+-->an *n /, +,*+i*n:* +,n:an:*= *:a6-,6i*4a4 6*-8i:* 4*ni- /a ,/+in. En *5*+:, 6,ni*n4, ?* C C

* na 5n+in Li6<+<i:@= 6a-a +a4a x0 *i:* na ni+a ,/+in 4*

$. A4*8á /a +a-a+:*->:i+a *:án g/,;a/8*n:* 4*ni4a n, 6*4*n+,-:a-* 6,- ni+i4a4 4* /a ,/+in 4* /a EDO $.P,- /:i8, 6,- +a4a

UNIVERSIDAD NACIONAL DEL CALLAOPágina 1$



6n:, 4*/ *6a+i, :i*86, 1

x1 , t ¿¿ 6aa na ni+a +a-a+:*->:i+a. Pa-a

+,86-,;a-/, ;a:a -*,/7*- /a *+a+inF

x 2 (t )=c ( x ( t )) l

C,n 4a:, ini+ia/ x (t 1 )= x1 ,;*-7a- ?* /a +a-a+:*->:i+a ,/+in 4*

$ 6aa 6,- 1

x1, t ¿¿ i */ 4a:, ini+ia/ x0 +,in+i4* +,n */ 7a/,- 4* *:a

,/+in *n :0= 0.

D* *:* 8,4, 7*8, ?* /a ,/+in 4* /a *+a+in 4* :-an6,-:* *:a g/,;a/8*n:* 4*ni4a 6,- 4a:, ini+ia/

u ( x ,0 )=f ( x ) (

II. LA ECUACI6N DEL CALOR

C,8, <*8, 8*n+i,na4, *n /a in:-,4++in /a *+a+in 4*/ +a/,- * */6-,:,:i6, 4* *+a+in 4* *7,/+in 4* :i6, 6a-a;/i+, +a 7a-ian:**:án 6-**n:* 4* 8an*-a i:*8á:i+a *n :,4, /, 8,4*/,8a:*8á:i+, 4* /a 4i5in 4* /a M*+áni+a 4* /i4,. S* :-a:a 4* n8,4*/, 5*-:*8*n:* i--*7*-i;/* *n :i*86, *n */ ?* /a in5,-8a+in *6-,6aga a 7*/,+i4a4 inni:a. E:a 6-,6i*4a4* ?*4a-an +/a-a8*n:* 4*8ani*:, a /, /a-g, 4* *:a *++in *n /a ?* -*+,-4a8, 6-in+i6a/* 6-,6i*4a4* ana/>:i+a.

E+ ,ro2+e/a *e 8a+ores (n($(a+es en R

C,ni4*-* */ 6-,;/*8a 4* 7a/,-* ini+ia/* 6a-a /a *+a+in 4*/ +a/,-.

UNIVERSIDAD NACIONAL DEL CALLAOPágina 1&



13 {ut u x=0 x Rϵ ,t >0u ( x ,0)=f ( x ) x Rϵ }

S* :-a:a *7i4*n:*8*n:* 4* na *+a+in +,n ,-4*n 4, /in*a/ +,n

+,*+i*n:* +,n:an:* 6,- :an:, +*6:i;/* 4* ?* * a6/i?* */:*,-*8a 4* C!H.

A 6*a- 4* ?* */ 6-,;/*8a 4* /, 7a/,-* ini+ia/* 1 n, *n:-a *n */8a-+, 4*/ TEOREMA DE LA C!H +a;* a;-4a/, a 6a-:i- 4*/ 89:,4, 4*Ca+< ;aa4, *n */ 4*a--,//, 4* /a ,/+i,n* *n *-i* 4* 6,:*n+ia.

S6,nga8, ?* */ 4a:, ini+ia/ 5 5 * na 5n+in ana/>:i+a 4* 8,4,?*F

f ( x )=∑k ..0

∞

f k x

k

( ;?*8, na ,/+in ana/>:i+a 4* /a 5,-8aF

u ( x , t )=∑k −0

∞

uk ( t ) xk

Pa-a ?* /a *+a+in 4*/ +a/,- * a:i5aga * n*+*a-i,.

u ( x , t )=∑k −0

∞

u2 k (t ) xk

∑k −0

∞

k (k −1)uk (t ) xk −2=0

E:a i4*n:i4a4 * 7*-i+a i ,/, iF

u2 k ( t ) ( k +2 ) ( k +1 )uk +2 (t )=0,∀k 40




P,- ,:-a 6a-:* */ /a4, ini+ia/ =05 4* /a EDP 6*-8i:* i4*n:i+a- */

4a:, ini+ia/ 4* +a4a n, 4* /, +,*+i*n:* uk .

uk (0 )=f k ,∀k 40

S* :-a:a 4* n i:*8a 4* *+a+i,n* 4i5*-*n+ia/* +,n n n8*-,inni:, 4* in+gni:a. S -*,/+in *+*4* ,;7ia8*n:* 4* /a :9+ni+a?* /a T*,->a EDO 6-,6,-+i,na.

)*4a 6*n4i*n:* *n:*n4*- i /, +,*+i*n:* ?* <*8, i4*n:i+a4, 4**:* 8,4, ga-an:i@an /a +,n7*-g*n+ia 4* /a *-i* 4* 6,:*n+ia+,--*6,n4i*n:*.

La ,/+in 6*4* *- +,n:-i4a 4* 8an*-a *6/>+i:a 8*4ian:* /a

,/+in 5n4a8*n:a/ , n+/*, 4* AUS." ( x , t )=4 πt

−1 /2*6 x

2/4 t ¿

E 5á+i/ +,86-,;a- ?* */ n+/*, 7*-i+a /a igi*n:* 6-,6i*4a4*.

• AUTOSEMEJANZAF * 4* /a 5,-8a

" ( x , t )= 1

√ t F ( x

√ t )Dn4*F

F ( z ) 4 πt −1/ 2

*6 z2/4 t ¿

• CONSERVACIÓN DE LA MASA.

∫" ( x , t )dx=1,∀t >0

• La 5n+in * 4* +/a* C ∞

4* <*+<, ana/>:i+a -*a/ 6a-a :,4, :

0.• La a86/i:4 8ái8a 4* /a 5n+in 4*+-*+* +,n */ 5a+:,-

8/:i6/i+a:i7, 4*/ ,-4*n t −1 /2

.

• * na 5n+in 6a- +,n -*6*+:, a /a 7a-ia;/* *6a+ia/ .




E a4*8á 5á+i/ +,86-,;a- ?* */ n+/*, 4* a * na ,/+in 4*/a *+a+in 4*/ +a/,-. En *5*+:, *n 7i:a 4* /a *:-+:-a a:,*8*an:* 4* :*n*8,.

t =¿ 1

2 t 3

2

F ( x

√ t ) x

2t 2 F 2 (

x

√ t )

"¿

( x /√ t

" zz= 1

t 3/2 F 2 2 ¿

N, ?*4a i4*n:i+a- */ 4a:, ini+ia/ ?* */ n+/*, 4* a :,8a *n :0.En 7i:a 4* *:-+:-a a:, *8*an:* * 5á+i/ +,86-,;a- ?* +an4,

:!!+¿0¿ = : +,n7*-g* a /a 8aa , 4*/:a 4* DIRAC *n 0 *n */ *n:i4,

?*F

x

√ t

ty√ ¿¿+¿

F (¿)) ( x ) dx=∫ F ( y ) )¿

∫" ( x , t )) ( x ) dx= 1

√ t ∫¿

Pa-a :,4a 5n+in +,n:ina a+,:a4a ) .

T,8an4, a ∆ x , ∆ y , ∆ t +,8, inni:9i8,= /a *+a+in 6*4**+-i;i-* a>F

∂5

∂ t =k

2 { ∂2t

∂ x2 }k

2=k

c … … … … %




E:a * /a KE+a+in 4* +,n4++in 4*/ Ca/,- K= */ 5a+:,-2

κ

* */K+,*+i*n:* 4* :-a8iin 4*/ +a/,- K.

En :-* 4i8*ni,n* /a *+a+in 1& 6*4* g*n*-a/i@a-* a>F

∂5

∂ t =k

2 { ∂2

∂ x2+

∂2

∂ y2+

∂2

∂ z2 }

Una 7*@ ,;:*ni4a /a *+a+in 4*/ +a/,- * 6-,+*4* a ,/+inana/>:i+al -*+,-4*8, ?* /, 6-i8*-, ?* <a+*8, * +,86-,;a-= ?*

-*a/8*n:*= * na *+a+in 6a-a;/i+a= :,8an4, /a 5,-8a g*n*-a/ 4*na *+a+in 4i5*-*n+ia/ 6a-+ia/ 4* *gn4, ,-4*n /in*a/.

A ∂

2u

∂ x2+B

∂2u

∂ 6 ∂7 +C

∂2u

∂ y2+ D

∂u

∂ x+ E

∂ u

∂ y+ Fu="

V*-i+a8, ?* 9 4* ?9 B2−4 AC =0 = *:a * 6*4* //*7a- ;a, /a

:-an5,-8a+in a n. A 5,-8a +anni+a

8 t =8 xx+ F (8 x ,8 , 6 ,t )

D,n4* * na -*/a+in 4* *:a +a:-, 7a-ia;/*.

N,:aF R*+,-4*8, ?* * 4i+* 6a-a;/i+a a ?* 5,-8a +anni+a

:i*n* 5,-8a 4* 6a-á;,/a= * 4*+i-= 8 t =8 xx = 7>a /a :-an5,-8a4a 4*

,-i*- *:, * S2= * *+a6a 4*/ a/+an+* 4* *:a n,:a= 6*-, /a:-an5,-8a4a 4* ,-i*- +a8;ia *n a/gn *n:i4, +a8;ia /a 4*-i7a4a




6,- 6,/in,8i, 7*- ?* F (∂nu )=(ik )n F (u) 4,n4* i * */ i8agina-i,

,/+in 4* i2=−1

A> ;a:a <a+*- n *:4i, 4* *:* +a, 6a-a /a *+a+i,n* 6a-a;/i+a.

EL PROBLE7A DE DIRIC9LET

E/ 6-,;/*8a 4* Di-i+</*: * n 6-,;/*8a ?* +,ni:* *n <a//a- na5n+in ?* * /a ,/+in 4* na *+a+in *n 4*-i7a4a 6a-+ia/* EDP

*n */ in:*-i,- 4* n 4,8ini, 4* Rn

, 8á g*n*-a/8*n:* na 7a-i*4a4

4i5*-*n+ia;/* ?* :,8* 7a/,-* 6-*+-i:, ,;-* */ +,n:,-n, 4* 4i+<,4,8ini,.

Pa-a n 4,8ini, D :*ni*n4, /a a7i4a4 +i*n:* ,;-* */ +,n:,-n,∂ D = /a ,/+in g*n*-a/ a/ 6-,;/*8a 4* Di-i+</*: *F

u ( x )=∫ 9 ∂"( x , 9)d9

∂ n

Dn4*F " ( x , 9 ) * /a *+a+in 4* -**n 6a-a /a *+a+in 4*n

4*-i7a4a 6a-+ia/*= :

∂" ( 6 ,:)∂ n

=n (∇9 " ( x , 9 )=∑i

❑

¿ ∂" ( x , 9 )

∂ 9i = * /a 4*-i7a4a 4* /a 5n+in 4*

-**n a /, /a-g, 4*/ 7*+:,- ni:a-i, n,-8a/ a6n:an4, a in:*-i,- n .

La in:*g-a+in * -*a/i@a ,;-* */ +,n:,-n, +,n /a 8*4i4a ∂ 9 . La

5n+in 7 *:á 4a4a 6,- /a ,/+in ni+a 4* /a *+a+in 4* -*4<,/84* *gn4a +/a*.

F ( x ) ! ( x)2 +∫ (9) ∂ "( x , 9)d9∂n = /a 5n+in 4* -**n a :i/i@a- *n /a

in:*g-a/ 4* a--i;a 4*a6a-*+* *n */ +,n:,-n,.

= 0




Pa-a ϵ∂D 7 xϵ D . Ta/ 5n+in 4* -**n a/8*n:* * na 8a 4*

/a 5n+i,n* 4* -**n 4*/ +a86, /i;-* na ,/+in a-8ni+a a /a*+a+in 4i5*-*n+ia/.

E/ 6-,;/*8a 4* Di-i+</*: 6a-a 5n+i,n* a-8ni+a i*86-* :i*n*,/+in= *a ,/+in * ni+a +an4, */ +,n:,-n, * +i*n:*8*n:*a7* 5 * +,n:ina. Má 6-*+ia8*n:*= :i*n* ,/+in +an4,F

∂ DϵC (1,∝)

= Pa-a 0m ∝ 4,n4* C (1,∝)

* /a +,n4i+in 4* ",/4*-.

La +,n4i+i,n* an:*-i,-* 6*4*n -*/aa-* 64i9n4,* 6-,;a- ?* */6-,;/*8a 4* Di-i+</*: a48i:* ,/+in +,n:ina 6a-a n 4,8ini, +,n7*,in -*?*-i- +,n4i+i,n* 4* a7i4a4 , +an4, +a4a 6n:, 4*/ +,n:,-n,6*-:*n*+* a na ;,/a +*--a4a >n:*g-a8*n:* +,n:*ni4a *n */ +,nn:,

+,86/*8*n:a-i, 4*/ in:*-i,- 4*/ 4,8ini, +,n4i+in 4* /a ;,/a +*--a4a*:*-i,-.

S6nga* /a *i:*n+ia 4* 4, 5n+i,n* a-8ni+a *n na -*gin Ri86/*8*n:* +,n*a ?* +86/*nF

∆ u1=∆u2 ∀ x ϵ R u1¿&R=u2¿&R= F ( x) ∀ x ϵ&R

S* +,n:-* /a igi*n:* 5n+in.

u0=

u1−

u2 ∀ x ϵ R

P,- +,n:-++in /a 5n+in 4a4 +86/*.

u

∇(¿¿1−u2)=∆ u1−∆u2=0 yu0¿&R=u1 ¿&R−u2¿&R=0

∆ u0=¿

In:*g-an4, /a n,-8a 4*/ g-a4i*n:* 4* u0 *n :,4, */ 7,/8*n

4*/i8i:a4, 6,- */ +,n:,-n, a6/i+an4, */ :*,-*8a 4* -**n * :i*n*?*F




u0

∇(¿∇u0)d−∫ u0 ∆ u0 d=∫u0∇u0 d9−∫ u0∆ u0 d

∫∇u0∇u0 d=∫¿

P*-,F

∇u0=0ent;nce9 :∫u0 ∆ u0 d=0

u0 ¿&R=0ent;nce9 :∫u0∇u0 d9=0

D* 8,4, ?*F

∫∇u0∇u0d=∫||∇u0||2

d=0 ent;nc e9∇u0=0∀ x ϵ R ent;nce9

u0=cte∀ x ϵ R

Da4, ?* u0 ¿&R=0 9etiene fin%$<ente :

u0=0 ∀ x ϵ R En:,n+*F u1=u2 ∀ x ϵ R

L, 6-,;/*8a 4* Di-i+</*: ,n :>6i+, 4* /a *+a+i,n* *n 4*-i7a4a

6a-+ia/* */>6:i+a= /a :*,->a 4*/ 6,:*n+ia/= /a *+a+in 4* La6/a+* *n6a-:i+/a-. O:-, **86/, ,n /a *+a+in ;i!a-8ni+a /a *+a+i,n*-*/a+i,na4a +,n /a :*,->a 4* /a */a:i+i4a4.

E:* * n, 4* /, 6-,;/*8a 4* 7a-i, :i6, 4* +/a* 6-,;/*8a 4*EDP 4*ni4, 6,- /a in5,-8a+in 4a4a *n */ +,n:,-n,= *n:-* /, +a/**:án :a8;i9n */ 6-,;/*8a 4* N*8ann */ 6-,;/*8a 4* Ca+<.

III. LA ECUACI6N DE BURGES

E:* **-+i+i, :i*n* na na/i4a4 6-a8*n:* 4i4á+:i+a. V*-*8, a/gna4* /a 6a-:i+/a-i4a4* 4* /, 89:,4, 6*4,*6*+:-a/* 6a-a /a-*,/+in 4* *+a+i,n* *n 4*-i7a4a 6a-+ia/* *n 4,8ini, 6*-i4i+,.E/ , 4* n 89:,4, *6*+:-a/= 4* 5,-8/a+in 8+<, 8á +,86/i+a4a=* :i+a 6,- *- 8+<, 8á -á6i4,. An?* 6a-*@+a ,-6-*n4*n:* /a:-an5,-8a4a -á6i4a 4* ,-i*- , T 6,- ig/a *n ing/9 a:




,-i*- T-an5,-8 * na ,6*-a+in *6*+ia/8*n:* *+i*n:* *n :9-8in,+,86:a+i,na/*. En *:* 6-i8*- **86/,= *n */ ?* * -**/7* na*+a+in +,n :9-8in, 4* +,n7*++in 4i5in +,n n 89:,4, *6/>+i:,=n, * <a+*n *7i4*n:* /a 7*n:aa. S* <a+*n ,;7ia +an4,= -*,/7i*n4,/a *+a+i,n* 4* Na7i*- S:,* *n 4,8ini, ;i4i8*ni,na/* ,

:-i4i8*ni,na/*= * an 89:,4, 4* in:*g-a+in :*86,-a/ n 6,+, 8á,:i+a4,. "* a?> /a *+a+in 4* B-g*-F

∂t u+u (∇u=∇2 u

En na 4i8*ninF

ut = u xx−uu x

La *+a+in * *?*8a:i@a 4* /a igi*n:* 8an*-aF∂t u= L (u )− D (u )

C,8, */ 4,8ini, * 6*-i4i+, 6,4*8, 6,n*- ?* /a 5,-8a 4* /a

,/+in * u=u exp (ikx ) , ?* * *?i7a/*n:* a 6/an:*a- /a *+a+in 6a-a

*/ *6*+:-, 4* 7*/,+i4a4 4 :-an5,-8a4a 4* ,-i*-.

u

∂t u= L(u)− D ¿

En *:* +a, /a 5,-8a 4*/ ,6*-a4,- /in*a/ *F

L ( u )=−k 2 u

E ,;7i, ?* in */ :9-8in, n, /in*a/ /a -*,/+in 4* /a EDO *->a :-i7ia/.Ba:a->a +,n :-an5,-8a- /a +,n4i+in ini+ia/ a/ *6a+i, 4* 5-*+*n+ia *in:*g-a- ,;-* 9/.

N, *i:* ningna 8an*-a *a+:a 4* +,n7*-:i- /a ,6*-a+in u ut *n n

,6*-a4,- D u = *:* +á/+/, 4*;* -*a/i@a-* *n */ *6a+i, -*a/. Pa-a

+a4a 6a, :*86,-a/ *-á n*+*a-i, an:i!:-an5,-8a- /a ,/+in= +a/+/a-*/ ,6*-a4,- D= :-an5,-8a-/, 8a-/, *n */ *6a+i, 4* 5-*+*n+ia.




La *+a+in 4* B-g*- n, -**-7a n 5*n8*n, n, *6*-a4,F */a/iaing. Can4, n ,6*-a4,- 4i5*-*n+ia/ * /in*a/ n, * 8*@+/a */*6*+:-, 4* ningna 8an*-a. P,- **86/,= */ ,6*-a4,- 4*-i7a4a *gn4a

∂ xx u=−k x2u n, in:-,4+* ningna ,6*-a+in *n:-* +a4a n, 4* /a

+,86,n*n:* 4* u l /, 8i8, +*4* +,n /, /:-, 6-,6,-+i,na/*

+,8, */ 6aa!a/:, , */ 6aa!;a,. E/ ,6*-a4,- D (u ) n, * /in*a/ *

in:-,4+i-á a/iaing *n u (

I. TRANSFOR7ADAS INTEGRALES DE FUNCIONES

TRANSORMADA DE LAPLACE

DEFINICION 1: Da4a na 5n+in 5: 4*ni4a a/ 8*n, 6a-a +a4a : 0= i * 6-**-* * 6*4* 6,n*- ?* 5: 0 6a-a +a4a : m 0 +atransfor/a*a *e La,+a$e *e 5 * ,:-a 5n+in ?* 4*igna8, 6,-L5 X, ?* 7i*n* 4a4a 6,-

L ( f ) (9 )= F ( 9)=∫0

+∞

f (t ) e−9t

dt ,

DEFINICION 0 Se dice que una f&n$()n f t es *e or*en e4,onen$(a+

γ $&an*o t \ h], si existen M > 0 y T > 0 tales que se verica

df t d Meγt para cada t > T.

EJE7PLO 0.1:1 T,4a 5n+in a+,:a4a * 4* ,-4*n *6,n*n+ia/ o +an4, : \ h]=

6a-a :,4, o p 0= 4a4, ?*d5:d M M*o: 6a-a +a4a : p 0.

En 6a-:i+/a-= /a 5n+i,n* +,n:an:*= *n, +,*n, ,n 4* ,-4*n*6,n*n+ia/ o +an4, : \ h]= 6a-a :,4, o p 0.

2 Si 5: q *bo: \ 0 +an4, : \ h]= *n:,n+* 5: 4* ,-4*n *6,n*n+ia/o +an4, : \ h]. En 6a-:i+/a-= +a/?i*- 6,/in,8i, * 4* ,-4*n*6,n*n+ia/ o +an4, : \ h]= 6a-a :,4, o 0.




3 Sin *8;a-g,= /a 5n+in 5: *:2 n, * 4* ,-4*n *6,n*n+ia/ o+an4, : \ h]= 6a-a ningn o p 0.

S6*:, ?* /, 5*-a= *i:i->an o IR= M 0 T 0 :a/* ?* * :2 M*o: 6a-a +a4a : T. En:,n+*= *::bo M 6a-a +a4a : T :,8an4,/i8i:* +an4, : \ h]= //*ga8, a /a +,n:-a4i++in 4* ?* h] M.

EJE7PLO 0.0:

1 P-,;a- ?* n, *i:* /a :-an5,-8a4a 4* La6/a+* L 5 4* /a5n+in 5: *:2 6a-a ningn 7a/,- 4* . In4i+a+inF P-,;a- ?* 6a-a+a4a IR * 7*-i+a

*:2b: p *b2W r : 0.

2 P-,;a- ?* n, *i:* ningna 5n+in 5: , 5n+in +,n:ina a :-,@,*n s0=Tt 6a-a :,4, T 0 4* ,-4*n *6,n*n+ia/ o +an4, : \ h] :a/?*

L ( f ) (9 )= 9

39−1 .

La 6-in+i6a/* 6-,6i*4a4* 4* /a :-an5,-8a4a 4* La6/a+* ?* 7a8, a:i/i@a- ,n /a igi*n:*F

i La :-an5,-8a4a 4* La6/a+* * /in*a/F 4a4a 4, 5n+i,n* 5 1 5 2=L5 1 h u5 2 L5 1 h uL5 2= r =u IR.

ii S6,nga8, ?* 5 * +,n:ina *n s0=h]= 4* ,-4*n *6,n*n+ia/ o

+an4, : \ h] ?* 5 X: * +,n:ina a :-,@, *n s0=Tt 6a-a +a4a T 0.

En:,n+*= i o= L5 X L5 b 50.

iii S6,nga8, ?* 5 5 v ,n +,n:ina *n s0=h]= 4* ,-4*n *6,n*n+ia/ o+an4, : \ h] ?* 5 vv: * +,n:ina a :-,@, *n s0=Tt 6a-a +a4a T 0.

En:,n+*= i o=

L5 vv 2L5 b 50 w 5 v0.

TEORE7A S6,nga8, ?* 5: g: ,n 5n+i,n* +,n:ina a :-,@,*n s0=Tt 6a-a :,4, T 0 4* ,-4*n *6,n*n+ia/ o +an4, : \ h].En:,n+*




L5 Y g L5 qLg= i o.

APLICACIÓN DE LA TRANSORMADA DE LAPLACE A LAS EDP

C,ni4*-*8, */ 6-,;/*8a 4* /a 7i;-a+i,n* 4* na +*-4a Kinni:a+,/,+a4a *n */ ** 0= 6*:a +,n,+i4a /a 6,i+in 4*/ *:-*8, 0 *n +a4a 8,8*n:,= n:, a /a +,n4i+i,n* ini+ia/* *n +a4a 6n:,

utt ( x , t )=u xx ( x ,t ) , x>0, t >0 Ec ( de =nd%9

u ( x ,0 )=f ( t )t >0C;ndici>nf?;nte?%

u ( x ,0 )=0, x>0 @;9ici>ninici%$ ( en?e;9; )

u ( x ,0 )=0, x>0.e$;cid%d inici%$ (nu$%)

D*4* n 6n:, 4* 7i:a 5>i+,= n, in:*-*an /a ,/+i,n* =: ?**:9n a+,:a4a +an4, \ h]. En /, ?* ig*= 7,/7*8, a 6,n*-?* :,4a /a 5n+i,n* ?* a6a-*+*n= 7*-i+an /a <i6:*i n*+*a-ia6a-a ?* :*ngan *n:i4, /, +á/+/, ?* 7a8, a //*7a- a +a;,.

A6/i+an4, a/ 6-,;/*8a /a :-an5,-8a4a 4* La6/a+* -*6*+:, 4* /a7a-ia;/* := //a8an4,

8 ( x , 9 )= L (u ( x , t ) ) ( 9 )=∫0

+∞

u ( x , t ) e−9t dt

:i/i@an4, /a 6-,6i*4a4 iii ,;-* /a :-an5,-8a4a 4* /a 4*-i7a4a*gn4a -*6*+:, 4* t = * ,;:i*n*

{928 ( x , 9 )−9u ( x ,0 )−u1

( x ,0 )=8 xx ( x , 9 ) , x>0,9> y

8 (0,9 )= L ( f ) (9 ) , 9> y

3.3

T*ni*n4, *n +*n:a /a +,n4i+i,n* ini+ia/* 4*/ 6-,;/*8a 3.33 n,?*4a




{928 ( x , 9 )=8 xx

( x , 9 ) , x>0, 9> y

8 (0,9 )= L (f ) (9 ) , 9> y

3.3#

ia4, s > 0= /a EDP an:*-i,- 6,4*8, -*,/7*-/a +,8, na EDO /in*a/4* *gn4, ,-4*n -*6*+:, 4* x = 6,- /, ?* in:*g-an4, -*/:a

U x, s c1sesx h c2sebsx ,

4,n4* c1 c2 ,n 5n+i,n* a-;i:-a-ia. Si u x,t *:á a+,:a4a +an4, x \h]= :a8;i9n 4*;* *:a-

U x, s a+,:a4a +an4, x \ h]= 6,- /, ?* ,;/iga:,-ia8*n:* c1s x 0.P,- *//,=

U x, s c2sebsx ,

:i/i@an4, /a +,n4i+in 5-,n:*-a= +,n+/i8, ?* c2s U0,s Lf s,4* 4,n4* -*/:a

U x, s Lf s q ebsx .

)*-*8, a<,-a a6/i+a- :-an5,-8a4a in7*-a ,;:*n*- /a ,/+in 4*/6-,;/*8a ini+ia/ 3.33. Pa-a *//,= 7,/7*8, a a- /a 6-,6i*4a4 in4i+a4a

*n */ a6a-:a4, an:*-i,-= 6*-, a<,-a +,n a x f yt f t = -*/:an4,

u( x ,t )={ 0 ,9i t <0f ( t −9 ) ,9i t >0

E:a ,/+in * 6*4* in:*-6-*:a- 4*/ igi*n:* 8,4,F /a 6,i+in 4*/*:-*8, x 0 *n +a4a in:an:* 4* :i*86, t * :-an8i:* a/ 6n:, x 0 > 04* /a +*-4a *n */ in:an:* t h x 0= /, +a/ igni+a *n 6a-:i+/a- ?* /a,n4a * 8*7* <a+ia /a 4*-*+<a +,n 7*/,+i4a4 1.




PROBLEMAS

ENUNCIADOE+a+in 4* La6/a+* *n 6,/a-*

Di:-i;+in 4* :*86*-a:-a *n /a +,-,na +i-+/a-F

∇2u=0 { u (1,) )=f () )=9en ( ) )u (2,) )=1 () )=) · (2π −) )

SOL.

O;*-7an4, */ *nn+ia4,= 6,4*8, 7*- ?* *n */ +,n:,-n, <a 4,+,n4i+i,n* n, n/a. E:, igni+a ?* *:* 6-,;/*8a n, 6*4* *--**/:, *n 5,-8a 4i-*+:a a6/i+an4, */ 89:,4, 4* *6a-a+in 4*




7a-ia;/*. P,- /, :an:,= /, -*,/7*-*8, a6/i+an4, */ 6-in+i6i, 4*6*-6,i+in. En *:* +a,= */ 8i8, +,ni:* *n 4*+,86,n*- */6-,;/*8a *n 4, 8á i86/*= -*,/7*- +a4a n, 4* *//, 6,- *6a-a4, /*g, ,;:*n*- /a ,/+in 4*/ 6-,;/*8a ,-igina/ <a+i*n4, /a 8a 4* /a,/+i,n* ,;:*ni4a 6a-a +a4a n, 4* *, 4, 6-,;/*8a.

En:,n+* 6-,6,n*8, /, igi*n:* 4, 6-,;/*8aF

Problema 1

∇2u1=0 {u1 (1,) )=f () )=9en () )u1 (2,) )=0

Problema 2

∇2

u2=0{ u2 (1,) )=0

u2 (2,) )=1 ( ) )=) ·(2 π −))

Una 7*@ -**/:, *:, 4, 6-,;/*8a ,;:*ni4a u1 (? ,) ) u2 (? , ) ) =

/a ,/+in 4*/ 6-,;/*8a 4*/ *nn+ia4, *-á /a igi*n:*F

u (? , ) )=u1 (? , ) )+u2 (? ,) )

SOL. ;Problema 1)

N,:aF 7a8, a //a8a- u1=u 6a-a n, a--a:-a- */ ;in4i+* 1 4-an:* /a

-*,/+in 4*/ 6-,;/*8a.

Pa-a -*,/7*- *:* 6-,;/*8a 7a8, a :i/i@a- */ 89:,4, 4* *6a-a+in 4*7a-ia;/*= ?* 6a-:* 4* 6,n*- ?* 6,4*8, *6-*a- /a 5n+in

u (? , ) ) +,8, */ 6-,4+:, 4* na 5n+in ?* 4*6*n4* /, 4* /a

7a-ia;/* ? ,:-a 5n+in ?* 4*6*n4* /, 4* /a 7a-ia;/* ) .




u (? , ) )= R (? ) · ( ) )(1)

E/ La6/a+ian, *6-*a4, *n +,,-4*na4a 6,/a-* * */ igi*n:*F

∇2u (? , ) )=u??+1? u?+

1

?2 u))(2)

E/ 6-,;/*8a ;+a <a//a- na 5n+in ?* +86/a +,n ∇2u=0 =

*n:,n+*F

u??+1

? u?+

1

?2 u))=0(3)

A +,n:ina+in *7a/a-*8, *n /a *+a+in 3 /a 4i:in:a 4*-i7a4a

6a-+ia/* 4* u (? ,) ) = a 6a-:i- 4* /a *6-*in 1F

u?= R (? ) · () )

u??= R (? ) · () )

u))

= R (? ) · () )

R**86/a@a8, *n 3F

R (?) · () )+

1

? R

(? ) · () )+ 1

?2 R (? ) · ( ) )=0

Di7i4i8, a8;, 8i*8;-, 6,- R (? ) ·∅

() ) 4,n4* R (? ) ()) 4*;*n

*- 4i:in:a 4* /a 5n+in n/a a ?* 4* /, +,n:-a-i, u (? , ) ) *->a /a

5n+in n/a.




R (? )· () )

R (? ) ·()) +

1

?

R (? ) · () )

R (? ) · ())+1

?2

R (? ) · () )

R (? ) ·())=0

R (? )

R (? ) +

1

?

R (? )

R (? ) +

1

?2

() )

())=0

S*6a-a8, 4* +a4a /a4, 4* /a iga/4a4 /a 5n+i,n* 4* 4i:in:a7a-ia;/*F

R (? )

R (? ) +

1

?

R (? )

R (? )=−1

?2

() )

())

?2 R

(?)

R (? )

+? R

(? )

R (? )

=−

() )

())

(4)

En /a *6-*in */ 8i*8;-, i@?i*-4, 4* /a iga/4a4 4*6*n4* /, 4*

/a 7a-ia;/* ? */ 8i*8;-, 4*-*+<, 4*6*n4* /, 4* /a 7a-ia;/* ) .

Pa-a ?* * +86/a /a iga/4a4 6a-a :,4, ?∈(1,2 ) 6a-a :,4,

)∈(0,2π ) = *n:,n+* a8;, 8i*8;-, 4*;*n *- iga/* a na 8i8a

+,n:an:* F

?2 R

(? )

R (? ) +?

R (? )

R (? ) =−

() )())

= (5)

D* a?> * 4*6-*n4*n /a igi*n:* *+a+i,n* 4i5*-*n+ia/*F

− () )

()) =

?2 R

(? ) R (? )

+ ? R

(? ) R (? )

= }⟹{

() )+ ())=0

?2 R

(? )+? R (? )− R (? )=0

(6)(7)




P,- /a +a-a+:*->:i+a 4*/ -*+in:, 4*/ 6-,;/*8a= n 6n:, +,n ?0∈ (1,2)

+,n n áng/, )=0 ,:-, +,n */ 8i8, ?0∈ (1,2) +,n )=2π

-*6-**n:an a/ 8i8, 6n:,.= P,- /, :an:, /a 5n+in 4*-i7a4a4*;*n :,8a- */ 8i8, 7a/,-. Ba, *:a +,n4i+in= :i/i@an4, /a *6-*in

1= 6/an:*a8,F

u (?0 ,0 )=u ( ?0 ,2π )⟹ R (?0 ) ·(0)= R ( ?0 )· (2π )

u) (?0,0 )=u) (?0 ,2 π )⟹ R (? 0 )· (0 )= R (? 0 )·

(2 π )

E:a +,n4i+i,n* * 4*;*n +86/i- 6a-a :,4, ?∈ (1,2 ) . Sa;*8, ?*

R (? ) n, 6*4* *- /a 5n+in n/a= a ?* *n ** +a, u (? ,) ) *->a /a

5n+in :-i7ia/ 9:a n, +86/* +,n /a +,n4i+in u (1,) )=f ( ) )=9en () ) .

En:,n+* 4*4+i8, ?*F

(0 )= (2π ) ( )

(0)=

(2π )( )

A4*8á= /a +,n4i+i,n* 4* +,n:,-n, u (1,) )=f ()) u (2,) )=0 +,n /a

*6-*in 1 -*/:anF

u (2,) )= R (2 ) · ( ) )⏟ 0

=0⟹ R (2)=0( )

u (1,) )=f ( ) )=9en () )( . )

La *+a+in % /a +,n4i+i,n* ( ) ( ) 5,-8an n 6-,;/*8a 4*

S:-8 Li,7i//* Si86/i+a4, PSLS. P,- /, :an:, /, 6,i;/* 7a/,-* 4* ,n 40




R*,/7*8, */ +a, =0 6a-a () )

La ,/+in 4* /a *+a+in 4i5*-*n+ia/ % +,n =0 *F

() )= A0 )+B0⟹ () )= A0

A6/i+an4, /a +,n4i+in 4* +,n:,-n, ( ) :*n*8,F

(0 )= (2π )⟹B0=2π A0+B0⟹ A0=0

A6/i+an4, /a +,n4i+in 4* +,n:,-n, ( ) ,;:*n*8,F

(0 )=

(2π )⟹ A0 ¿ A0

N, :*n*8, -*:-i++i,n* ,;-* B0 6,- /, :an:,F

0 () )=B0(8)

C,n+/i8, *n:,n+* ?* =0 * a:,7a/,- *:á a,+ia4, a /a

a:,5n+in 0 () )=1 .

R*,/7*8, */ +a, >0 6a-a () )

La ,/+in 4* /a *+a+in 4i5*-*n+ia/ % +,n >0 *F

() )= A cos (√ ) )+B9en (√ ) )(9)




() )=− A√ 9en (√ ) )+B√ cos (√ ) )

U:i/i@an4, /a +,n4i+in 4* +,n:,-n, ( ) F

(0 )= (2π )⟹ A= A cos (√ 2 π )+B 9en (√ 2π )⟹

A [cos (√ 2π )−1 ]+B9en (√ 2π )=0(10)

U:i/i@an4, /a +,n4i+in 4* +,n:,-n, ( ) F

(0 )=

(2π )⟹B√ =− A √ 9en (√ 2π )+B√ cos (√ 2π )

>0⇒

B=− A 9en (√ 2π )+B cos (√ 2π )⟹

− A 9en (√ 2π )+B [cos (√ 2π )−1 ]=0(11)

La *+a+i,n* 10 11 5,-8an n i:*8a 4* 4, *+a+i,n*

<,8,g9n*, +,n in+gni:a A B . E6-*a4, *n 5,-8a 8a:-i+ia/

?*4aF

([cos (√ 2 π )−1 ] 9en (√ 2π )

−9en (√ 2π ) [cos (√ 2π )−1 ])( AB )=(00)

Da4, ?* * :-a:a 4* n i:*8a <,8,g9n*,= i <a ,/+in ni+a= 9:a* /a ,/+in :-i7ia/. Pa-a ?* <aa na ,/+in 4i5*-*n:* 4* /a :-i7ia/ */4*:*-8inan:* 4* /a 8a:-i@ 4*;* *- +*-,. En:,n+* ;+a8, ?*F

det

|[cos (√ 2π )−1 ] 9en (√ 2 π )−9en (√ 2π ) [cos (√ 2 π )−1 ]|

=0

[cos (√ 2π )−1 ] · [cos (√ 2π )−1 ]+9en (√ 2π )·9en (√ 2 π )=0⟹




⟹cos2 (√ 2π )−2cos (√ 2 π )+1+9en

2 (√ 2 π )=0⟹

⟹cos2 (√ 2π )+9en

2 (√ 2π )⏟

1

−2cos (√ 2π )+1=0⟹

⟹1−2cos (√ 2π )+1=0⟹2−2cos (√ 2π )=0⟹⟹cos (√ 2π )=1⟹√ 2π =2nπ ⟹

√ =n

=n2(12)

U:i/i@an4, /a *6-*in 12 *n /a *+a+in & ,;:*n*8,F

() )= A cos (n) )+B9en ( n) )

Pa-a +a4a 7a/,- 4* n=1,2,3…( = *i:* n 6a- 4* 7a/,-* A B

?* 4a /ga- a /a ,/+in n () ) +,--*6,n4i*n:*. P,- /, :an:, :*n*8,F

n () )= An cos (n) )+Bn 9en (n) )∀nϵ G (13)

A<,-a ;+a-*8, /a ,/+in R (? ) 4* /a *+a+in 4i5*-*n+ia/ .

R*,/7*8, */ +a, =0 6a-a R (? )

En *:* +a, /a *+a+in 4i5*-*n+ia/ * -*4+* aF

?2 R

(? )+? R (? )=0

La ,/+in 4* *:a *+a+in * /a igi*n:*F

R0(? )=C 0 ln(? )+ D0




A6/i+a8, /a +,n4i+in 4* +,n:,-n, ( ) F

R0 (2 )=C 0 ln(2)+ D0=0⟹ D 0=−C 0 ln(2)⟹ R0 (? )=C 0 ln (? )−C 0 ln(2)

R0 (? )=C 0[ ln( ?

2 )](14 )

R*,/7*8, */ +a, >0 6a-a R (? )

La ,/+in g*n*-a/ 4* /a *+a+in 4i5*-*n+ia/ 6a-a >0 *F

R (? )=C ? √ + D ?−√

R**86/a@a8, √ a 6a-:i- 4* 12F

Rn (? )=C ?n+ D ?

−n

A6/i+a8, /a +,n4i+in 4* +,n:,-n, ( ) F

Rn (2 )=C 2n+ D 2

−n=0⟹ D 2−n=−C 2

n⟹ D=−C 22n

P,- /, :an:,F

Rn (? )=C ?n−C 2

2n?−n

Sa+an4, 5a+:,- +,8n C ·2n

F




Rn (? )=2nC (2−n

?n−2

n?−n )= E(( ?

2 )n

−( ?2 )−n

)

Pa-a +a4a

n

*i:* na ,/+in

R (? )

= *n:,n+* :*n*8,F

Rn (? )= En(( ?

2 )n

−( ?

2 )−n

)(15)

La ,/+in u (? , ) ) *-á na +,8;ina+in /in*a/ 4* /a un (? ,) )

,;:*ni4a.

u (? , ) )= R0 (? ) ·

0( ) )+∑

n=1

∞

Rn (? ) · n () )

U:i/i@an4, /a ,/+i,n* $= 13= 1 1# *n /a *6-*in an:*-i,-F

u (? , ) )=B0C 0 ln( ?2 )+∑n=1

∞

[ Ancos (n) )+Bn 9en (n) ) ] En(( ?

2 )n

−( ?

2 )−n

)

Ag-6an4, /, +,*+i*n:* ?*4aF

u (? , ) )=B0C 0⏟ F

0

ln( ?2 )+∑n=1

∞

[ An En⏟

F n

cos (n) )+Bn En⏟"

n

9en (n) )]((?

2 )n

−( ?

2 )−n

)

u (? , ) )= F 0 ln( ?

2 )+∑n=1

∞

[ F ncos ( n) )+"n 9en (n) ) ](( ?

2 )n

−( ?

2 )−n

)(16)

Pa-a <a//a- /, +,*+i*n:* F 0 = F n "n :i/i@a8, /a +,n4i+in

( . ) F




u (1,) )=f ( ) )=9en())

u (1,) )= F 0 ln( 12 )⏟ H 0

+∑n=1

∞

[ F n(( 12 )

n

−( 12 )−n

)

⏟

H n

cos (n) )+"n(( 12 )n

−(12 )−n

)

⏟

n

9en ( n) )

]=9en())

H 0+∑

n=1

∞

H n cos (n) )+ n 9en (n) )=9en ())(17)

H 0 = H n = * n 6*4*n in:*-6-*:a-* +,8, /, +,*+i*n:* 4* n

4*a--,//, *n S*-i* T-ig,n,89:-i+a 4* ,-i*- DST 4* /a 5n+in

9en) .

R*+,-4*8, ?* {1 3cos (n) ) 39en ( n) ) } nϵ G ,) I [0 , 2π ) () )=1 5,-8an na

;a* ,-:,g,na/ 6a-a */ DST. C,8, /a 5n+in ?* ;+a8, 4*a--,//a- 9en()) * n, 4* /, */*8*n:, 4* *a ;a*= */ 4*a--,//, *n *-i*

+,ni:* *n ?* :,4, /, +,*+i*n:* ,n +*-,= *+*6:, */+,--*6,n4i*n:* a /a a:,5n+in. P,- /, :an:,F

H 0=0 I F 0= H 0

ln 1

2

=0

H n=0 I F n= H n

(( 12 )n

−( 12 )−n

)=0

n=

{9i n=1⟹ 1=1 ⟹"1=

1

((12 )

1

−(12 )

−1

)=

1

−32

=−2

3

9i n>1⟹ n=0 ⟹"n= n

(( 12 )n

−( 12 )−n

)=0




R*86/a@a8, F 0 = F n "n *n /a *+a+in 1%= ,;:*ni*n4,F

u (? , ) )=

−2

3

((?

2

)1

−( ?

2

)−1

)9en(1))

R*+,-4a8, ?* *:a * /a ,/+in 4*/ 6-,;/*8a 1 *n:,n+*-*:,8a8, */ ;>n4i+* +,--*6,n4i*n:* 4* /a ,/+inF

u1 (? ,) )=( 43 1

? −

1

3? ) 9en())(18)

SOL. ;Problema 2)

A6/i+an4, */ 89:,4, 4* *6a-a+in 4* 7a-ia;/* 6-,6,n*8, ?*F

u2 (? ,) )= R (? ) · () )

R*,/7i*n4, *n 5,-8a aná/,ga +,8, 6a-a u1 //*ga8, a /a 8i8a

*+a+i,n* % = /a +,n4i+i,n* 4* +,n:,-n, ( ) ( ) .

{ () )+ ())=0

?2 R

(? )+? R (? )− R (? )=0

(6)(7)

(0 )= (2π ) ( )

(0)=

(2π )( )

La +,n4i+i,n* 4* +,n:,-n, -*:an:* 6a-a u2 ,nF




u2 (1,) )= R (1) · () )⏟0

=0⟹ R (1 )=0(. )

u2 (2,) )=1 () )=) · (2 π −) )(. )

N*7a8*n:* /a *+a+in % /a +,n4i+i,n* ( ) ( ) -*6-**n:an

n PSLS /a -*,/+in= iga/ a /a 4*a--,//a4a *n */ 6-,;/*8a an:*-i,-=n, 4aF

=0⟹ 0 () )=B0

>0⟹n () )= Ancos (n) )+Bn 9en (n) )

C,n =n2

.

La ,/+in a /a *+a+in 6a-a =0 *F

R0(? )=C 0 ln(? )+ D0

A6/i+a8, (. ) F

R0 (1 )=C 0 ln (1 )+ D0=0⟹ D0=0

R0(? )=C 0 ln(? )

La ,/+in 4* /a *+a+in 6a-a >0 *F

Rn (? )=C n?n+ D n?

−n

A6/i+a8, (. ) F

Rn (1 )=C n1n+ Dn1

−n=0⟹ Dn=−C n

Rn (? )=C n ( ?n−?−n )




E+-i;i8, /a ,/+in g*n*-a/ u2 (? ,) ) *n 5,-8a aná/,ga a /, -*a/i@a4,

*n */ 6-,;/*8a 1F

u2 (? ,) )= R0 (? ) ·0 () )+∑n=1

∞

Rn (? ) · n () )

R**86/a@an4, n, ?*4aF

u2 (? , ) )=B

0C

0ln(? )+∑

n=1

∞

[ An cos (n) )+B n 9en (n) ) ] C n (?n−?

−n )

u2

(? ,) )=B0

C 0⏟

E0

ln(?)+

∑n=1

∞

[ A

n

C n⏟

En

cos (n) )+Bn

C n⏟

F n

9en (n) )

](? n−?

−n )

u2 (? ,) )= E0 ln(?)+∑n=1

∞

[ Encos (n) )+ F n 9en (n) ) ] (?n−?−n )(19)

Ua8, /a +,n4i+in (. ) 6a-a *n+,n:-a- /, +,*+i*n:* E0 = En

F n F

u2 (2,) )=) · (2π −) ) ,0≤ )<2π

u2 (2,) )= E0 ln (2 )⏟"

0

+∑n=1

∞

En(2n−2

−n )⏟

"n

cos (n ) )+ F n (2n−2−n )⏟

H n

9en (n))

u2 (2,) )="

0+∑

n=1

∞

"ncos (n ) )+ H n 9en(n ))=) · (2 π −) )(20)

La /:i8a *+a+in 6*4* in:*-6-*:a-* +,8, n DST 4* /a 5n+in1 () )=)· (2π −) ) . P,- /, :an:, a +,n:ina+in +a/+/a8, /, +,*+i*n:*

"0 = "n H n . C,8*n@a8, +,n "0 F




) · (2π −) ) d)=¿ 1

2π (2π )

2

2 −

)3

3 )|02 π

=¿ 1

2π (2 π (2π )

2

2 −

(2π )3

3 )= (2π )3

2π (12−1

3 )=4 π 2

6 =

2 π 2

3

"0=1

5 ∫0

5

1 () ) d)= 1

2π ∫0

2π

¿

"0= E0 ln (2 )=2π

2

3

E0= 2 π

2

3ln (2 )(21)

A +,n:ina+in +a/+/a8, "n:

"n=2

5 ∫0

5

1 ( ) )cos (n) ) d)=1

π ∫0

2π

) · (2π −) ) ·cos (n) ) d)

"n=1

π ∫0

2π

[2π ) cos (n) )−)2cos (n) ) ] d)

"n=2∫0

2π

) cos (n) )d)−1

π ∫0

2π

)2cos (n) ) d)

R*,/7i*n4, 6,- :a;/aF

"n=2 ( cos (n) )

n2 +

)9en(n))n )|

0

2π

−1

π (2) cos (n) )

n2 +( )

2

n −

2

n3 )9en(n)))|

0

2π

"n=2( cos (2nπ )⏞1

n2 +

2π 9en (2nπ )⏞0

n −

cos (0 )⏞1

n2 )− 1

π (4 π cos (2nπ )⏞1

n2

+( 4 π 2

n −

2

n3 )9en (2nπ )⏞

0 )




"n=−1

π ( 4 π

n2 )=−4

n2

"n= En (2n−2−n )=−4

n2

En= −4

n2 (2n−2

−n)(22)

ina/8*n:* +a/+/a8, H n F

H n=2

5 ∫0

5

1 ( ) ) 9en (n) ) d)=1

π ∫0

2π

)· (2π −) ) ·9en (n) ) d)

H n=1

π ∫0

2π

[2π) 9en (n) )−)2·9en (n) ) ] d)

H n=2∫0

2π

)9en (n) ) d)− 1π ∫0

2π

)2 ·9en (n) ) d)

R*,/7*8, 6,- :a;/aF

H n=2 ( 9en (n) )

n2 −

)cos(n))n )|

0

2π

−1

π (2) 9en (n) )

n2

−()2

n −

2

n3 )cos(n)))|

0

2π

H n=2( 9en (2nπ )⏞0

n2 −

2 π cos(2nπ )⏞1

n )−1

π (4 π 9en (2nπ )⏞0

n2

−(4 π 2

n −

2

n3 )cos (2nπ )⏞

1

+(−2

n3 )cos(0)⏞

1 )




H n=−4 π

n −

1

π (−( 4 π 2

n −

2

n3 )+(−2

n3 ))=−4 π

n +

4 π

n =0

H n= F n (2n

−2

−n

)=0

F n=0(23)

R**86/a@an4, 21= 22 23 *n /a *+a+in 1& :*n*8,F

u2 (? ,) )= 2π

2

3 ln (2 ) ln (? )+∑

n=1

∞ (−4 )

n2 (2n−2

−n ) cos (n) ) (?n−?

−n )(24)

S,/+in g*n*-a/ 4*/ 6-,;/*8a

ina/8*n:* :i/i@an4, /a *6-*i,n* 1$ 2 ,;:*n*8, /a ,/+in4*/ **-+i+i,F

u (? , ) )=u1 (? , ) )+u2 (? ,) )

u (? , ) )=( 43 ?−

?

3 ) 9en())+ 2π

2

3 ln (2 ) ln (?)+∑

n=1

∞ (−4 )

n2 (2n−2

−n ) cos (n) ) (?n−?

−n )

Problema 3

Can4, na +*-4a 7i;-an:* * ,8*:* a na 5*-@a 7*-:i+a/ *:án4a-?* 7a->a +,n /a 4i:an+ia <,-i@,n:a/ 4*4* */ *:-*8, i@?i*-4,= /a*+a+in 4* ,n4a 4* /a 5,-8aF

%2 d

28

dx2 + Ax=

d28

dt 2

UNIVERSIDAD NACIONAL DEL CALLAOPágina $



D,n4* A * na +,n:an:* -**/7a /a *+a+in 4i5*-*n+ia/ 6a-+ia/*:a aF

U 0=:0 l U1=:0 : 0

U= 00 ld8

dt

⎪t =0=0 : m 0

50 l

A/ n, *- <,8,g9n*a * 6-,6,n* n +a8;i, 4* 7a-ia;/*F

%2 d

28

dx2 + Ax=

d28

dt 2

U= : = : hzLa *gn4a 4*-i7a4a +,n -*6*+:, 4* *F

d28

dx2 =

d2-

dx2 +J ( x)

d28

dt 2 =

d2-

dt 2

- t ( x , t )=ut ( x , t )

- t ( x ,0 )=u t ( x ,0 )=0

- ( x ,0 )=8 ( x ,0 )−J ( x )

( a;*8, ?* 8 ( x ,0 )=0 6,- /, ?*F

- ( x ,0 )=0−J ( x )

S:i:*n4, /, 7a/,-* *n /a *+a+in ,-igina/F

UNIVERSIDAD NACIONAL DEL CALLAOPágina &



d2-

dt 2 =% ²( d

2-

dx2 +J ( x ))+ Ax

A> * ,;:i*n* /a *+a+in <,8,g9n*a.

d2-

dt 2 =% ²

d2-

dx2

P,- /, ?*

%2J

( x )+ Ax=0

En+,n:-an4, */ 7a/,- 4* J ( x )

J ( x )=

− Ax

%2

J ( x )=

− Ax ²

2%2 +0

J ( x )=− Ax ³

2 ⋅3%2

+0x+ K

D*F U=: =: hz * 4*6*a =: * a6/i+a /a 6-i8*-+,n4i+inF

=: U=: wz

0=: 0 U0=: ! z0

S:i:*n4, */ 7a/,- 4* +*-, *n /a *+a+i,n* ,;:*ni4a 6a-a U z-*6*+:i7a8*n:* * ,;:i*n*F

'0=: 0 U0=: ! z0 0 h u 0

P,- /, :an:, u 0

UNIVERSIDAD NACIONAL DEL CALLAOPágina #0



A6/i+an4, /a *gn4a +,n4i+inF

'1=: 0 U1=: ! z1

S:i:*n4, */ 7a/,- *n /a *+a+i,n* ,;:*ni4a 6a-a U z-*6*+:i7a8*n:* * ,;:i*n*F

'1=: 0 U1=: ! z1 w A

6 %2 0 6,- /, ?*

A

6 %2

S:i:*n4, *n /a *+a+in *n+,n:-a4a 6a-a z ,;:*n*8,F

J ( x )=− Ax ³

6%2 +

Ax ³

6 %2 x

J ( x )= A6 %

2 ( x− x3 )

d2-

dt 2 =% ²

d2-

dx2

- ( x ,t )=(C 1cos ( x )+C 2 9en ( x ) )(C 3cos (%t )+C 4 9en (%t ))

• A6/i+an4, /a +,n4i+inF 0=:0

C 1 cos (0 )+C 2 9en (0)=0 3C 1 cos (0 )=0 3C 1=0

P,- /, ?* /a *+a+in * -*4+*F

- ( x ,t )=(C 2 9en ( x ) )(C 3 cos (%t )+C 4 9en (%t ))

•

A6/i+an4, /a +,n4i+inF

t

( x ,0 )=0

- t ( x , t )=(C 2 9en ( x )) (−C 3 9en (%t )+C 4 cos (%t ))




−C 3 9en (0 )+C 4 cos (0 )=0 3C 4cos (0)=0 3C 4=0

- t ( x ,0 )=¿

P,- /, ?* /a *+a+in * -*4+*F

- ( x ,t )=C 2 9en ( x )C 3 cos (%t )

Ag-6an4, /a +,n:an:*F

- ( x ,t )=Mn 9en ( x ) cos (%t )

• A6/i+an4, /a +,n4i+in L=:0

9en ( L)=0 a;*8, ?* 9en (nπ )=0

En:,n+*F {L n 6,- /, ?* {nWL l +,8, L1 *n:,n+*F{n

S:i:9n4,/a *n /a *+a+in n, ?*4aF

- ( x ,t )=Mn 9en (nπx ) cos (nπ%t )

P*-, +,8, n 6*4* :,8a- +a/?i*- 7a/,- *n:,n+* * +,n7i*-:* *n na8a:,-iaF

- ( x ,t )=∑n=1

∞

Mn 9en (nπx ) cos (nπ%t )

A6/i+an4, =0

- ( x ,0 )=∑n=1

∞

Mn 9en (nπx )=0

- ( x ,0 )=8 ( x ,0 )−J ( x )=0

- ( x ,0 )=− A

6%2 ( x− x

3 )=f ( x )=0

S* -**/7* /a *-ia 4* ,-i*- *n,




Mn=2

L∫0

1

f ( x ) 9en (nπx )dx

S* a6/i+a /a /:i8a +,n4i+in 4* 5-,n:*-a 50 * :i:* *n 7a/,-

4* z 6a-a /a *-i* 4* ,-i*-FL1

Mn=− A

3% ²∫0

1

( x− x ³ ) 9 en (nπx ) dx

Mn=− A

3% ²

∫0

1

x9en (nπx ) dx+ A

3% ²

∫0

1

x ³ 9en (nπx ) dx

PRIMERA PARTE DE LA INTERALF

− A

3% ²∫0

1

x9en (nπx ) dx

P,- 6a-:*F

44

47*nn 7!+,n1Wn

− x

nπ cos (n πx )

1

⎪0

+ 1

nπ ∫0

1

cos (nπ x)dx

−1

nπ cos (nπ )+( 1nπ )

2

9en (n π x )1

⎪0

=−(−1 )n

n π

− A

3% ²⋅−(−1)n

n π =

A (−1 )n

3 % ²n π ?e9u$t%d; de$% ?i<e? inte1?%$




SEUNDA PARTE DE LA INTERALF

A

3%2∫

0

1

x39en (nπx )dx

P,- 6a-:*F

| 4 3}4

47 *nn 7!+,n1Wn

− x ³

nπ cos (n πx )

1

⎪0

+ 3

nπ ∫0

1

x ²cos (n π x)dx

P,- 6a-:*F} 424

47+,n 7*nn1Wn

−(−1 )n

nπ +

3

nπ ( x2

nπ 9en (n πx )

1

⎪0

− 2

nπ ∫0

1

x 9en (n π x ) dx )

P,- 6a-:*F

44

47*nn 7!+,n1Wn

−(−1 )n

nπ

+ 3

nπ

(−2

nπ

(− x

nπ

cos (n πx )1

⎪0

1

nπ

∫0

1

cos (n π x ) dx

))




−(−1 )n

nπ −

6

(nπ )² (−(−1 )n

nπ +

1

nπ (( 1nπ )2

9en (nπ x )1

⎪0))

−(−1 )n

nπ +

6 (−1 )n

(nπ )³

¿ A

3%2 (−(−1 )n

n π +

6 (−1 )n

(nπ ) ³ )

¿− A (−1 )n

3%2 nπ +2 A (−1)n

%2(nπ )³?e9u$t%d; de$%9e1und% %?tede$%inte1?%$

Mn= A (−1 )n

3% ²n π −

A (−1 )n

3%2nπ +2 A (−1 )n

%2(nπ ) ³

Mn=2 A (−1)n

%2(nπ )³

S:i:*n4, *n /a *-i* 4* ,-i*- a6/i+an /a +,n4i+in =0F

- ( x ,0 )=∑n=1

∞2 A (−1 )n

%2(nπ ) ³9en (nπ )

- ( x ,0 )= 2 A

%2 π ³∑n=1

∞ (−1 )n

n ³ 9en (nπ )

- ( x ,t )= 2 A

%2

π ³∑n=1

∞ (−1 )n

n ³

9en ( nπx ) cos (nπ%t )

=: U=: wz

U=: =: hz :i:*n4, 6a-a *n+,n:-a- */ -*/:a4, 5ina/F

UNIVERSIDAD NACIONAL DEL CALLAOPágina ##



8 ( x , t )= 2 A

%2π ³∑n=1

∞ (−1 )n

n ³ 9en (nπx )cos (nπ%t )+

A

6%2 ( x− x

3 )

Problema 4

a En */ 6-,;/*8a 1= 6,nga ?* %=M=π f ( x )=100 x (π − x) . Sin

a- /a ,/+in == ;,?** /a a6a-i*n+ia ?* :*n4->a /a

6*-+i* *n /a -*gin -*+:ang/a- 4*ni4a 8*4ian:* 0≤ x ≤ π =

0≤ y ≤ π (

; ~Cá/ * */ 7a/,- 8ái8, 4* /a :*86*-a:-a 6a-a 0≤ x ≤ π =

0≤ y ≤π •

a

C,n4i+i,n* 4* -,n:*-aF

u (0, y )=0 u (% , y )=0 u ( x ,0 )=0 u ( x , M )=f ( x )

UNIVERSIDAD NACIONAL DEL CALLAOPágina #%



∂2u

∂ x2+

∂2u

∂ y2=0

La ,/+in g*n*-a/ 4* /a *+a+in 4* +a/,- 4* *:* +a, *F

u ( x , y )=(c1cos ( x )+c2 9en ( x ))(c3e y+c4 e− y )

A6/i+an4, /a +,n4i+i,n*

u (0, y )=0,c1 (c3 e y+c4 e

− y )=0∴c1=0 u (% , y )=0,c2 9en ( % )=0∴ %=nπ , =nπ

%

u ( x ,0 )=0,c2 9en

(nπ

% x

)(c3+c 4 )=0

∴

−c3=c4

u ( x , y )=(c2 9en( nπ

% x))(9enN ( nπ

% y )) u ( x , y )=∑

n=1

∞

Mn 9en( nπ

% x) 9enN ( nπ

% y )

u ( x , M )=f ( x )=∑n=1

∞

Mn 9en( nπ

% x) 9enN ( nπ

% M)

2

S6,ni*n4, ?* %=M=π f ( x )=100 x (π − x)

u ( x , M )=f ( x )=∑n=1

∞

Mn 9en( nπ

% x) 9enN ( nπ

% M)

S* in:*g-a 6a-a ,;:*n*- */ 7a/,- 4* Mn F

Mn=2

%∫0

%

f ( x ) 9en ( nπ

% x)dx




Mn= 2

%9enN( nπ

% M)∫0

%

100 x (π − x )9en( nπ

% x)dx

Mn= 200

%9enN( nπ

% M) [

π ∫0

π

x9en (nπ %

x)dx−∫0

π

x2 9en (nπ %

x)dx

]

1= 200

%9enN ( nπ

% M)∫0

π

x 9en ( nπ

% x )dx=

200

%9enN ( nπ

% M) [

− x%

nπ cos( nπ

% x )+ %

nπ ∫cos( nπ

% x)dx ]

1= 200

%9enN ( nπ

% M)∫0

π

x9en ( nπ

% x )dx=

200

%9enN( nπ

% M)

[− x%

nπ cos( nπ

% x)+( %

nπ )2

9en( nπ

% x)]

0

π

E7a/an4, a6/i+an4, /a +,n4i+in %=π

1= −200π

9enN(nπ )ncos (n π )

Pa-a 2

2= 200

%9enN( nπ

% M)∫0

π

x29en ( nπ

% x)dx=

200

%9enN ( nπ

% M) [

− x2%

nπ c; 9

nπx

% +2( %

nπ )∫ xc;9 nπx

% dx ]

2= 200

%9enN

(nπ

% M

)∫0

π

x29en ( nπ

% x)dx=

200

%9enN

(nπ

% M

) [

− x2%

nπ cos

nπx

% +2( %

nπ )[

x%

nπ 9en

nπx

% −

%

nπ ∫ 9en

nπ

%

UNIVERSIDAD NACIONAL DEL CALLAOPágina #$



2= 200

%9enN( nπ

% M)∫0

π

x29en ( nπ

% x)dx=

200

%9enN ( nπ

% M) [

− x2%

nπ cos

nπx

% +2 x ( %

nπ )2

9en nπx

% +

2%3

n3π

3 cos

nπx

% ]

E7a/an4, a6/i+an4, /a +,n4i+in %=π

2=

200 π

n9enN (n π )cos (n π )−

400

π n3

9enN(nπ )cos (n π )+

400

π n3

9enN(nπ )

P,- /, :an:,F

Mn= 1+ 2

Mn= −200π 9enN(nπ )n cos (n π )+ 200π n9enN(nπ ) cos (n π )− 400π n

39enN(nπ )

cos (n π )+ 400π n

39enN (nπ )

Mn= 400

π n3

9enN (nπ )[1−(−1)n ]

P,- /, :an:, = *F

u ( x , y )=∑n=1

∞400

π n39enN(nπ )

[1−(−1)n ] 9en (nx ) 9enN (ny )

S* -*a/i@a /a 4*-i7a4a 6a-+ia/ +,n -*6*+:, a K K= 6a-a +,86-,;a-*n ?9 6n:, * ,;:i*n* na :*86*-a:-a 8ái8a 4* a+*-4, a /ag-á+a 4* /a 4*-i7a4a 6a-+ia/*.

u x ( x , y )=∑n=1

∞400

π n39enN (nπ )

[1−(−1)n ] cos (nx ) 9enN (ny )

u y ( x , y )=∑n=1

∞400

π n39enN(nπ )

[1−(−1)n ] 9en ( nx ) cosh (ny )

UNIVERSIDAD NACIONAL DEL CALLAOPágina #&



. APLICACIONES DE LAS EDP LINEALES DE PRI7ERORDEN:

ECUACI6N DE TRANSPORTE

C,ni4*-*8, n i4, ?* * 8*7* +,n na 7*/,+i4a4 +,n:an:* V *nn :;, -*+:,= n, +,n *++in :-an7*-a/ A. S6,nga8, ?* */i4, +,n:i*n* n +,n:a8inan:* +a +,n+*n:-a+in *n */ 6n:, K x *n*/ in:an:* Kt 4*n,:a-*8, 6,- ux, t!. Pa-a i86/i+a- 6,n4-*8,?* n, <a ,:-a 5*n:* 4* +,n:a8inan:* *n */ :;, ?* */+,n:a8inan:* n, 6*4* *+a6a- a :-a79 4* 6a-*4*. En:,n+*= *n*/ in:an:* := /a +an:i4a4 4* +,n:a8inan:* *n /a *++in 4*/ :;, *n:-*

/a 6,i+i,n* x " y x # 7i*n* 4a4a 6,-

∫ x1

x2

u( x ,t ) Adx

P,- ,:-, /a4,= 6,4*8, *6-*a- /a +an:i4a4 4* +,n:a8inan:* ?* *a :-a79 4* n 6/an, i:a4, *n */ 6n:, K x = 4-an:* */ in:*-7a/, 4*:i*86, st ", t #t= +,8,

∫t 1

t 2

u( x ,t ) A.dt

E:a8, a<,-a *n +,n4i+i,n* 4* <a+*- */ igi*n:* ;a/an+*F /a +an:i4a44* +,n:a8inan:* *n /a *++in s x ", x #t *n */ in:an:* :2 * iga/ a /a+an:i4a4 *n 4i+<a *++in *n */ in:an:* an:*-i,- :1 8á /a +an:i4a4 ?*

*n:-, a :-a79 4*/ 6/an, *n /a 6,i+in x " 4-an:* */ in:*-7a/, 4* :i*86,st ", t #t :*n*8, /a +an:i4a4 ?* a/i a :-a79 4*/ 6/an, *n /a 6,i+in 24-an:* */ 8i8, in:*-7a/, 4* :i*86,. E:, *=

UNIVERSIDAD NACIONAL DEL CALLAOPágina %0



u( x ,t 1) Adx+¿∫

t 1

t 2

u( x1,t ) A.dt −∫

t 1

t 2

u( x2, t ) A.dt

∫ x1

x2

u( x ,t 2) Adx=∫

x1

x2

¿

P,- */ T*,-*8a n4a8*n:a/ 4*/ Cá/+/, a;*8, ?*

∫ x1

x2

u( x ,t 2) Adx−∫

x1

x2

u( x ,t 1) Adx=∬

t 1 3 x1

t 2 3 x2

u t ( x ,t ) Adxdt

∫t 1

t 2

u( x1,t ) A.dt −∫

t 1

t 2

u( x2, t ) A.dt =−∬

x1 3t 1

x2 3t 2

u x( x ,t ) A.dtdx

C,8;inan4, *:a i4*n:i4a4* //*ga8, a ?*

u

(¿¿ t ( x ,t )+. u x ( x , t )) Adxdt =0

∬t 1 3x 1

t 2 3x 2

¿

S6,ni*n4, ?* /a iga/4a4 an:*-i,- * 7*-i+a *n +a4a *g8*n:, 4*/:;, *n +a4a in:*-7a/, 4* :i*86, ?* /a 5n+in ux, t!

4*-i7a4a 6a-+ia/* 4* 6-i8*- ,-4*n ,n +,n:ina= ,;:*n*8,

ut ( x , t )+. u x ( x ,t )=0.




LA ECUACI6N DE TRANSPORTE SI7PLE

T,8a /a 5,-8a=

ut +cu x=0 a

A48i:* +,8, ,/+i,n* *n R2 a /, //a8a4, 5-*n:* 4* ,n4a K:-a7*//inga7*€=

u( x ,t )=N( x−ct ) ;

D,n4* * +,n,+* a Kc$ +,8, 7*/,+i4a4 4* 6-,6aga+in.

C,8, *n */ +a, 4* /a EDOS= n 6-,;/*8a 4* Ca+< 6*-8i:*4*:*-8ina- :,4a /a ,/+i,n* 4* /a 6-i8*-a *+a+in a. A :a/ *5*+:,* 8á g*:i7, *+-i;i- a *n /a 5,-8a=

ut =−cu x +

E i8agina-* ?* */ 7a/,- ini+ia/ * :,4a na 5n+in 4* 8i*n:-a ?**/ K/ga- 4* /, 4a:, in>+ia/* *= *n 7*@ 4* n 6n:,= :,4, */ ** K x$.




TEORE7A ;a

Pa-a +a4a ‚ C1R */ 6-,;/*8aF

{u t +cu x=0

u( x ,0 )=)( x )

S,/, a48i:* u=)( x−ct ) +,8, ,/+in.

Ee/,+o 1. A6/i+a+in 4*/ 89:,4, 4* /a +a-a+:*->:i+a 6a-a -*,/7*- */6-,;/*8a 4* Ca+< 6a-a /a *+a+in 4* :-an6,-:*

{u t +cu x=0,

u( x ,0 )=)( x ) , x∈ R ()∈C 1 ( R ))

S,/+in ux, t! % &x ' ct F 6-,6aga+in 4* a /, /a-g, 4* /a+a-a+:*->:i+a= ,n4a a ?* * 8*7* a 7*/,+i4a4 +,n:an:*.

Ee/,+o 0. P-,;/*8a 4* Ca+< 6a-a /a *+a+in 4* :-an6,-:* +,n+,n4i+in ini+ia/ ,;-* na 6-,*++in +a-a+:*->:i+aF

{u t +cu x=0,

u( ct , t )=N(t ) , t ∈ R (N∈C

1 ( R ))

Inni:a , ningna ,/+in 4*6*n4i*n4, 4* i < * +,n:an:* , n,=-*6*+:i7a8*n:*.




Ee/,+o 3. E/ 6-,;/*8a 4*/ :-á+, *n na a:,6i:a "a;*-8an s1t.D*4++in 4*/ 8,4*/,. La i86/i+a+in 4*/ 8,4*/, +,n /a <i6:*i 4*?* /a 7*/,+i4a4 4*6*n4* *+/i7a8*n:* 4* /a 4*ni4a4 4*/ :-á+, (x,

t! //*7a a /a *+a+inF

Ot +c ( O ) O x=0

M,4*/, 6a-a +a/?i*- 5*n8*n, 5>i+, *n */ ?* * +,n*-7a /a 8aa */ , 4* 8a:*-ia * 5n+in 4* /a 4*ni4a4. A6/i+a+in 4*/ 89:,4, 4*

/a +a-a+:*->:i+a *n */ 6-,;/*8a 4* Ca+< n, /in*a/ 4a4, 6,- /a*+a+in 4* B-g*-=

{u t +uu x=0,

u( x ,0 )=N( x ) , x∈ R

P-i8*- +a,= +,n < na 5n+in +-*+i*n:*F ,n4a *6ani7a. S*gn4,+a,= +,n < na 5n+in 4*+-*+i*n:* *n a/gn in:*-7a/,= 6,- **86/,

N( x ) { 11− x0

9i x≤0

9i 0≤ x ≤1

9 i x41

C,-:* *n /a 6-,*++i,n* +a-a+:*->:i+a. On4a +,86-*i7a ,n4a 4*+<,?*F ing/a-i4a4= n, *i:* ,/+in +/ái+a g/,;a/8*n:* 4*ni4a.Di+,-4an+ia *n:-* /a *+a+in /a -*a/i4a4 */ , +,n:ina= /a*+a+in * 4*8aia4, *ig*n:*. C,n+*6:, 4* ,/+in 49;i/ +,8,,/+in 4* /a /* 4* +,n*-7a+inF




∂

∂t ∫ x1

x2

u( x ,t )dx+1

2u( x2 ,t )2−

1

2u( x1 ,t )2=0

D*a--,//, :*-i+, 6a-a na *+a+in g*n*-a/ :i6,

A (u)t +B(u) x=0

C,n /* 4* +,n*-7a+in a,+ia4a

u

A (¿¿ ( x , t ))dx+B(u( x2,t ))−B(u( x

1,t ))=0

∂

∂ t ∫ x

1

x2

¿

Si ux, t! * na ,/+in -*g/a- a/7, *n na +-7a -*g/a- 4*/ 6/an, x

% )t! a :-a79 4* /a +a/ 6-**n:a na 4i+,n:ini4a4 4* a/:, ni:,= *<a 4* +86/i- /a +,n4i+in 4* +<,?*F

UNIVERSIDAD NACIONAL DEL CALLAOPágina %#



+¿ (t )u¿

−¿ ( t )¿¿¿

u¿+¿ (t )−¿ ( t )

u¿

¿¿

u¿−B ¿B ¿

P Q (t )=¿

D,n4*

x*P (t )−¿u ( x , t )−¿ (t )=lim

¿¿

u¿

x* P (t )+¿u ( x , t )+¿ (t )=lim

¿¿

u¿

En */ **86/, an:*-i,-= /a 7*/,+i4a4 4* /a +-7a 4* +<,?* <a 4* *-*n:,n+* 1W2= /, ?* 6*-8i:* 4*:*-8ina- /a +-7a ƒ: :h1W2 4*5ini-g/,;a/8*n:* /a ,/+in 49;i/

u ( x , t )={10 9i x<P (t )9 i x 4 P (t )

UNIVERSIDAD NACIONAL DEL CALLAOPágina %%



I. DISTRIBUCIONES " ESPACIOS DE SOBOLE

Problema 1

D*8,:-a- ?* /a 4*iga/4a4 4* P,in+a-9 n, * +86/* *n :,4a /a -*+:a-*a/.

SOL.

T*n*8, ?* 6-,;a- ?* n, *i:* ningna +,n:an:* 6,i:i7a C 0 :a/?*

∫ R

)2dx≤C ∫

R

|)Q |2

dx ,∀)∈ H 1( R)

Pa-a *//, 7a8, a +,n:-i- na +*in 4* 5n+i,n* :*: n C +]R

:a/* ?*

∫ R

)2dx /∫

R

|) Q |2

dx*∞

Da4a na 5n+in :*: n, :-i7ia/ C +] R 4*ni8,

)n ( x )=)

( xn )

En:,n+*




∫ R

❑

)n

2 ( x ) dx=∫ R

❑

)( xn )dx=∫ R

❑

)2( y)dy

Mi*n:-a ?*

∫ R

|)Q

n( x)|2

dx= 1

n2∫

R

()Q )2( xn )dx=1

n∫ R

()Q )2( y )dy

P,- /, :an:,

∫ R

❑

)n

2 ( x ) dx

∫ R

❑

()Q

n )2( x )dx

=∫ R

❑

)2( y)dy

∫ R

❑

() Q )2( y)dy

n2*∞,n*∞

Problema 2

C,86-,;a- ?* /a 4*iga/4a4 4* P,in+a-9 :a86,+, * +86/* *n nin:*-7a/, *8i!inni:,.

SOLF P,- :-a/a+in i8*:->a ;a:a +,ni4*-a- */ +a, 4*/ in:*-7a/, 0=] 4,n4* /a 8i8a 6-*;a 4*/ 6-,;/*8a an:*-i,- * a6/i+a.

Problema 3

S*a * % +, "! +, "! „ R2 . D*8,:-a- ?*

UNIVERSIDAD NACIONAL DEL CALLAOPágina %$



∫

❑

)2dxdy ≤∫

❑

|) x|2dxdy ,∀ )∈C c

1()

SOL. Pa-a +a/?i*- 6n:, x, y! …= +,8, +, y! †… :*n*8, ?*-+, y! % +. In:*g-an4, *n /a 7a-ia;/* x *n */ *g8*n:, <,-i@,n:a/ ?*n* +, y! +,n x, y! :*n*8, *n:,n+*

)( x , y)=∫0

x

∂ x )(9 , y)d9

P,- /, :an:,

|)( x , y )|≤∫0

x

|∂ x )(9 , y)|dx

≤

(∫0

x

|∂ x )(9 , y)|2dx

)

1

2 √ x

≤(∫0

1

|∂ x )(9 , y)|2

dx)1

2

En:,n+*

|)( x , y )|2≤∫

0

1

|∂ x )(9 , y)|2dx ,∀( x , y)∈

UNIVERSIDAD NACIONAL DEL CALLAOPágina %&



In:*g-an4, *n x, y! … ,;:*n*8, ?*

∫

|)( x, y )|2dxdy ≤∫

|∂ x )(9 , y)|2dxdy

Pro2+e/a <. C,ni4*-a8, /a ;an4a inni:al 4*8,:-a- ?* * +86/*/a 4*iga/4a4 4*/ **-+i+i, an:*-i,-.

… 0= 1 ‡ R y „ R2

SOL. La 6-*;a 4*/ **-+i+i, an:*-i,- * a6/i+a in ningna a/:*-a+in6* n, *ga 8á ?* */ 6a6*/ 4* n 6a-á8*:-, 4* in:*g-a+in.

Problema 5

C,ni4*-a8, */ 6-,;/*8a 4* N*8ann

{−u xx=f , 0< x<1u x (0 )=u x (1 )=0 1

• P-,;a- ?* /a +,n4i+in n*+*a-ia 6a-a ?*1 :*nga ,/+in *

?* ∫0

1

f S 0

• P-,;a- ?* /a ,/+in 4* 1 n, 6*4* *- ni+a.• P-,;a-= an4, /a 4*iga/4a4 4* P,in+a-9 !'i-:ing*-= ?* /a

,/+in 4* 1 4* 8*4ia +*-, * ni+a.




SOL

• In:*g-an4, /a *+a+in 4* 1 an4, /a +,n4i+i,n* 4*+,n:,-n, :*n*8,

∫0

1

fdx=−∫0

1

−u xx dx=−u x (1)+u x (0)=0

• En *5*+:,= i * na ,/+in 4* 1 +a/?i*- ,:-a 5n+in 4* /a5,-8a u % u c +,n + R +,n:an:* * :a8;i9n ,/+in 4*16*:, ?*

ucQ Q =−(u+c)Q Q =−u

Q Q =f

?*

uc

Q ( x)=(u+c )Q ( x)=uQ ( x)=0, x=(0,1 )

• S6,nga8, ?* u" u# ,n 4, ,/+i,n* 4* 1. En:,n+* v %u" ' u# * ,/+in 4*

∫0

1

x

2dx=0

A<,-a ;i*n= i u" u# ,n 5n+i,n* 4* 8*4ia n/a :a8;i9n v /, * 6,- /,

?*= 6,- /a 4*iga/4a4 4* P,in+a-*!'i-:ing*-= i ∫0

1

x2 dx=0 :a8;i9n *




:i*n* ∫0

1

2dx=0 = 6,- +,nigi*n:*= v / +. E 4*+i-= u" / u# /, +a/

ga-an:i@a */ -*/:a4, 4* ni+i4a4 *nn+ia4,.

BIBLIOGRAF=A

>1? R. 9a2er/an! E$&a$(ones en Der(8a*as Par$(a+es $on Ser(es*e Fo&r(er 5 Pro2+e/as *e Contorno! Prent($e 9a++! 7a*r(*!0@@3.

>0? I. Pera+! Pr(/er C&rso *e E$&a$(ones en Der(8a*as Par$(a+es!A**(sones+e5! 1. ;D(s,on(2+e en +a ,g(na e2 *e+ ,rofesorIreneo Pera+

>3? ". P(n$o8er! J. R&2(nste(n! An Intro*&$t(on to Part(a+D(Herent(a+ E%&at(ons! Ca/2r(*ge Un(8ers(t5 Press! 0@@.

Solucion Del Examn Parcial

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