Inženjerska matematika III

7/23/2019 Inženjerska matematika III

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UNIVERZITET U SARAJEVUGRAĐEVINSKI FAKULTETODSJEK ZA GEODEZIJU

PREDMET: INŽINJERSKA MATEMATIKA 3

ZADAĆA 12

Sarajevo 1!"1" Zora# Pe$rov%&



Zadatak 1:Me$o'o( )o#a*#%+ ra,-%)a r%je.%$% r/0#% ro0-e(

Δu=−3,

u x , y =0,

x , y ∈∂ Ω ,

Ω={ x , y ∣ x2 y

21} ,

h=1

3

Rješenje: za čvor i , j : u i−1, jui 1, ju i , j−1ui , j 1−4⋅ui , j =h2⋅ f

h2⋅ f =

1

32

⋅−3=−1

3

Oblast je simetrična , pa je dovoljno računati samo za prvi kvadrant.

0,0:

u1,0u0,1u0,−1u−1,0−4⋅u0,0=−1

3

−4⋅u0,02⋅u0,12⋅u1,0=−1

3

=−4⋅u0,04⋅u0,1

0,1

u0,2u0,0u1,1u−1,1−4⋅u0,1=−1

3

u0,0−4⋅u0,1u0,22⋅u1,1=−1

3

0,2

u0,3u0,1u1,2u−1,2−4⋅u0,2=−1

3

0u0,1−4⋅u0,22⋅u1,2=−1

3

1,1

u1,0u1,2u0,1u2,1−4⋅u1,1=−1

3

2⋅u0,1−4⋅u1,12⋅u1,2=−1

3

1,2

u1,1u1,3u0,2u2,2−4⋅u1,2=−13

u0,2u1,1−4⋅u1,2=−1

3

u0,0=3

4 u0,1=

2

3 u0,2=

5

12 u1,1=

7

12 u1,2=

1

3

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1



Zadatak 2:Da$% / %#$e4ra-%:

A= ∫1

2

e− x

2

dx

B= ∫

6

4

x⋅tan x dx

a!5 Kor%$e&% o.$e $rae,#o % o.$e S%(o#ovo rav%-o %,ra*/#a$% %#$e4ra-e A % 6 a $a*#o.&/78"!""""9!0!5 I,ra*/#a$% %#$e4ra-e A % 6 o(o&/ Ga/ov%+ )va'ra$/r#%+ or(/-a; re'a % 3!

Rješenje:a!5 %! O.$e $rae,#o rav%-o:

Integral A:

a=1, b=2

f x=e− x

2

f ! ! x =4⋅ x2⋅e

− x2

−2⋅e− x

2

" 2=∣ f ! !

ab

2 ∣=0.73779457193305

n= b−a3⋅ " 2

12⋅# =35.066

n≈36

h=b−a

n =

1

36

A1=h

2⋅ f x0 f x362⋅∑

i =1

35

f x i

x i= x0i⋅h , i=0...n

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a



A1≈0,135300

n=72

h=b−a

n =

1

72

A2=h

2⋅ f x0 f x722⋅∑i=1

71

f x i


i

0 1, 000000 0, 367879

1 1, 027778 0, 347731

2 1, 055556 0, 328179

3 1, 083333 0, 309248

4 1, 111111 0, 290960

5 1, 138889 0, 273332

6 1, 166667 0, 256376

7 1, 194444 0, 240101

8 1, 222222 0, 224512

9 1, 250000 0, 209611

10 1,277778 0,195398

11 1,305556 0,181868

12 1,333333 0,169013

13 1,361111 0,156825

14 1,388889 0,145292

15 1,416667 0,134399

16 1,444444 0,124131

17 1,472222 0,114470

18 1,500000 0,105399

19 1,527778 0,096897

20 1,555556 0,088944

21 1,583333 0,081517

22 1,611111 0,074595

23 1,638889 0,068156

24 1,666667 0,062177

25 1,694444 0,056634

26 1,722222 0,051506

27 1,750000 0,046771

28 1,777778 0,042405

29 1,805556 0,038387

30 1,833333 0,034697

31 1,861111 0,031313

32 1,888889 0,02821533 1,916667 0,025385

34 1,944444 0,022803

35 1,972222 0,020452

36 2,000000 0,018316

xi

f(xi)



A2≈0,135268

∣ A1− A2∣# A≈0,1353

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a <

i

0 1,000000 0, 367879

1 1,013889 0, 357732

2 1,027778 0, 347731

3 1,041667 0, 337878

4 1,055556 0, 328179

5 1,069444 0, 318634

6 1,083333 0, 309248

7 1,097222 0, 300023

8 1,111111 0, 290960

9 1,125000 0, 282063

10 1, 138889 0, 273332

11 1, 152778 0, 264769

12 1, 166667 0, 256376

13 1, 180556 0, 248153

14 1, 194444 0, 240101

15 1, 208333 0, 232220

16 1, 222222 0, 224512

17 1, 236111 0, 216976

18 1, 250000 0, 209611

19 1, 263889 0, 202419

20 1, 277778 0, 195398

21 1, 291667 0, 188548

22 1, 305556 0, 181868

23 1, 319444 0, 175357

24 1, 333333 0, 169013

25 1, 347222 0, 162837

26 1, 361111 0, 156825

27 1, 375000 0, 150977

28 1, 388889 0, 145292

29 1, 402778 0, 139766

30 1, 416667 0, 134399

31 1, 430556 0, 129188

32 1, 444444 0, 124131

33 1, 458333 0, 119226

34 1, 472222 0, 114470

35 1, 486111 0, 109862

36 1, 500000 0, 105399

37 1, 513889 0, 101078

38 1, 527778 0, 096897

39 1, 541667 0, 092853

40 1, 555556 0, 088944

41 1, 569444 0, 085166

42 1, 583333 0, 081517

43 1, 597222 0, 077994

44 1, 611111 0, 074595

45 1, 625000 0, 07131746 1, 638889 0, 068156

47 1, 652778 0, 065110

48 1, 666667 0, 062177

49 1, 680556 0, 059352

50 1, 694444 0, 056634

51 1, 708333 0, 054020

52 1, 722222 0, 051506

53 1, 736111 0, 049091

54 1, 750000 0, 046771

55 1, 763889 0, 044543

56 1, 777778 0, 042405

57 1, 791667 0, 040354

58 1, 805556 0, 038387

59 1, 819444 0, 036502

60 1, 833333 0, 034697

61 1, 847222 0, 03296862 1, 861111 0, 031313

63 1, 875000 0, 029729

64 1, 888889 0, 028215

65 1, 902778 0, 026768

66 1, 916667 0, 025385

67 1, 930556 0, 024064

68 1, 944444 0, 022803

69 1, 958333 0, 021600

70 1, 972222 0, 020452

71 1, 986111 0, 019358

72 2, 000000 0, 018316

xi

f(xi)



Integral B:

a=

6, b=

4

f x= x⋅tan x

f ! ! x=2⋅cos x x⋅sin x

cos x 3

" 2=∣ f ! ! ab

2 ∣=4.77340836264294

n=

b−a

3

⋅ " 2

12⋅# =11.94790084035262

n≈12

h=b−a

n =0.021911747619712

B=h

2⋅ f x0 f x12 2⋅∑

i =1

11

f x i

xi= x0i⋅h , i=0...n

B1≈0,135138

n=24

h=b−a

n =0,010908

B2=h

2⋅ f x0 f x242⋅∑

i=1

23

f xi


i0 0,523599 0, 302300

1 0,545415 0, 330966

2 0,567232 0, 361367

3 0,589049 0, 393590

4 0,610865 0, 427732

5 0,632682 0, 463902

6 0,654498 0, 502214

7 0,676315 0, 542800

8 0,698132 0, 585802

9 0,719948 0, 631378

10 0, 741765 0, 679702

11 0, 763582 0, 730970

12 0, 785398 0, 785398

x i f(xi)



B2≈0,135100

∣ B1− B2∣#

B≈0,1351

a!5 %%! S%(o#ovo rav%-o: Integral A:

a=1, b=2

f x=e− x

2

f ! ! ! ! x =16⋅ x4⋅e

− x2

−48⋅ x2⋅e

− x2

12⋅e− x

2

" 4=∣ f ! ! ! ! ab

2 ∣=1.580988368427965

n=b−a⋅4

b−a⋅ "

4

180⋅# =3.640588067797857

n≈4

h=b−a

n =

1

4

A1=

h

3⋅ f x

0 f x

42⋅∑

i=1

1

f x2⋅i4⋅∑

i=1

2

f x2⋅i−1

x i= x0i⋅h , i=0...n

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a =

i

0 0,523599 0,302300

1 0,534507 0,316421

2 0,545415 0,330966

3 0,556324 0,345944

4 0,567232 0,361367

5 0,578140 0,377244

6 0,589049 0,393590

7 0,599957 0,410415

8 0,610865 0,427732

9 0,621774 0,445557

10 0,632682 0,463902

11 0,643590 0,482782

12 0,654498 0,502214

13 0,665407 0,522215

14 0,676315 0,542800

15 0,687223 0,563990

16 0,698132 0,585802

17 0,709040 0,608258

18 0,719948 0,631378

19 0,730857 0,655185

20 0,741765 0,679702

21 0,752673 0,704955

22 0,763582 0,730970

23 0,774490 0,757775

24 0,785398 0,785398

xi

f(xi)

i

0 1, 000000 0, 367879

1 1, 250000 0, 209611

2 1, 500000 0, 105399

3 1, 750000 0, 046771

4 2, 000000 0, 018316

xi

f(xi)



A1≈0,135210

n=8

h=b−a

n =

1

8

A2=h

3⋅ f x0 f x82⋅∑

i=1

3

f x2⋅i4⋅∑i=1

4

f x2⋅i−1

A2≈0,135254

∣ A1− A2∣#

A≈0,1352

Integral B :

a=

6 , b=

4

f x = x⋅tan x

f ! ! ! ! x=8⋅ x⋅ se$ x 2⋅tan x 316⋅ se$ x 2⋅tan x

216⋅ x⋅ se$ x4⋅tan x8⋅ se$ x 4

" 4=∣ f ! ! ! ! ab

2 ∣=59.20340303524299

n=b−a ⋅4

b−a⋅ "

4

180⋅# =1.686501856759905

n≈2

h=b−a

n =0.15523219660257

B1=

h

3⋅ f x0 f x24⋅∑

i=1

1

f x2⋅i−1

x i= x0i⋅h , i=0...n

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a >

i

0 1, 000000 0, 367879

1 1, 125000 0, 282063

2 1, 250000 0, 209611

3 1, 375000 0, 150977

4 1, 500000 0, 105399

5 1, 625000 0, 071317

6 1, 750000 0, 046771

7 1, 875000 0, 029729

8 2, 000000 0, 018316

xi

f(xi)

i

0 0, 523599 0, 302300

1 0, 654498 0, 502214

2 0, 785398 0, 785398

xi

f(xi)



B1≈0,135113

n=4

h=b−a

n =0,130900

B2=h

3⋅ f x0 f x42⋅∑

i =1

1

f x2⋅i4⋅∑i=1

2

f x2⋅i−1

B2≈0,135088

∣ B1− B2∣#

B≈0,13510!5 Ga/ove )va'ra$/r#e or(/-e:

Integral A:

a=1, b=2

f x =e− x

2

t i =b−a

2 ⋅ xi

ba

2 , i=1...n

kvadraturna formula reda 2n=2:

x1=−1

3

, t 1=1.211324865405187

x2= 1

3 , t 2=1.7886751345948137

A=b−a

2 ⋅ f t 1 f t 2≈0.1357

kvadraturna formula reda3n=3:

x1=− 3

5 , t 1=1.112701665379258

x2=0, t 2=1.5

x3=

35

, t 3=1.887298334620742

A=b−a

2 ⋅

5

9⋅ f t 1

8

9⋅ f t 2

5

9⋅ f t 3≈0.1353

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a ?

i

0 0,523599 0,302300

1 0,589049 0,393590

2 0,654498 0,502214

3 0,719948 0,631378

4 0,785398 0,785398

xi

f(xi)



Integral B:

a=

6 , b=

4

f x= x⋅tan x

t i=b−a

2 ⋅ xi

ba

2 , i=1...n

kvadraturna formula reda2 n=2:

x1=

−1

3 , t 1=0.57892349598811

x2= 1

3 , t 2=0.73007344300763

B=b−a

2 ⋅ f t 1 f t 2≈0.1351

kvadraturna formula reda3 n=3 :

x1=− 3

5 , t 1=0.55310400259853

x2=0, t 2=0.65449846949787

x3= 3

5 , t 3=0.75589293639721

B=b−a

2 ⋅

5

9⋅ f t 1

8

9⋅ f t 2

5

9⋅ f t 3≈0.1351

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a @



Zadatak 3:Kor%$e&% o.$e S%(o#ovo % o.$e $rae,#o rav%-o; %,ra*/#a$% o0%( e-%e:

x2

2

y2

3 =1

a $a*#o.&/ 78"!""9!Rješenje:

x=a⋅cost , x= 2⋅cos t , ˙ x=− 2⋅sin t y=b⋅sin t , y= 3⋅sin t , ˙ y= 3⋅cost

l =∫0

2

˙ x2 ˙ y

2dt =∫

0

2

2⋅sin t 23⋅cos t

2dt

O=4⋅l

S%(o#ovo rav%-o:

a=0, b=

2

f x = 2⋅sint 23⋅cost 2

f ! ! ! ! x=

4⋅sin x2

2⋅sin x23⋅cos x 2

4⋅cos x2

2⋅sin x23⋅cos x2 −

3⋅sin x4


2

22⋅cos x2⋅sin x 2

2⋅sin x 23⋅cos x 23

2−

3⋅cos x 4


2

18⋅cos x 2⋅sin x

4

2⋅sin x 23⋅cos x 25

2−

18⋅cos x 4⋅sin x 2

2⋅sin x 23⋅cos x25

2−

15⋅cos x 4⋅sin x4


2

" 4=∣ f ! ! ! ! ab

2 ∣=0.97398151933186

n=b−a⋅4

b−a ⋅ " 4

180⋅# =1.793605380472007

n≈2

h=b−a

n =0.78539816339745

l 1=h

3⋅ f x0 f x 24⋅

∑i=1

1

f x2⋅i −

1

xi = x0i⋅h , i=0...n

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1"

i

0 0, 000000 1, 732051

1 0, 785398 1, 581139

2 1, 570796 1, 414214

xi

f(xi)



l 1≈2,479455

n=4

h=b−a

n =0,392699

l 2=h

3⋅ f x0 f x42⋅∑

i=1

1

f x2⋅i 4⋅∑i=1

2

f x2⋅i−1

l 2≈2,477385

∣l 1−l 2∣#

l ≈2,48

O=4⋅l =9,92

O.$e $rae,#o rav%-o:

a=0, b=

2 f x = 2⋅sint 23⋅cost 2

f ! ! x = sin x 2

2⋅sin x 23⋅cos x 2−

cos x 2

2⋅sin x 23⋅cos x 2−

cos x 2⋅sin x 2

2⋅sin x 23⋅cos x23

2

" 2=∣ f ! ! ab

2 ∣=0.063245553203368

n= b−a

3⋅ " 2

12⋅# =2.021246068550205

n≈3

h=b−a

n =

6

l 1=h

2⋅ f x0 f x32⋅∑

i =1

2

f x i

xi = x0i⋅h , i=0...n

l 1≈2.477379

n=6

h=b−a

n =0.523599

l 2=

h

2⋅ f x0 f x6∑

i=1

5

f x i


i

0 0,000000 1,732051

1 0,392699 1,689246

2 0,785398 1,581139

3 1,178097 1,465076

4 1,570796 1,414214

xi

f(xi)

i

0 0, 000000 1, 732051

1 0, 523599 1, 6583122 1, 047198 1, 500000

3 1, 570796 1, 414214

xi

f(xi)



l 2≈2.477379

∣l 1−l 2∣#l ≈2,48

O=4⋅l =9,92


i

0 0,000000 1,732051

1 0,261799 1,712604

2 0,523599 1,658312

3 0,785398 1,581139

4 1,047198 1,500000

5 1,308997 1,437702

6 1,570796 1,414214

xi

f(xi)



Zadatak 4:R%je.%$% '%ere#2%ja-#e je'#a*%#e Ko.%jeve ro0-e(e5:

A : y ! x⋅ y= x⋅ y2 , y 0=2, x∈[0,1]

B : y ! = x2 y

2 , y 1=0, x ∈[1,2]

a!5 Oj-erovo( (e$o'o(; +8"!1!0!5 Po0o-j.a#o( Oj-erovo( (e$o'o(; +8"!!2!5 RK (e$o'o(; +8"!!

Rješenje:

a! A!5n=

x− x0

h =10

xi= x0i⋅h

yi = yi−1h⋅ f x i−1 , yi−1

x1=0.1 y10.1=2.0

x2=0.2 y20.2=2.02

x3=0.3 y30.3=2.061208

x4=0.4 y40.4=2.1268291

x5=0.5 y50.5=2.222692

x6=0.6 y60.6=2.3585754 x7=0.7 y7 0.7=2.5508336

x8=0.8 y80.8=2.8277479

x9=0.9 y9 0.9=3.2412207

x10=1.0 y10 1.0=3.8950068

a! 6!5

n= x− x0

h =10

xi = x0i⋅h

y i= y i−1h⋅ f xi −1 , y i −1 x1=1.1 y11.1=0.1

x2=1.2 y21.2=0.222

x3=1.3 y31.3=0.370928

x4=1.4 y41.4=0.553687

x5=1.5 y51.5=0.780344

x6=1.6 y61.6=1.0662378

x7=1.7 y71.7=1.4359241

x8=1.8 y81.8=1.931112

x9=1.9 y91.9=2.6280313 x10=2.0 y102.0=3.6796862




0! A!5

n= x− x0

h =5

x i= x0i⋅h

Ai = y i−1h⋅ f xi−1 , y i−1

yi= yi−1h

2⋅ f x i−1 , yi−1 f xi−1h , Ai

x1=0.2 y

10.2=2.04

x2=0.4 y20.4=2.1780393 x3=0.6 y30.6=2.4784821

x4=0.8 y40.8=3.1461645

x5=1.0 y51.0=5.0500302

0! 6!5

n= x− x0

h =5

xi= x0i⋅h

Ai = yi−1h⋅ f xi−1 , y i−1 yi= y i−1

h

2⋅ f x i−1 , yi−1 f xi−1h , Ai

x1=1.2 y11.2=0.248

x2=1.4 y21.4=0.624214

x3=1.6 y31.6=1.2348928

x4=1.8 y41.8=2.388412

x5=2.0 y52.0=5.4278587

2! A!5

n= x− x0

h =5

x i= x0i⋅h

k 1=h⋅ f x i −1 , yi −1

k 2=h⋅ f xi−1h

2 , y i−1

k 1

2

k 3=h⋅ f x i −1

h

2 , yi −1

k 22

k 4=h⋅ f x i−1h , y i−1k 3

yi = y i−11

6⋅ k 12⋅k 22⋅k 3k 4

x1=0.2 k%=0 k&=0.04 k'=0.04121 k(=0.08501 y10.2=2.04124

x2=0.4 k%=0.08502 k&=0.1355 k'=0.1403 k(=0.2062 y20.4=2.18172

x3=0.6 k%=0.2063 k&=0.2936 k'=0.3093 k(=0.4457 y30.6=2.49135

x4=0.8 k%=0.4459 k&=0.6514 k'=0.7166 k(=1.1333 y40.8=3.21056

x5=1.0 k%=1.13554 k&=1.88954 k'=2.36006 k(=5.09225 y51.0=5.66506

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1<



2! 6!5

n= x− x0

h =5

xi = x0i⋅h

k 1=h⋅ f xi −1 , yi −1

k 2=h⋅ f xi−1h

2 , y i−1

k 1

2

k 3=h⋅ f xi −1h

2

, y i−1k 2

2

k 4=h⋅ f x i−1h , y i−1k 3

yi= yi−11

6⋅ k 12⋅k 22⋅k 3k 4

x1=1.2 k%=0.2 k&=0.244 k'=0.245 k(=0.3 y11.2=0.2463

x2=1.4 k%=0.3001 k&=0.3694 k'=0.3752 k(=0.4692 y21.4=0.6228

x3=1.6 k%=0.4696 k&=0.5971 k'=0.6198 k(=0.8208 y31.6=1.24342

x4=1.8 k%=0.8212 k&=1.12516 k'=1.23032 k(=1.87188 y4 1.8=2.47742

x5=2.0 k%=1.87553 k&=3.0547 k'=3.92965 k(=9.01011 y52.0 =6.61981




Zadatak 5:RK (e$o'o( r%je.%$% -je'e&e Ko.%jeve ro0-e(e:

a. y ! ! −5⋅ y ! 6⋅ y=0, y 0=0, y ! 0=1

x ∈[0,0.4] , h=0.2

b. y ! ! =2− y , y 0=2, y ! 0=−1

x ∈[0,0.2] , h=0.1

$. y ! =− x2⋅ y z , y 0=2

z ! = x2⋅ y3⋅ z , z 0=−2

x ∈[0,0.4] , h=0.2

Rješenje:a!5

y ! ! =5⋅ y ! −6⋅ y

z 0 x = y x

z 1 x = y ! x z

1! x = y ! ! x =5⋅ z

1 x −6⋅ z

0 x

y ! x = z 1 x

z 1 ! x =5⋅ z 1 x−6⋅ z 0 x

f 1 x , y

1, y

2= y

1! = y

2, y

10=0

f 2 x , y

1, y

2= y

2! =5⋅ y

2 x −6⋅ y

1 x , y

20=1

k 11=h⋅ f 1 x , y1, y2 ,

k 12=h⋅ f

2 x , y1,

y2 ,

k 21=h⋅ f 1 xh

2 , y1

k 112

, y2k 122

,

k 22=h⋅ f 2 xh

2 , y1

k 11

2 , y2

k 12

2 ,

k 31=h⋅ f 1 xh

2 , y1

k 212

, y2k 222

,

k 32=h⋅ f 2 xh

2 , y1

k 21

2 , y2

k 22

2 ,

k 41=h⋅ f 1 xh , y1k 31, y2k 32 ,

k 42=h⋅ f 2 xh , y1k 31, y2k 32 ,

yi ,1= y i −1,11

6

⋅k 112⋅k 212⋅k 31k 41 ,

yi ,2= yi −1,21

6⋅ k 122⋅k 222⋅k 32k 42

n= x− x1,0

h =2

xi= x0i⋅h

x1,1=0.2 k 11=0.2 k 21=0.3 k 31=0.338 k 41=0.502 y1,10.2=0.329667

x1,2=0.2 k 12=1.0 k 22=1.38 k 32=1.51 k 42=2.1044 y1,20.2=2.4807333

x2,1=0.4 k 11=0.496147 k 21=0.70466 k 31=0.779148 k 41=1.1116153 y2,10.4=1.0922296

x2,2=0.4 k 12=2.0851333 k 22=2.830012 k 32=3.0773433 k 42=4.2274992 y2,20.4=5.5019572

0!5

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1=



y ! ! =2− y

z 0 x = y x

z 1 x= y ! x z 1 ! x = y ! ! x =2− y

y ! x = z 1 x

z 1 ! x=2− z 0 x

f 1

x , y1,

y2

= y1

! = y2,

y1

0=2

f 2 x , y1, y2= y2 ! =2− y% , y20=−1

k 11=h⋅ f 1 x , y1, y2 ,

k 12=h⋅ f 2 x , y1, y2 ,

k 21=h⋅ f 1 xh

2 , y1

k 112

, y2k 122

,

k 22=h⋅ f 2 xh

2 , y1

k 11

2 , y2

k 12

2 ,

k 31=h⋅ f 1 x h2

, y1 k 212

, y2 k 222

,

k 32=h⋅ f 2 xh

2 , y1

k 212

, y2k 222

,

k 41=h⋅ f 1 xh , y1k 31, y2k 32 ,

k 42=h⋅ f 2 xh , y1k 31, y2k 32 ,

yi ,1= y i −1,11

6⋅k 112⋅k 212⋅k 31k 41 ,

yi ,2= yi −1,21

6⋅ k 122⋅k 222⋅k 32k 42

n= x − x1,0

h =2

x i= x0i⋅h

x1,1=0.1 k 11=−0.1 k 21=−0.1 k 31=−0.09975 k 41=−0.0995 y1,10.1=1.9001667

x1,2=0.1 k 12=0 k 22=0.005 k 32=0.005 k 42=0.009975 y1,20.1=−0.995004

x2,1=0.2 k 11=−0.0995004 k 21=−0.0990012 k 31=−0.0987525 k 41=−0.0980071

y2,10.2=1.8013308

x2,2=0.2 k 12=0.00998333 k 22=0.0149584 k 32=0.0149334 k 42=0.0198586

y2,20.2=−0.980067

2!5 y1 ! =− x2⋅ y1 y2, y10=2

y2 ! = x2⋅ y13⋅ y2, y20=−2

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1>



k 11=h⋅ f 1 x , y1, y2 ,

k 12=h⋅ f 2 x , y1, y2 ,

k 21=h⋅ f 1 xh

2 , y1

k 112

, y2k 122

,

k 22=h⋅ f 2 xh

2 , y1

k 112

, y2k 122

,

k 31=h⋅ f 1 xh

2 , y1

k 212

, y2k 222

,

k 32=h⋅ f 2 x h2

, y1 k 212

, y2 k 222

,

k 41=h⋅ f 1 xh , y1k 31, y2k 32 ,

k 42=h⋅ f 2 xh , y1k 31, y2k 32 ,

yi ,1= y i −1,11

6⋅k 112⋅k 212⋅k 31k 41 ,

yi ,2= yi −1,21

6⋅ k 122⋅k 222⋅k 32k 42

n= x− x

1,0

h =2

x i= x0i⋅h x

1,1=0.2 k 11=0.4 k

21=0.42 k 31=0.422 k

41=0.4444 y1,1 0.2=2.4214

x1,2=0.2 k 12=−0.4 k 22=−0.42 k 32=−0.422 k 42=−0.4444 y1,2 0.2=−2.4214

x2,1=0.4 k 11=0.44428 k 21=0.468708 k 31=0.471151 k 41=0.49851 y2,10.4=2.891818

x2,2=0.4 k

12=−0.44428 k

22=−0.468708 k

32=−0.471151 k

42=−0.49851

y2,20.4=−2.891818

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1?



Zadatak 6:Me$o'o( )o#a*#%+ e-e(e#a$a r%je.%$% -je'e&e r/0#e ro0-e(e:

a. y ! ! =−3⋅ x , y 0=0, y ! 1=2, Δx=h=1

3

b. y ! ! x⋅ y=0, y 0=1, y ! 1=0, Δx=h=0.25

$. y ! ! −4⋅ y3=0, y ! 0=1, y ! 0.6=1, Δx=h=0.2

Rješenje:

a!5 y ! ! )⋅ y= f

za čvor

y j−2⋅1) j−1⋅ Δx

2

6−2⋅ y j −1⋅1−

) j−1)

j

6⋅ Δx

2 y j⋅1) j ⋅ Δx

2

6=

f j−1 f

j

2⋅ Δx

2

za rubove

− y0⋅1−

)1 ⋅ Δx

2

3 y

1⋅1

)1⋅ Δx

2

6=

f 1⋅ Δx

2

2 y ! a ⋅ Δx

yn−1⋅1

)n⋅ Δx

2

6− yn⋅1−

)n ⋅ Δx

2

3=

f n⋅ Δx

2

2− y ! b⋅ Δx

a=0 b=1

[a , b]=[0,1]

[0,1

3][1

3 ,2

3] [2

3 ,1]

)=0, f =−3⋅ x

−2⋅ y1 y2=−0.16666667

y1−2⋅ y2 y3=−0.27777778

y2− y

3=−0.83333333

y1=1.277777778

y2=2.388888886

y3=3.222222216

0!5 y ! ! )⋅ y= f

za čvor

y j−

2⋅1)

j−1⋅ Δx2

6 −2⋅ y

j −1⋅1−

) j −1)

j

6 ⋅ Δx

2 y j

⋅1)

j ⋅ Δx2

6 =

f j−1 f

j

2 ⋅ Δx

2

za rubove

− y0⋅1−)

1 ⋅ Δx2

3 y1⋅1

)1⋅ Δx

2

6 =

f 1⋅ Δx

2

2 y ! a ⋅ Δx

yn−1⋅1)

n⋅ Δx2

6 − yn⋅1−

)n ⋅ Δx

2

3 =

f n⋅ Δx

2

2 − y ! b⋅ Δx

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a 1@



a=0 b=1

[a , b]=[0,1][0,0.25][0.25,0.50] [0.50,0.75][0.75,1.00]

)= x , f =0⋅ x

−1.984375⋅ y11.005208333⋅ y2=−1.002604167

1.00520833⋅ y1−1.973958333⋅ y21.0078125⋅ y3=0

1.0078125⋅ y2−1.963541667⋅ y31.010416667⋅ y4=0

1.01041667⋅ y3−0.979166667⋅ y 4=0

y1=1.21633329109

y2=1.40374603073

y3=1.53626574241

y4=1.58529550484

2!5 y ! ! )⋅ y= f

za čvor

y j−2⋅1) j−1⋅ Δx2

6 −2⋅ y j −1⋅1−

) j −1) j

6 ⋅ Δx

2 y j⋅1) j ⋅ Δx2

6 =

f j−1 f j

2 ⋅ Δx

2

za rubove

− y0⋅1−)

1 ⋅ Δx2

3 y1⋅1

)1⋅ Δx

2

6 =

f 1⋅ Δx

2

2 y ! a ⋅ Δx

yn−1⋅1)

n⋅ Δx2

6 − yn⋅1−

)n ⋅ Δx

2

3 =

f n⋅ Δx

2

2 − y ! b⋅ Δx

a=0 b=0.6

[a , b]=[0,1]

[0,0.20] [0.20,0.40][0.40,0.60])=−4, f =−3⋅ x

−1.05333333⋅ y00.97333333⋅ y1=0.14

0.97333333⋅ y0−2.10666667⋅ y10.97333333⋅ y2=−0.12

0.97333333⋅ y1−2.10666667⋅ y20.97333333⋅ y3=−0.12

0.97333333⋅ y2−1.0533333⋅ y3=−0.26

y0=0.481811074641

y1=0.665247601952

y2=0.834752235001

y3=1.01818875092

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a "



Zadatak 7:Me$o'o( )o#a*#%+ ra,-%)a r%je.%$% -je'e&e r/0#e ro0-e(e:

a. Δu = y , x , y ∈Ω=[0,1]×[0,1]

u x ,0= x3 , u x ,1= x

3 , x∈[0,1]

u 0, y =0, u 1, y =1, y∈[0,1]h=0.2

b. Δu=−4, x , y ∈Ω={ x , y ∣ x2 y

21}u x , y =0, x , y ∈∂ Ω

h=1

3

Rješenje:




a.)

0u420.008u31−4⋅u41=h2⋅0.8

u41u430.064u32−4⋅u42=h2⋅0.8

u42u440.216u33−4⋅u43=h2⋅0.8

u4310.512u34−4⋅u44=h2⋅0.8

0u32u41u21−4⋅u31=h2⋅0.6

u31u33u42u22−4⋅u32=h2⋅0.6

u32u34u43u23−4⋅u33=h2⋅0.6

u331u44u24−4⋅u34=h2⋅0.6

0u22u31u11−4⋅u21=h2⋅0.4

u21u23u32u12−4⋅u22=h2⋅0.4

u22u24u33u13−4⋅u23=h2⋅0.4

u231u34u14−4⋅u24=h2⋅0.4

0u12u210.008−4⋅u11=h2⋅0.2

u11u13u220.064−4⋅u12=h2⋅0.2

u12u14u230.216−4⋅u13=h2⋅0.2

u131u240.512−4⋅u14=h

2

⋅0.2

u11=0.0645152, u12=0.172303, u13=0.354848, u14=0.632879,

u21=0.0857576, u22=0.213848, u23=0.406212, u24=0.672667,

u31=0.0806667, u32=0.207121, u33=0.399485, u34=0.667576,

u41=0.0537879, u42=0.158485, u43=0.34103, u44=0.622152

Zora# Pe$rov%& Za'a&a 1 S$ra#%2a



0!5

za čvor i , j : u i−1, jui 1, ju i , j−1ui , j 1−4⋅ui , j =h2⋅ f

h2⋅ f =

1

32

⋅−4=−4

9

Oblast je simetrična , pa je dovoljno računati samo za prvi kvadrant.

0,0:

u1,0u0,1u0,−1u−1,0−4⋅u0,0=−4

9

−4⋅u0,02⋅u0,12⋅u1,0=−4

9 =−4⋅u0,04⋅u0,1

0,1

u0,2u0,0u1,1u−1,1−4⋅u0,1=−4

9

u0,0−4⋅u0,1u0,22⋅u1,1=−4

9

0,2

u0,3u0,1u1,2u−1,2−4⋅u0,2=−49

0u0,1−4⋅u0,22⋅u1,2=−4

9

1,1

u1,0u1,2u0,1u2,1−4⋅u1,1=−4

9

2⋅u0,1−4⋅u1,12⋅u1,2=−4

9

1,2

u1,1u1,3u0,2u2,2−4⋅u1,2=−4

9

u0,2u1,1−4⋅u1,2=−4

9

u0,0=1 u0,1=8

9 u0,2=

5

9 u1,1=

7

9 u1,2=

4

9

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