УрМатФиз / УрМатФиз с теорией

x + v x = 0

x 2 D:

3:11

3.9 3.8 ,

T t

@v

+ v

!

= 0 ,

@v + v

!

= 0:

3:12

@n

@D

@n

@D

6 , " 4 # 3.9 $":

un x t = vn x Tn t = Ane na2tvn x :

3:13

1

1

Ane na2tvn x

u x t =

X

Tn t vn x =

X

3:14

n=1

n=1

1

g x =

X

An vn x :

3:15

1

g x vk x dx = nX=1 An ZZZ vn x vk x dx = Ak k =

: 3:16

D

1 1

D

& 3.8 .

7 ) ! " 3.6 , 3.8, 3.2

"-

& & 2 * ' ( & vn x n = 1 1

1

u x t = X un t vn x

3:17

n=1

1

f x t = X fn t vn x

3:18

n=1

1

g x = X gn vn x

3:19

n=1

fn t = ZZZ

f x t vn x dx

3:20

D

gn = ZZZ

g x vn x dx n =

:

3:21

1 1

D

1 , &

3.17

'' ( )

" t " x. 1

3.17 , 3.18 3.6

dun

vn x a2 4vn x un t fn t vn x = 0:

1

dt

:

n1 dudtn + a2 n un t

fn t vn x = 0

3:22

X

=1

, 3.11 4vn x =

nvn x :

dun t + a2 n un t = fn t n =

3:23

1 n:

dt

* 3.17 3.19 3.2

1

X

fun0

gng vn x = 0

un0 = gn n =

3:24

1 n:

n=1

. /

0 / 1 un t 3.23 , 3.24 , -

, ( :

t

un t = Z e a2 n t

fn d + gne a2 nt:

3:25

0

t

u x t = Z

ZZZ

f y G x y: t dy d + ZZZ g y G x y: t 0 dy

3:26

0

D

D

6

1

e a2

n t vn y vn x

G x y: t =

X

3:27

n=1

@u

+ u

!

= x t x 2 @D

j j + j j 6= 0 3:28

@n

@D

(-'

f~ x t 0 (

g x 0 -

&

u x t =

@

t v x t x d = Zt @v x t

x d =

@t

@t

t:

@n@u + u!

@D= 1

j j + j j 6= :0

3.1.

@u

2 @2u

@t

= a @x2

D

= fx : 0 x =4g 0 t

:1:1

@D

@u

u x= =4= 0 0 t

@x x=0= 0

:1:2

(

u

= g x = 2 cos 2x

3 cos 6x

0 x =4:

:1:3

t=0

.

-

! "

u x t = X x T t 60:

:1:4

#

$ %

Xx

T 0 t = a2T t X x

,

T t

=

X x

= :

a2T t

X x

&

% &'(

T t + a2T t = 0

:1:5

X x + X x = 0:

:1:6

#

"

%-

T t X

= 0 X

= 0

:1:7

T t X =4 = 0 X =4 = 0:

, $ -|/

%

% .

1

0 . 0 $

! 0-

1 23 % l = =4

!

n = 4 2n + 12

Xn x = cos

n + 1x

n =

:

0 1

7 % 8 &'(

% = n:

3.1.1 | 3.1.3

-

!

1

na2t cos 2 2n + 1x

u x t =

un x t =

Ane

3:1:8

0

n=0

& & !, (

& & -

t

& & x.

)

3.1.8 3.1.3 &

1

g x =

An cos 2 2n + 1x :

3:1:9

=4

Z cos 2 2n + 1x cos 2 2k + 1x dx = nk:

0

8

5 (

3.1.9 cos 2 2k + 1x, &

-

& x 00 =43 &

=4

1

=4

Z g x cos 2 2k+1x dx= X An Z cos 2 2n+1x cos 2 2k+1x dx = Ak

:

0

n=0

0

8

6 .

= 8

=4

An

Z g x cos 2 2n + 1x dx:

3:1:10

0

) , , / 3.1.8 , & -

! .

7 ! g x 3.1.3 , & !.8

, / - 3.1.10 , & ! ..

) & 3.1.8 3.1.3 & 3.1.9

1

2 cos 2x

3 cos 6x = X An cos 2 2n + 1x :

3:1:11

n=0

-

3.1.11 ,

A0 = 2

A1 = 3 An = 0

n 6= 0 1:

3:1:12

" # 3.1.12

3.1.8 ,

% -

& .

. u x t = 2e

4a2t cos 2x 3e

36a2t cos 6x:

G x y+ t = 8

e a24 2n+1 2 t cos 2 2n + 1x cos 2 2n + 1y

n=0

3.26

=4

u x t = Z

g y G x y$ t 0 dy:

0

3.1.1.

& -

@2u

@u

= a2

@t

@x2

1.

u x=0= u x= = 0

u t=0= 3 sin 2x sin 3x:

2.

u x=0= ux x= =2= 0

u t=0= 2 sin x

3 sin 5x:

3.

ux x=0= u x= =2= 0

u t=0= 2 cos x + 3 cos 3x:

4.

ux x=0= ux x= = 0

u t=0= 3 + cos x

5 cos 2x:

5.

u x=0= u x=2 = 0

u t=0= 2 sinx=2

sin x:

6.

u x=0= ux x= = 0

u t=0= 3 sinx=2

sin 3x=2:

28.

ux x=0= ux x= = 0

u t=0= 2 cos x

cos 3x:

29.

u x=0= u x=2 = 0

u t=0= sin x 3 sin 5x:

30.

u x=0= ux x= = 0

u t=0= sin 3x=2

sin 5x=2:

@u

= a

2

@2u

+ f x t D = fx : 0 x =4g

0 t

3:1:13

@t

@x2

$ () " () $!)

@u

@x x=0= 0

u x= =4= 0

0 t

3:1:14

() $)

u t=0= g x = 2 cos 2x 3 cos 6x 0 x =4

3:1:15

" * + !

f x t = cos 2t cos 10x:

3:1:16

4 2n + 12

3:1:17

Xn x = cos 2 2n + 1x n = 0 1:

1

u x t =

un t Xn x = un t cos 2 2n + 1x

3:1:18

0

n=0