Задачи по мат.физике

(dy)2 ; 5 dy dx ; 6(dx)2 = 0

y + x = C1 y ; 6x = C2

I ZAPI[EM EGO OB]EE RE[ENIE

(

u y=6x

=

f(7x) + g(0)

=

'(x)

(16)

uy

=

f0(7x) + g0(0)

=

(x):

y=6x

'(x) = 7x2

(x) = 2x:

rE[ENIE ZADA^I NEEDINSTWENNO:

1

u(t x ) = 7 (x + y)2 + g(y ; 6x)

GDE g( ) 2 C

2

R

( ) | L@BAQ FUNKCIQ, UDOWLETWORQ@]AQ USLOWIQM

g(0) = g0(0) = 0:

B) pRIMER NA^ALXNYH USLOWIJ, PRI KOTORYH ZADA^A kO[I

NE IMEET RE[ENIQ:

'(x) = x2

(x) = 2x:

nAJTI RE[ENIE u(x t ), x = (x1

2

3), W R3 R+ ZADA^I:

utt = xu

u

= 0

ut

= x 7:

t=0

t=0

j j

x =

@2

+ n ; 1

@

:

@r2

r @r

utt = urr + 2 ur

r > 0>

0

u

t=0

= 0 ut

t=0

= r7:

r

dOMNOVIM URAWNENIE NA r :

rutt = rurr + 2ur

I SDELAEM ZAMENU v(r

) = ru(r ): tOGDA vtt = rutt

r = rur + u

vrr = rurr + 2ur: tAK KAK u(0

) OGRANI^ENA, TO v(0

) = 0:

pOLU^IM ZADA^U DLQ FUNKCII v(r )

vtt = vrr r > 0>

0

v

t=0

= 0 t

= r8

v

r=0

= 0:

t=0

8

1

r+t

1

2rZt

8 d =

(r + t)9

; (r ; t)9

r > t

v(r ) =

18

>

;

1

r+t

1

<

8

9

; (t ; r)

9

>

;

d =

(r + t)

r < t:

:

2tZr

18

82

A

;e;(x+t)

2

2

+ cos !(x ; t):

u(x t ) = 2

; e;(x;t)

fUNKCIQ u(x t ) PRINADLEVIT KLASSU C2(

+

+), ESLI ONA IME-

R

R

ET DWE NEPRERYWNYE PROIZWODNYE NA UGLOWOJ HARAKTERISTIKE

x = t. dLQ \TOGO FUNKCIQ f( ), ZADAWAEMAQ f( ) = Ae; 2 =2 PRI

> 0

I f( ) =

;

Ae; 2 =2 + cos ! PRI < 0, DOLVNA BYTX KLAS-

SA C

2

W NULE,

TO ESTX

f(+0) = f(;0)

()

A

= 1 ;

A

()

A = 1

2

2

f0(+0) = f0(;0)

(WYPOLNENO WSEGDA),

! = p

f00(+0) = f00(;0)

()

;1 = 1 ; !2

()

2

TAK KAK

f0(+0) = ; e; 2

=0 = 0

f0(

;

0) =

e; 2

! sin

!t

= 0

;

=0

f00

(+0) =

;

e;

+ 2 2e;

=0

=

;

1

2

2

f00

(;0) = e;

; 2 2e;

; !2 cos ! =0 = 1 ; !2

pRI NAJDENNYH ZNA^ENIQH

A

I

!

POLU^IM DWAVDY NEPRERYW

-

NO DIFFERENCIRUEMOE RE[ENIE ZADA^I:

e;(x+t)2 + e;(x;t)2

=2

x > t

u(t x ) = (;e;(x+t)2

e;(x;t)2

=2 + cos

;

p

(x

2

t)

x < t:

;

;

;

zADA^A 3.33.

pUSTX u(x t ) | RE[ENIE W [0 1]

+ SME[ANNOJ ZADA^I

R

utt = 4uxx

u x=0 = u x=1 = 0

u

t=0

= 4 sin3 x

ut

t=0

= 30x(1

x):

84

;

Z0

1

A) nAJTI f

1

, GDE f(t) =

ut2(x t ) + 4u2x(x t )

dx.

3

B) nAJTI u;(x

2).

rE[ENIE. A)

f0(t) = Z01

2ut(x t )utt(x t ) + 8ux(x t )utx(x t ) dx =

= f

IZ URAWNENIQ utt = 4uxx g =

1

= 8

Z0

ut(x t )uxx(x t ) + ux(x t )utx(x t )

dx =

x=1

= fPO ^ASTQMg = 8ut(x t )ux(x t )

x=0

;

;

8

Z0

1 utx(x t )ux(x t )dx + 8

1 ux(x t )utx(x t )dx = 0

Z0

PODSTANOWKA RAWNA NUL@ IZ GRANI^NYH USLOWIJ:

u x=0 = u x=1 = 0 =) ut

x=0 = ut

x=1 = 0:

tAK KAK f

0(t) = 0 TO f(t) const, I

f;

1

= f(0) = Z01

ut2(x 0) + ux2(x 0) dx:

3

dLQ TOGO, ^TOBY NAJTI ux(x 0)

PRODIFFERENCIRUEM NA^ALXNOE

USLOWIE u(x 0) = 4 sin3 x PO x. pOLU^IM

;

1

Z0

1

x))2 + 4(12 sin2 x cos x)2

dx = 30 + 36 2:

f

=

(30x(1

;

3

B) nAJDEM OB]EE RE[ENIE ZADA^I METODOM fURXE:

X

u(x t ) =

1

An cos 2 nt + Bn sin 2 nt

sin nx:

n=1

tOGDA RE[ENIE u(x t ) 1{PERIODI^NO PO WREMENI, I

u(x 2) =

X

An cos 4 n + Bn sin 4 n

sin nx =

1

1

n=1

=

X

An sin nx = u(x 0) = 4 sin3 x:

'(x) 2 C1((0

))

SU]ESTWUET NEOGRANI^ENNOE RE[ENIE W

[0 ]

R+ ZADA^I

utt = 9uxx

u x=0 = (ux ; ku) x= = 0

u t=0

= 0

ut

t=0 = '(x):

B

)

dLQ

k =

1

OPISATX WSE FUNKCII

'(x) 2 C1((0 )),

DLQ

KOTORYH RE[ENIE u(t x ) \TOJ ZADA^I OGRANI^ENO.

X00(x) = jXj(x)

Xj(0) = 0

X0 ( )

;

kXj( ) = 0

(17)

j

j

A FUNKCII Tj(t) | RE[ENIQ ZADA^I

Z0

Z0

T 00 = 9 jTj Tj(0) = 0

0(0) =

'(x)Xj(x)dx

.

X2

(x)dx:

j

j

j

(18)

1 ; k = 0 ()

k = 1= :

:

ut = uxx

(0 1)>

0

:

ut = uxx

(0 1)> 0

x2=1

A) 8 u x=0

= ux

= 0

!

B) 8 ux

x=0 = ux 2x=1

= 0

= '(x)

!

1

<

u

t=0

<

u

t=0

= '(x)

OBLADAET SWOJSTWOM u(t x )

0 PRI t

+

?

rE[ENIE.

A) nAJDEM RE[ENIE ZADA^I METODOM fURXE

1

2(n+

1

)2t

1

X

2

u(t x ) =

'n e;

sin

n + 2

x

n=0

sin

;

n +

1

2

0

;;!

x n

= 0 1:: : sLEDOWATELXNO, u(t x )

0 PRI

2

t

L@BOJ FUNKCII '(x)

C1(0 1):

!1

u(t x ) = '0 +

1

'n e; 2n2t cos( nx)

X

n=1

ZISU 1 cos( nx) = 1 2::

: sLEDOWATELXNO, lim u(t x ) = '0

A KO\FFICIENT

t!1

'0 = 0 PRI SLEDU@]EM USLOWII NA FUNKCI@ '(x) :

Z1

'(x) dx = 0:

0

pUSTX u(t x ) | RE[ENIE W Q1

)

ZADA^I

(0

ut = uxx

u x=0 = ux x= = 0

u t=0 = '(x)

GDE

'(0) = '0( ) = 0.

j

A) dOKAZATX, ^TO

sup

u(1

)

6

sup

'(x)

:

0<x< j

0<x< j

j

B) wERNO LI, ^TO

sup

u(1

)

j

6

1

sup

'(x)

j

?

0<x< j

2

0<x< j

REZ TO^KU NA MNOVESTWO x 2

( 2 ), TO ESTX POLOVIM u~(t x ) =

u(t 2 ; x) PRI x 2 ( 2 ). pOSTROENNAQ FUNKCIQ u~(t x ) QWLQ-

ETSQ RE[ENIEM KRAEWOJ ZADA^I

u~t = u~xx

x 2 (0 2 ) t > 0

sup

u(1

)

j

=

sup

u~(1 )

j

6

sup

'~(x)

j

=

sup

'(x)

:

0<x< j

0<x<2 j

0<x<2 j

0<x< j

j

sup

u(1

)

= e;1=4

sup

'(x)

j

= 1

0<x< j

j

0<x< j

e;1=4 > 1=2, TAK KAK e < 24.

zADA^A 4.33.

pUSTX u(x t ) | RE[ENIE W R R+ ZADA^I kO[I

x2 + sin x

ut = 4uxx

u

t=0 = 1 + 2x2

:

nAJTI

lim u(x t ):

t!+1

zADA^A 4.34.

pUSTX u(x t ) | RE[ENIE W R R+ ZADA^I kO[I

ut = uxx

u

t=0

= arcctg x:

nAJTI

lim u(x t ):

t!+1

ut = uxx

u t=0 = '(x) 2 C(R) \ L1(R):

1

l

nAJTI

lim u(0 ),

lim

'(x) dx = A:

ESLI

t!+1

l!+1 l

Z;l

ut = uxx

W

R

R

+

( u t=0 = '(x)

x

2

R

'(x) 2 C(R) \ L1(R): tOGDA

1. eSLI

lim

'(x) = A

lim

'(x) = B

(20)

x!+1

x!;1

TO

lim

u(x t ) =

A + B :

t!+1

2

2. eSLI

lim 1 l

'(x)dx = A

(21)

l!+1 l ;Zl

TO

lim

u(x t ) =

A:

t!+1

2

u(x t ) = '0

3. eSLI

'(x) | PERIODI^ESKAQ FUNKCIQ, TO lim

STAWLQ@]IH '+ =

'(x) + '(;x)

=

'(x) ;

'(;x)

: w SILU

2

;

2

FORMULY pUASSONA POLU^IM,

^TO

u(x t ) =

1

1 '+( ) exp

;

( ; x)2

d +

2p t Z

4t

;1

+

1 1

'

( ) exp

;

( ; x)2

d =

=

;

x

=

2p t Z

;

4t

2pt

;1

90