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^{; ? +,*:)02.1',* &) -. ;5.*1'1% &)} 9,52)9)*1<A #

divu = 0

+,*&'1',* &!'*+,930)::'/'-'1% ;

?+,*:)02.1',*&) -. 9.::)< ,

N = 1

^A

# %H.8.

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^A#

%0JA .0

.; AA<

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f (x)

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u(x)

^{;< ,}

; D<

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∆ (∆u) = f Ω

u = 0

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∂n = 0

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; <

R

∂u

∂n = ∇ u · n

^.

n

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x t

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∆ x j n ∆ t

W7 ()

* 6A 3.: &!):3.+)

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, ;.2<

(t _n , x _j ) = (n∆t, j∆x)

n ≥ 0, j ∈ Z.

@

u ⁿ _j

^{. ,}

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u(t, x)

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(∆x) ²

^{; "<}

R,P8 5#

− u(t, x − ∆x) + 2u(t, x) − u(t, x + ∆x) = − (∆x) ² ∂ ² u

∂x ² (t, x)

− (∆x) ⁴ 12

∂ ⁴ u

∂x ⁴ (t, x) + O (∆x) ⁶

^{; $<}

∆x

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,?8 H . @

8 ( ) J A 9

8LM

9 A ()

∂u

∂t (t _n , x _j ) ≈ u ⁿ⁺¹ _j − u ⁿ⁻¹ _j 2∆t

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n

j

^;

:+4%9. &) '+4.0&:,*<

u ⁿ⁺¹ _j − u ⁿ⁻¹ _j

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(∆x) ² = 0.

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, 2 "<V / I)

? *,A6

:. (

) .

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u ⁿ _j − u ⁿ⁻¹ _j

∆t + V u ⁿ _j+1 − u ⁿ _j−1

2∆x + ν −u ⁿ _j−1 + 2u ⁿ _j − u ⁿ _j+1

(∆x) ² = 0.

^{; <}

# J ()

.;. Y :+4%9.&!5-)0 30,(0)::'=<

∂u

∂t (t _n , x _j ) ≈ u ⁿ⁺¹ _j − u ⁿ _j

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u ⁿ⁺¹ _j − u ⁿ _j

∆t + V u ⁿ _j+1 − u ⁿ _j−1

2∆x + ν − u ⁿ _j−1 + 2u ⁿ _j − u ⁿ _j+1

(∆x) ² = 0.

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3-'+'1)8 # , .

(u ⁿ _j ) _j∈Z

^{8 .}

(u ⁿ⁻¹ _j ) _j∈Z

^{A ;< )>}

3-'+'1), .

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⁸

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,

n

^{; < ;< , .}

,, .;< 8

u ⁿ _j − u ⁿ⁻¹ _j

∆t + V u ⁿ⁻¹ _j+1 − u ⁿ⁻¹ _j−1

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1 . )A # 0 G

n

^{J .}

(u ⁰ _j ) _j∈Z

^)A

A

u ⁰ _j = u ₀ (j∆x)

^R

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: J

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.

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" #

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V = 0

ν = 1

^{A,6?6 -!,* 0%:,51}*59%0';5)9)*1-!%;5.1',*&) -. +4.-)50C

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^{A ,6?6 A}

n ≥ 0

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^{A ,}

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R

L M

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¹

/ . ; ) 6

6 < / ) , ?

∆x = 0.05

^{J #}

.

(u ⁿ _j ) −200≤j≤+200

^?+.

u ⁿ _j

^{, , ,}

. J A A

, 0

( 0. )

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^A'*:1./-)A,6

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∆x

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0I)

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3 , A ; < L0M

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^@

0

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⁶

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^;<

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+,*&'1',* ,5 +,*&'1',*&) ,50.*1H0')&0'+4:H )1)IB

#

&W .

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^;<

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^;0<

/I)0. 0;#6

(<+.8

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(∆x) ² u ⁿ _j−1 +

1 − 2 ν∆t (∆x) ²

u ⁿ _j + ν∆t

(∆x) ² u ⁿ _j+1 .

^{; <}

'2 .)A ; <

u ⁿ⁺¹ _j

⁰

. .

u ⁿ _j−1 , u ⁿ _j , u ⁿ _j+1

^{; :}

0 ; <8 .< 9 A

u ₀

⁰

m

M

m ≤ u ⁰ _j ≤ M

j ∈ Z,

8 H .

m ≤ u ⁿ _j ≤ M

j ∈ Z

n ≥ 0.

^;<

;<H , 0J

0 '2 ;<

J , ,.&':+0)1 +,*1'*5

..?+

'2.)A,6?6

2ν ∆t > (∆x) ² .

3A 0;H0

LM J A

u ₀ ≡ 0

^{V< *}

)

u ⁰ _j = (−1) ^j

080

u ⁿ _j = ( − 1) ^j

1 − 4 ν∆t (∆x) ²

_n

.,)

n

^.,)

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0 '2,8

>)0+'+) CC % 2/(

V = 0

⁽

6% ($ + 2/

1 ≤ j ≤ J

u ⁿ ₀ = u ⁿ _J+1 = 0

n ∈ N

⁾

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m ≤ u ⁰ _j ≤ M

1 ≤ j ≤ J

^$

u ⁿ⁺¹ _j

u ⁿ _j

^"

n ≥ 0

*

m ≤ u ⁿ _j ≤ M

1 ≤ j ≤ J

∆t ∆x

. ? 0 A

,. .A,6?6 ?6

/ ?

+0A0GA

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^..

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.0;..) 6

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^{; .}

0.5

^A

0.1

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0.05

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t = 1

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9( ,-!%;5.1',*&) +,*

2)+1',*&'A5:',* ; &< . .

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^/

H .

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^{/ ,D . (6}

ν

^{; . 0 *< 0 2}

0

ν = 1

^A0

ν = 0.01

^A

.

ν = 0.1

^A00

( @ 0 ,.

0 *

ν

^{A .8}

(8 * A '2 ;<A 0 .

V

#

W , .6 ( .

ν∆t =

0.4(∆x) ²

V = 1

^{9 A}

ν = 1

^A

ν = 0.1

^A0

ν = 0.01

A .08?

ν

* A -!%;5.1',* &!.&2)+1',* ,06

?

ν = 0

^{+ ,0 '2 ;<}

8

ν = 0

^;

∆t

∆x

^{<A 0 6}

. 0 2

#

W , , . .

∆t = 0.9∆x

^A

V = 1

^A

ν = 0

* , , . ;,6?6 ; &< .

ν = 0

^<A

;<

u ⁿ⁺¹ _j = V ∆t

2∆x u ⁿ _j−1 + u ⁿ _j − V ∆t

2∆x u ⁿ _j+1 .

^;<

' 2 H 6

0 2 @ . 0

u ⁿ⁺¹ _j

^{, I ;}

∆t

^{< 0 .}

u ⁿ _j−1

^A

u ⁿ _j

^A

u ⁿ _j+1

^{# .}

A 6

0; . <

, 0A ;<A.

.8 /.

,.8 .10J

. . .

V

^0GJ

V > 0

^{;# .0}

V < 0

^<*

V > 0

^{A ?} &%+)*10)9)*1.2.-J0

V ∂u

∂x (t _n , x _j ) ≈ V u ⁿ _j+1 − u ⁿ _j

∆x

L,8M .' ?

.L M

u ⁿ⁺¹ _j − u ⁿ _j

∆t + V u ⁿ _j+1 − u ⁿ _j

∆x = 0

^;"<

0 3 &%+)*10)9)*1

.9,*1;,6?6 ?

V > 0

^<A.L,8M

V ∂u

∂x (t _n , x _j ) ≈ V u ⁿ _j − u ⁿ _j−1

∆x

u ⁿ⁺¹ _j − u ⁿ _j

∆t + V u ⁿ _j − u ⁿ _j−1

∆x = 0

^;$<

2@.);$<

W , , . .

∆t = 0.9∆x

^A

V = 1

0 . '2 ; ( '2

;<<

|V |∆t ≤ ∆x.

^;&<

9(A;$<8

u ⁿ⁺¹ _j = V ∆t

∆x u ⁿ _j−1 +

1 − V ∆t

∆x

u ⁿ _j ,

A ;&<8A

u ⁿ⁺¹ _j

^0.

u ⁿ _j−1

u ⁿ _j

^{* A ;$< .)}

AP0 , &%+)*10)

9)*1 .9,*1 ):1 5*) .510) '&%) 9.<)50) &) -!.*.-B:) *59%0';5) 9

0 D R8

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.<AP 0 ,

, , .

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ν

^A80

.,.8. '2;&<X

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$ $

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∆t ≤ ∆x

0 . D

5 ,0 A L0M

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;. < #

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"$ $ "$&" """ $"+ $

" ),

*A; <03A,

(

_dy

dt = f (t, y)

0 < t < T

y(t = 0) = y ₀

^;-<

, 0 ,

(0, T )

^{A .}

0 < T ≤ + ∞

^{A, L0 M ,}

t = 0

^;

t = T

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" .' ) $ $ -/ 0$$" ")

t

^{' $ "$ 1}

/ 2 $" $ "$ $" " $ )"$" $ /

t = 0

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t = T

^,

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x

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10 ?8 0 0

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t

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x

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f

^{; 0A}

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u

^A

A

^L,M

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^{, 0A}

A

^{? 8 ,}

. # 0

.

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A(u) = f

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f

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u

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u

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J,0 ? ?

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Ω = (0, 1) ×(0, 2π)

0%:

x

y

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

− ∂ ² u

∂x ² − ∂ ² u

∂y ² = 0

Ω

u(x, 0) = u(x, 2π) = 0

0 < x < 1 u(0, y) = 0, ∂u

∂x (0, y) = −e ^{− √ n} sin(ny)

0 < y < 2π

u(x, y) = ^{e −}

√ n

n sin(ny)sh(nx)

^"

x = 0

^{* ;( (}

x > 0

⁽

u(x, y)

n

$0

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t

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x

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u(x, y)

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a ∂ ² u

∂x ² + b ∂ ² u

∂x∂y + c ∂ ² u

∂y ² + d ∂u

∂x + e ∂u

∂y + f u = g.

^{# !}

"

a, b, c, d, e, f

b ² − 4ac < 0

b ² − 4ac = 0

b ² − 4ac > 0

%''-""

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ax ² + bxy + cy ² + dx + ey + f = 0

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b ² − 4ac < 0

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