Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

m = M − 1

˜

˜

yM = LM yM−1 + KM ;

˜

˜

yM =

+ KM

.

˜

1 − LM LM−1

˜

˜

LM

KM

β1KM−1 + hγ2(tn+1)

yM =

.

L0 K0

Lm Km

δumn+1 − δumn

−

a2 δumn

+1 − 2δumn + δumn −1

= 0.

h2

δumn+1 − δumn

n+1

n+1

= 0.

a2 δum+1 − 2δumn+1 + δum−1

−

h2

λneiwh

λ − 1

a2

eiwh

− 2 + e−iwh

= 0.

τ

h2

eiwh +e−iwh = 2 cos wh2

1−cos wh = 2 sin2 wh/2

τ a2

wh

λ = 1 − 4

sin2

.

h2

2

τ /h2

λ 1

sin2 wh/2 = 1

r

1

.

2a

2

t x

τ

−

h2

λ − 1

a2

= 0.

.

λ =

1

|λ| 1

1 + 4ra2 sin2 wh/2

r

0.0001

ORIGION := 0

a := 1

A :=

−

3

B := 7

n := 0..N

m := 0..M

h :=

B − A

M

t0 := 0

t4 := 5

otherwise

100

3

Ψ(x)

if

(τ

·

x)2

2

·

a

·

√π

·

t 1

·

·

u(xm, tn)

U <0>

100

U <1>

80

U <2>

60

U <3>

40

U <4>

20

U <5>

0

-1

1

3

5

7

-3

x

A

B

0x

|

| = |

|

| = x1

|

1 + ux2dx.

M M1

x

"

¯

¯

| = T

|T

| = |T1

− sin α) = T (tg α1 − tg α).

∂2u

F1 = T

dx.

∂x2

0u

F2 = g(x, t)|

M M1

| = g(x, t)|

| = g(x, t)dx.

M M1

dm = ρ|M M1| = ρdx

utt(x, t)

0u

∂2u

∂2u

∂2u

dm

= F1 + F2

ρdx

= T

dx + g(x, t)dx.

2

∂t

2

∂x

2

∂t

ρdx

∂2u

T ∂2u

g(x, t)

=

+

,

2

2

ρ

∂t

ρ ∂t

∂2u

= a

2 ∂2u

+ F (x, t).

2

∂x

2

∂t

g(x, t) = 0 F (x, t) = 0

∂2u

2 ∂2u

= a

.

2

∂x

2

∂t

u

|t=0

(x, t)

= ψ(x)

ϕ(x) ψ(x)

t

x

u(x, t)

0

x

x1=x+dx

l

∂2u

= ES

∂2u

∂2u

= a2

∂2u

ρSdx

dx,

,

2

2

2

2

∂t

∂x

∂t

∂x

x = l ESux(l, t) = 0

x = l

Q

∂u

Q