dx × dy

(1)

"!$#&%(')'+*

,.-0/013254760896;:<4>=@?BAC:EDGFH2I8J1)4LKBMON36P64Q=6SR6<T0UV602WT;UV6YXY/Z-02C1)4LK[FO:Z:360\]\^6_

`a4b4/dcfeeVghghgi_jKk2WF@_lNZ8mFO:<4n2W6o1p\q_rTs10e

∼

8tK[uO:3F;4b476VeZKvDW47wpxywZz

{}|~YV@~YrHZbIV]I5;~ZIEVjlE

(2)

9 !"$#%&

'(*),+

u

-/.101-/2435.16

+57

6 + 87 +

.19 02:-

7+<;

-/. = 7

6>3 8@?

2 8

219 .16

+

=&.

+A7B)C8D8

.

dx × dy

^E ^8@? ² ^+A7+ ³ ^+57F&8 ^. ^G ⁺ ^0>3

→ ∞

H

•

I 8J(1+

=. K1L

78

. ; -MK

)

-/K

;N8J7

6 + = 7

6O3 8P?

2 8

219 .16

+

.16

x

−

GRQ

(

6:=

; K

+S),TB)U+

2OHVGXW

)

-/.OHVGRYZ-

7 = )

.16

+ H

= − k A ∂u

∂x

= − k (t dy) ∂u

∂x

•

I 8J(1+

=. K1L

78

. ; -MK

)

-/K

;N8J7

6 + = 7

6O3 8P?

2 8

219 .16

+

.16

y

− k A ∂u

∂y = − k (t dx) ∂u

∂y

[\]%&^"5& `_aB ^b#c&

I

8J(1+

K1L 78

. ; -M.B6

+ - 7 6 + = 7

6O3 8@?

2 8

219 .:6

+ d

I

8e(B+

=. K1L

78

. ; -f3

( - +57

6 + g I

8J(1+

=. K1L

78

. ;

-h0.1-/=

;jikiki

GAl>H

(3)

I =&. KBL . - .16 - 6 GR-/.430 3 - 6 H = 6>3 2 219 .:6

d

I

8e(B+

=&. K1L 78

. ; - .16

+ - 7 6 + G - .430

i 3 ( -

+57

6 + H = 7

6O3 87

= )

-/. K + )J(

6 =&.43

g I

8e(1+

=&. K1L 78

. ; - .16

+ - 7 6 +

GR- .430

i 3 ( -

+57

6 + H = 7

6O3 8 7

= )

-/. K + )J(

6 =&.43

•

I 8J(1+

=. K1L

78

. ; -f3

( - +57

6 +

=.

8@?

2 8

219 .16

+ g

0N.1- = ;

− k ( t dy ) h _∂u

∂x + ∂ ² u

∂x ² dx i

− k ( t dx ) h _∂u

∂y + ∂ ² u

∂y ² dy i

+ Q ( dx dy )

I ) 6 78

.:9 .16

+

G

;17+S)J(

6 G5l>H H

k t ∂ ² u

∂x ² + ∂ ² u

∂y ²

!

( dx dy ) = Q ( dx dy )

(!;

∂ ² u

∂x ² + ∂ ² u

∂y ²

!

= Q k t

'%) 8@? (F

. + E K (

6O3

)

=N21- .1-h.43

+

.16 + -

()

3 = ) 9 .:6O3

)J(

6

; 6 =2

T

.

8e(

00.:9 .16

+ 3 ) 9

)b8e7B)

-/. K ( 6 = ; )U+

E

∂ ² u

∂x ² + ∂ ² u

∂y ² + ∂ ² u

∂z ²

!

= Q

kt

(4)

∂ ² u

∂x ² + ∂ ² u

∂y ²

!

= Q k t

∇ ² u = Q k t

7 _ ] & ] CS^D]

•

l i

;:7+S)J(

6 3 7;

=

)

21- .16 K .43

6 7

0&0:-

( ) 9 . 8

.43 =&21- ) T

243 3 . K ( 6 =N.43 0 7 -

87 (

-9

;8

. =.43

=

)

21-/.16 K . 3 K .16

+ - 2 .43 GXK i

K1L 7 0

i

0

7

.$H

7 T . K ) K )

h = ∆x

∂ ² u

∂x ² = u L − 2 u 0 + u R (∆ x ) ²

(

u L

^. ⁺

u R

³ ⁽ ⁶ ⁺ ⁸ ^.43 ⁺ ^.19 ^0N21- ^7+V; ^-/.43E ^$7; ^KBL ^. ^. ⁺ ^E ^=- ^()U+ ^.

= ; 6 ( . ; = = ( 6 + 87 +

.19 0N21-

7+V;

-/. .3

+

u 0

ⁱ ^. ⁹ ¹⁹ ^.

∂ ² u

∂y ² = u A − 2 u 0 + u B (∆y) ²

(

u A

^. ⁺

u B

³ ⁽ ⁶ ⁺ ⁸ ^.43 ⁺ ^.19 ^0N21- ^7+<; ^-/.43 ^.16 ^L ^7; ⁺ ^. ⁺ ^.16 ^F17 ³

= ; 6 ( . ; = = ( 6

+87 +

.19 0N21-

7+V;

-/. .43

+

u 0

ⁱ ⁶ ^01- ^.16 ⁷ ⁶ ⁺

∆ x =

∆ y = h

∇ ² u = uL + uR + uA + uB − 4 u 0

h ²

(5)

'(*),+ ; 6 . 0 87

; . - . K

+57

6

!;N8J7B)

-/. G

20

^KB9

10

^K19 ^H ⁱ ^.43

FN(

-/=B3\3

;

0N21-

) . ; - 3

$7;

K1L .43

FB7

3 3 ( 6 + 9

7B)

6 + .16

; 3 E

0 ^o

^. ⁺ ⁸ ^.

FN(

-/= =-

(j)U+

.43

+ 9

7B)

6 + .16

; E

100 ^o

^iN'(),+ ^=&.43 ⁶ ⁽ ^. ^; ⁼ ³ ^.4340 ⁷ ^K ²⁴³

=.

h = ∆ x = ∆ y = 2 . 5 cm

ⁱ ^'() ⁺

u i

87 +

.19 0N21-

7+<;

-/.

7;

6 ( . ; =

i

ⁱ

'%)

Q = 0

∂ ² u

∂x ² + ∂ ² u

∂y ²

!

= 0

W T . K

∂ ² u

∂x ² = (uL − 2u0 + uR) 0.25 ²

∂ ² u

∂y ² = (uA − 2u0 + uB )

0.25 ²

(6)

(uL + uR + uA + uB − 4u0)

0.125 = 0

∇ ² u = 1 h ²

( ₁ 1 − 4 1

1 )

u 0 = 0

6

(!F + )

.16

+

21

² ^;17+S)J( ^6O3 ⁰ ^(!; ^- ⁸ ⁷ ^- ²⁴³ ⁽⁸ ^; ⁺ ^)J( ⁶ ^=. ^K ^. ^0:- ⁽

F8

9 . .16 .

01-

) 9 7 6 + 8 .

X7B),+

; . 8 .

87

0

87

K )

.16 0

(;

- K1L

7 ; .

6 ( . ; = .43

+ 6

;&8"8

. G

∇ ² u = 0, ∀ u

^H

(!;

- 8 . 6 ( . ; =

1 − 4u 1 + u 2 + u 8 = 0

(!;

- 8 . 6 ( . ; =

7 u 6 − 4u 7 + u 14 = − 100

(!;

- 8 . 6 ( . ; =

9 u 2 + u 8 − 4u 9 + u 10 + u 16 = 0

(;

- 8 . 6 ( . ; =

14 u 7 + u 13 − 4 u 14 + u 21 = − 100

(;

- 8 . 6 ( . ; =

18 u 11 + u 17 − 4u 18 + u 19 = 0

'(*),+

21

^;17+S)J( ^6O3 ⁷ ^T ^. ^K

21

⁾ ⁶ ^K ⁽ ⁶⁶ ^; ³ ⁱ ^. ³ ³ ⁺ ⁹ ^. ^.43 ⁺

3

(8,T 7F8

. 0 7 - 2 8 ) 9 ) 6

7+S)J(

6 Y

7;

3 3 )

.16N6 . G 0 7 - .

.19 0 8

.OH .

+

( 6 + -

(!;OT

.

Q

(!8e(

6N6 .

) 6 . l

) 6 .

l i

i

l l i

i

l

i

`l i

i

`l

i %i

l i l

i l l l

i i l l

l i

`l i l

`i

`l

i ll

i l

i ll

(7)

I

Q

( + K

78

K

;8J7+ ()

-/. 2 8 . T 2

8e(

- 3

; . L .3

+ X7B) F&8

.

I

Q

( +

=&. 3 + ( K

7

. 9 219

(j)

-/. 2 8 . T 2 8e(

- 3

; . L .43

+ X7B) F&8

.

•

i

2 + L (

=N.

),+

21-

7+ )T . 3

u 0 = 1 4

( ₁ 1 0 1

1 )

;

u i,j = (u _i ⁻ _1,j + u _i+1,j + u _i,j ⁻ 1 + u _i,j +1 ) 4

u ^[n+1] _i,j = (u ^[n] _i,j − 1 + u _i,j ^[n] ₊₁ + u ^[n] _i − 1,j + u ^[n] _i+1,j )

4

(8)

•

⁷⁸ . ; - 3

u i,j

) 6 +

21-

) . ; - 3 ) 6

)U+S)78 )

352 .43 E 21-

(

•

⁷⁸ . ; - 3

u _i,j

⁾ ⁶ ⁺ ^21- ⁾ ^. ^; ^- ³ ⁾ ⁶ ^)U+S) ⁷⁸ ⁾³⁵²⁴³ ⁰ ⁷ ^- ⁸ ⁷ ⁹ ⁽ ^4.16N6 ^. ^=.43

T 78

. ; - 3 - ( 6 + )

-/.43

1 S^D] ] 1* 1

I

2 + L (

=. =. 3 ; - -/.

8J7

7+S)J(

6 3 ; K K .43 3 ) T .

u ^[n+1] _i,j = u ^[n] _i,j + w (u ^[ _i,j ⁿ ^] − 1 + u _i,j+1 ^[ ⁿ ^] + u ^[ _i ⁿ − ^] 1,j + u ^[ _i+1,j ⁿ ^] − 4u ^[ _i,j ⁿ ^] ) 4

7 T . K

i = 1 , . . . , 3 j = 1 , . . . , 7

. +

.1-9 .

w

^.3 ⁺ ⁸ ^. ^X7 ^K ⁺ ^. ^; ^- ^=. ³ ^; ^- ^A-/. ^8J7 ⁷⁺ ⁾⁽ ⁶

w

^F*i ⁼ ^? ^),+ ^21- ⁱ

l i

l i l

l i

%l

.19

7 - ; .

7 T:78

. ; - ( 0 + ) 9 ; 9 =.

w

⁶ ^? ^.43 ⁺ ⁰ ⁷ ³ ⁺ ^(; ^(*; ^- ³

.43 +S)

9

7F8

. 9

7B)

3 0

(;

- ; 6 . -/2

*)J(

6 - . K

+A7

6

!;N8J7B)

-/. fK

( 9 9 .

87

0

87

; . K (

6O3

)

=N21-/2 .

w

^F ^/E4+K ^8J1p\ ^d ⁰ ^8D; ³ ^0. ⁺ ^),+ ^. ^- ⁷ ^K ⁾ ⁶ ^. ^=&. ^8@? ² ^;:7+S)J( ⁶

"

cos π

p

+ cos π

q #

w ² − 16 w + 16

(9)

•

2 L =N. 9 0 K . =. = -/. K 6 .1-X6 2 . G WZH

7

9 2 + L (

=N. l i

G

;17+S)J(

6O3 7;

=

)

21- .16 K .43 H =.19

7 6 =.

87

-/23

(!8"; + )J(

6 =

?@;

6 . 9

7+

- ) K . K1-/.

;

35. G ) i . i F

K10 =.

0

^H ⁼ ⁽ ⁶ ⁺ ⁸ ^.43

T 78

. ; - 3 6 ( 6 6

;8

3 3 ( 6 + K ( 6 K .16

+

-/2 .43 3 ; - ; 6 .

F17

6:=N. = ) 7 N(

6

78

. =&.

87

9

7+

- ) K .

H

$(

- 3 ; . 8 7 9

7+

- ) K . .43 ++

- ) = ) 7 (

6

78

. 8

.439 2 + L (

=N.43 =&. -/2

3

(8D; + )J(

6 =&. K . +

0N. =. 9

7+

- ) K . 3 ( 6 + = ) - . K +

.43 . +

. K 7 K .43

I

7

9 2 + L (

=N.

i i W 0N.1-9 . +

=. 3 . - 7 9 .16 .1- E ; 6 01-

(

F8

9 . =&. - 243

(!8"; + )J(

6 =. 9

7+

- ) K . + - ) =

)7 N(

6

78

.

∇ ² u = u L − 2 u 0 + u R

(∆ x ) ² + u A − 2 u 0 + u B

(∆ y ) ² = 0

$(

- 3 ; .

∆x = ∆y = 1

(u L − 2 u 0 + u R) ^[ ^k ^+1] = − (u A − 2 u 0 + u B ) ^[ ^k ^] ( u A − 2 u 0 + u B ) ^[k+2] = − ( u L − 2 u 0 + u R ) ^[k+1]

6 - .

7

--

7 6 . 7 6 + ( 6

(!F + )

.16

+ 8 . 3 3

+

9 . + - ) = ) 7 (

6

78

(10)

− u L ^[k+1] + 2 u 0 ^[k+1] − u R ^[k+1] = h

u A − 2 u 0 + u B i [k]

− u A ^[k+2] + 2u 0 ^[k+2] − u B ^[k+2] = h

u L − 2u 0 + u R i [k+1]

77 ^b] ^ %*D^*X& & ^CC & $] X 1S % $^ &

'%) 87 -

)b8"8

. 6 ?

.43

+ 0 7 3 -/. K

+57

6

*;8e7B)

-/. . + ; . 8

.43

u i,j

^6:.

0N.

;OT

.16

+ 0 7 3 + - . - 2

!;8 )

-/.19 .16

+

.4340

7 K 243

)C8

. ) 3 + . + -

()

3

9 2 + L (

=N.43 0 ( 3 3

)bF8

. 3

•

l i 87

K .B9 .16

+

=.43 6 ( . ;

=B3 3 ; -

87 - ( 6 +S)

-/.

(11)

_∂u

∂x

L, 0 = (u0 − uL) hL

_∂u

∂x

0 ,R = (uR − u0) hR

∂ ² u

∂x ² =

"

∂u

∂x

0 ,R

− ∂u

∂x

L, 0

#

(hL+hR) 2

= 2

(hL + hR)

"

uL

hL − (hL + hR)

(hL × hR) × u0 + uR hR

#

•

i )

3 + ( - 3 )J(

6 =&.

8 7 - ( 6 +S)

-/.

(12)

•

3 6 =.43 K -/= 6N6 2 .43 0 -/.43

∇ ² u = ∂ ² u

∂r ² + 1 r

∂u

∂r + 1 r ²

∂ ² u

∂θ ²

6 + -

(!;OT

.

∇ ² u = ( uL − 2 u 0 + uR )

(∆ r ) ² + ₁ r

×

h ₍ _uR ₋ _uL ₎ (2∆ r )

i

+ 1 r ² ×

h _(uA ₋ _{2u0 +} _uB) (∆θ) ²

i

(13)

A ∂ ² u

∂x ² + B ∂ ² u

∂x∂t + C ∂ ² u

∂t ² + f x, t, u, ∂u

∂x , ∂u

∂t

!

= 0

∆ = B ² − 4AC

•

^'%)

∆ < 0

^? ² ^;B7+ ⁾⁽ ⁶ ⁼ ⁾ ^21- ^.:6 ⁺ ⁾ ^. ^8"8 ^. ^.43 ⁺ ^. ^8D8 ⁾ ⁰ ^+S) ^; ^.

•

^'%)

∆ = 0

^? ² ^;B7+ ⁾⁽ ⁶ ⁼ ⁾ ^21- ^.:6 ⁺ ⁾ ^. ^8"8 ^. ^.43 ⁺ ⁰ ⁷ ^- ^7FN(!8 ⁾ ^; ^.

•

^'%)

∆ > 0

^? ² ^;B7+ ⁾⁽ ⁶ ⁼ ⁾ ^21- ^.:6 ⁺ ⁾ ^. ^8"8 ^. ^.43 ⁺ ^L ^0N.1- ^F(*8 ⁾ ^; ^.

;17+S)J(

6 =&.

8 7 Q L 78

. ; -

k ∇ ² u = cρ ∂u

∂t

;17+S)J(

6 =&.

8 7 K ( - =N.

TB) F - 7 6 + .

∂ ² u

∂t ² = _{T g} w

∇ ² u

(14)

•

l i

2 + L (

=N.

0 8 ) K )U+

.

I

K ( 6O3

) 3 + . E -/.B9 0

8J7

K .1-

8

.43Z=&21- )T

243 0:- .19

)

-/. 3Z.

+ 3 . K ( 6 =N.43

0 7 - 8 . ; - 7

0&01- ( ) 9

7+S)J(

6

∂u

∂t = u ^[j+1] _i − u ^[j] _i

∆t

^G ⁰

+ 3

x _i

^. ⁺ ⁺ ^0O3

t _j

^H

∂ ² u

∂x ² = u ^[j] _i ₊₁ − 2 u ^[j] _i + u ^[j] _i − 1

(∆x) ²

6 ; +S) 8 ) 3 7 6 + K .43

7

0&01- ( ) 9

7+S)J(

6O3

k ∂ ² u

∂x ²

!

= cρ ∂u

∂t H

u ^[j+1] _i = r × (u ^[j] _i ₊₁ + u ^[j] _i ₋ ₁ ) + (1 − 2r) × u ^[j] _i

7 T . K

r = k∆t

cρ(∆x) ²

#%

∂u

∂t

^.43

+ =

?"(

-/=!-/.

o(∆t)

^. ⁺

^∂ _∂x ² ^u 2

^.43

+ = ? (

-/=*-/.

o(∆x) ²

I

3 (!;

-/K . = ?

.1--/.

; - = ?

.43 +S)

9

7+ )(

6

(15)

•

2 L =N. =&. Q - 6 K1L 3 6

I

6 ; +S) 8 )

35.

8

.3 9 19 .43

(

-/=!-/.43 =

?D7

0&0:- ( ) 9

7+ )(

6 0

(;

-

.43 +S)

9 .1-

∂u

∂t

^.

+

∂ ² u

∂x ²

k ∂ ² u

∂x ²

!

= cρ ∂u

∂t H

u ^[ _i ^j ^+1] − u ^[ _i ^j ^]

∆t = 1 2

k cρ

h _u ^[j]

i+1 − 2 u ^[j] _i + u ^[j] _i − 1

(∆x) ² + u ^[j+1] _i+1 − 2 u ^[j+1] _i + u ^[j+1] _i − 1

(∆x) ²

i

Qf.

;)

= (

6N6 . 7

01- 3 - 2

7

-X-

7 6

.:9 .16

+

− ru ^[j+1] _i+1 + (2 + r)u ^[j+1] _i − ru ^[j+1] _i ₋ ₁ = ru ^[j] _i+1 + (2 − r)u ^[j] _i + ru ^[j] _i ₋ ₁

7 T . K

r = k∆t

cρ(∆x) ²

#%

. 0N.

; + + - . -/243

(!8"; ),+

21- 7+ ) T

.19 .16

+

I

² ^K ^.43 ³

)U+

.

87

-/243

(!8"; +S)J(

6 =

?P;

6 3 3

+

9 . + - ) = ) 7 (

6

78

(16)

6 =&. K - =N. - 6 .

∂ ² u

∂t ² = _{T g} w

∇ ² u = _{T g} w

_∂ ² _u

∂x ²

I

6 - .19 0

8J7O7

6 + 8

.43 =21-

) T

243 0 7 - 8 . ; - 7

001- ( ) 9

7+S)J(

6 6 ;

9 21-

) ; . ( 6 + -

(;OT

.

u ^[j] _i ₊₁ − 2 u ^[j] _i + u ^[j] _i − 1

(∆x) ² = w T g

u ^[j+1] _i − 2 u ^[j] _i + u ^[j _i ⁻ ^1]

(∆t) ²

( 8@?C)

6 = ) K . =&243

) 6 . 8

.43

T 78

. ; - 3 =&.

x

^. ⁺ ^8@? ^. ⁰ ⁽ ³ ⁷ ⁶ ⁺ ⁸ ^.3 ^T ⁷

8 . ; - 3 =.

t

ⁱ ⁶ ^- ² ⁷ ^-X- ⁷ ⁶ ^. ⁷ ⁶ ⁺ ⁽ ⁶ ⁺ ^- ^(!;OT ^.

u ^[ _i ^j ^+1] = T g (∆ t ) ²

w (∆x) ² ( u ^[ _[i+1] ^j ^] + u ^[ _i − ^j ^] 1 ) − u ^[ _i ^j ⁻ ^1] + 2

1 − T g (∆ t ) ² w (∆x) ²

u ^[ _i ^j ^]

'%) ( 6 -/.B6 =

T g (∆t) ² /w(∆x) ²

² ^7N8 ^E

1

⁸ ^. ^01- ^(!FN8 ⁹ ^. ³ ^.

3 ) 9 0 8 )

. K (

6O3

)

=2:- 7F8

.19 .:6

+

u ^[j+1] _i = u ^[j] _i ₊₁ + u _i ^[j] ₋ ₁ − u ^[j _i ⁻ ^1] ∆t = ∆x p (T g/w)

G5lOH

#%

? 2

;B7+S)J(

6 0N.

; +

35. -/243

(!;

=!-/.

),+

21- 7+ ) T

.19 .16

+ 3

)(

6 + -

(!;OT

.

;

6:.

7 3 +V;

K . 0

(!;

- .43

+S)

9 .1-

u ^[ ⁻ ^1] _i

^T ⁷⁸ ^. ^; ^- ³ ^=.

u i

;

6:.

),+

21-

7+ )(

6 7 T 7 6 + 8 . =&210

7 - +

(17)

I

3 K 6 =. K - =. - 6 . .43 021- = . 6

; +S) 8 ) 3 7 6 + 8@?D7

0&0:- ( ) 9

7+ )(

6 =&.43 = )

21- .16 K .43 K .16

+

-/2 .43

u ^[1] _i − u ^[ _i ⁻ ^1]

2∆t = ∂u

∂t

^.16

x i

^.

+

t = 0

(∂u/∂t) = 0

^.43 ⁺ ^K ⁽ ⁶⁶ ^; ^GK ⁽ ⁶ ⁼ ^)U+ ^)J( ⁶ ⁾ ⁶ ^)U+ ⁾ ⁷⁸ ^.OH ⁱ

'(),+

g(x) = (∂u/∂t)

^E

t = 0

⁽ ⁶ ^0N. ^; ⁺ ² ^K1- ⁾ ^-/.

u ^[ _i ⁻ ^1] = u ^[1] _i − 2 g ( x ) ∆ t

+

.16 3

;F

3

+S)U+V;17

6 + = 7

6>3 8@?

2

;:7+S)J(

6 G5lOH ( 6 + -

(;OT

. 0

(!;

-

dx × dy

∼

u

dx × dy

→ ∞

•

x

−

= − k A ∂u

∂x

= − k (t dy) ∂u

∂x

•

y

− k A ∂u

∂y = − k (t dx) ∂u

∂y

•

− k ( t dy ) h ∂u

∂x + ∂ 2 u

∂x 2 dx i

− k ( t dx ) h ∂u

∂y + ∂ 2 u

∂y 2 dy i

+ Q ( dx dy )

k t ∂ 2 u

∂x 2 + ∂ 2 u

∂y 2

!

( dx dy ) = Q ( dx dy )

∂ 2 u

∂x 2 + ∂ 2 u

∂y 2

!

= Q k t

∂ 2 u

∂x 2 + ∂ 2 u

∂y 2 + ∂ 2 u

∂z 2

!

= Q

kt

∂ 2 u

∂x 2 + ∂ 2 u

∂y 2

!

= Q k t

∇ 2 u = Q k t

•

h = ∆x

∂ 2 u

∂x 2 = u L − 2 u 0 + u R (∆ x ) 2

u L

u R

u 0

∂ 2 u

∂y 2 = u A − 2 u 0 + u B (∆y) 2

u A

u B

u 0

∆ x =

∆ y = h

∇ 2 u = uL + uR + uA + uB − 4 u 0

h 2

20

10

0 o

100 o

h = ∆ x = ∆ y = 2 . 5 cm

u i

i

Q = 0

∂ 2 u

∂x 2 + ∂ 2 u

∂y 2

!

= 0

∂ 2 u

∂x 2 = (uL − 2u0 + uR) 0.25 2

∂ 2 u

− k ( t dy ) h _∂u

∂x + ∂ ² u

∂x ² dx i

− k ( t dx ) h _∂u

∂y + ∂ ² u

∂y ² dy i

k t ∂ ² u

∂x ² + ∂ ² u

∂y ²

∂ ² u

∂x ² + ∂ ² u

∂y ²

∂ ² u

∂x ² + ∂ ² u

∂y ² + ∂ ² u

∂z ²

∂ ² u

∂x ² + ∂ ² u

∂y ²

∇ ² u = Q k t

∂ ² u

∂x ² = u L − 2 u 0 + u R (∆ x ) ²

∂ ² u

∂y ² = u A − 2 u 0 + u B (∆y) ²

∇ ² u = uL + uR + uA + uB − 4 u 0

h ²

0 ^o

100 ^o

∂ ² u

∂x ² + ∂ ² u

∂y ²

∂ ² u

∂x ² = (uL − 2u0 + uR) 0.25 ²

∂ ² u

∂y ² = (uA − 2u0 + uB )

0.25 ²

∇ ² u = 1 h ²

( ₁ 1 − 4 1

∇ ² u = 0, ∀ u

( ₁ 1 0 1

u i,j = (u _i ⁻ _1,j + u _i+1,j + u _i,j ⁻ 1 + u _i,j +1 ) 4

u ^[n+1] _i,j = (u ^[n] _i,j − 1 + u _i,j ^[n] ₊₁ + u ^[n] _i − 1,j + u ^[n] _i+1,j )

u _i,j

u ^[n+1] _i,j = u ^[n] _i,j + w (u ^[ _i,j ⁿ ^] − 1 + u _i,j+1 ^[ ⁿ ^] + u ^[ _i ⁿ − ^] 1,j + u ^[ _i+1,j ⁿ ^] − 4u ^[ _i,j ⁿ ^] ) 4

w ² − 16 w + 16

∇ ² u = u L − 2 u 0 + u R

(∆ x ) ² + u A − 2 u 0 + u B

(∆ y ) ² = 0

(u L − 2 u 0 + u R) ^[ ^k ^+1] = − (u A − 2 u 0 + u B ) ^[ ^k ^] ( u A − 2 u 0 + u B ) ^[k+2] = − ( u L − 2 u 0 + u R ) ^[k+1]

− u L ^[k+1] + 2 u 0 ^[k+1] − u R ^[k+1] = h

− u A ^[k+2] + 2u 0 ^[k+2] − u B ^[k+2] = h

_∂u

_∂u

∂ ² u

∂x ² =