NAMA : Rahmad Hidayat
NIM : 090401010
Sebuah batang AB dengan panjang 10 dijaga pada suhu 0 oC. Pada t = 0, temperatur batang naik seragam dari ujung-ujungnya sampai 2 oC pada titik tengah batang AB. Tentukan persamaan temperatur u(x,t) pada setiap titik pada batang, jarak x dari kiri pada suatu waktu,t.
Penyelesaian :
Tc= 2 oC
Tb= 0 oC
Ta= 0 oC
L = 10
L/2 = 5
C
B
A
Persamaan panas pada satu dimensi :
= Dimana k = konduktivitas termal material
σ = panas spesifik material
ρ = massa jenis benda
u(0,t) = 0 , u(10,t) = 0 untuk t
u(x,0) = f(x) untuk 0
Persamaannya :
u(x,t) = n = 1,2,3,….,dst
x=l/2 f(l/2) = = 2
sin n = 1
Maka persamaannya menjadi :
u(x,t) = n = 1,2,3,….,dst
=
= dan c = 3
Tentukan persamaan solusi u = f(x,t) jika kondisi batas
u(0,t) = 0 dan u(2,t) = 0 untuk t
u(x,0) = x(2-x) dan
Penyelesaian :
u(x,t) =
= 0
Maka
u(x,t) =
u(x,0) = x(2-x) = f(x)
u(x,0) = = f(x)
dx
dx
= dx
= -
= ]
=
Maka persamaannya menjadi
u(x,t) =
4. Buktikan
I =
Penyelesaian :
Misalkan y = x =
dy = n. dx dx =
I = =
I =
I =
I =
I =
I =
I =
v-1 =
v = + 1
v =
I = =
I = =
( terbukti )
5. Selesaikan dengan metode runge kutta
y" = x-y-y' ; x = 0 ; y = 0 ; y' = 1 ; x = 0(0,2)1,0
n
x
K1
K2
K3
K4
y
y'
y"
h=
0.2
p
q
0
1
-0.0200000
0.0217000
0.0262870
0.0348346
0.0000000
1.0000000
-1.0000000
0.009329
0.036936
1
1.2
-0.0168373
0.0350846
0.0418345
0.0546467
0.2093290
1.1846810
-0.8418668
0.020027
0.063883
2
1.4
-0.0167290
0.0548639
0.0656028
0.0857782
0.4662925
1.5040936
-0.8364482
0.034579
0.103327
3
1.6
-0.0189959
0.0854558
0.1032126
0.1360838
0.8016904
2.0207310
-0.9497966
0.056558
0.164808
4
1.8
-0.0238160
0.1343947
0.1644547
0.2194888
1.2623941
2.8447724
-1.1908013
0.091678
0.264457
5
2
-0.0320555
0.2149192
0.2667838
0.3611999
1.9230263
4.1670582
-1.6027725
0.149883
0.43085
y" = (1-x)y' – y ; x = 0 ; y = 1 ; y' = 1 ; x = 0(0,2)1,0
n
x
K1
K2
K3
K4
y
y'
y"
h=
0.2
p
q
0
0
0.0000000
0.0240000
0.0242400
0.0294544
1.0000000
1.0000000
0.0000000
0.01608
0.041978
1
0.2
-0.0049633
0.0338270
0.0349907
0.0423394
1.2160800
1.2098907
-0.2481675
0.021285
0.058337
2
0.4
-0.0115679
0.0469695
0.0498964
0.0605976
1.4793429
1.5015764
-0.5783971
0.028433
0.080921
3
0.6
-0.0209124
0.0650923
0.0711126
0.0870857
1.8080909
1.9061790
-1.0456193
0.038431
0.112861
4
0.8
-0.0346732
0.0906709
0.1019518
0.1262762
2.2277575
2.4704839
-1.7336607
0.05265
0.158949
5
1
-0.0554901
0.1274743
0.1476004
0.1852926
2.7745041
3.2652312
-2.7745041
0.073195
0.226651
y" = 1+x - y2 ; x = 0 ; y = 2 ; y' = 1 ; x = 0(0,1)0,5
n
x
K1
K2
K3
K4
y
y'
y"
h=
0.1
p
q
0
0
-0.0150000
0.0104438
0.0105074
0.0111576
2.0000000
1.0000000
-3.0000000
0.001984
0.012687
1
0.1
-0.0165917
0.0114916
0.0117022
0.0124928
2.1019837
1.1268661
-3.3183355
0.002201
0.014096
2
0.2
-0.0185726
0.0127307
0.0131220
0.0140793
2.2168710
1.2678287
-3.7145172
0.002427
0.015737
3
0.3
-0.0210205
0.0141867
0.0148028
0.0159595
2.3460806
1.4252026
-4.2040943
0.002656
0.017639
4
0.4
-0.0240318
0.0158895
0.0167878
0.0181844
2.4912572
1.6015960
-4.8063625
0.002882
0.019836
5
0.5
-0.0277265
0.0178742
0.0191282
0.0208147
2.6542986
1.7999531
-5.5453012
0.003092
0.022364
y" = (x+2)y - 2y' ; x = 0 , y = 1, y' = 0 ; x = 0(0,2)1,0
n
x
K1
K2
K3
K4
y
y'
y"
h=
0.1
p
q
0
0
0.0100000
0.0050375
0.0050251
0.0050754
1.0000000
0.0000000
2.0000000
0.006688
0.011734
1
0.1
0.0093969
0.0052330
0.0052018
0.0053395
1.0066875
0.1173352
1.8793734
0.006611
0.011869
2
0.2
0.0089151
0.0056018
0.0055604
0.0057918
1.0250316
0.2360215
1.7830266
0.006692
0.012344
3
0.3
0.0085417
0.0061557
0.0061140
0.0064504
1.0553262
0.3594588
1.7083325
0.006937
0.013177
4
0.4
0.0082662
0.0069154
0.0068851
0.0073434
1.0982092
0.4912301
1.6532418
0.007356
0.014404
5
0.5
0.0080809
0.0079116
0.0079069
0.0085108
1.1546877
0.6352654
1.6161884
0.007966
0.016076
5
0.6
0.0079801
0.0091863
0.0092255
0.0100069
1.2261807
0.7960278
1.5960144
0.008797
0.01827
5
0.7
0.0079595
0.0107963
0.0109026
0.0119039
1.3145808
0.9787302
1.5919079
0.009886
0.021087
5
0.8
0.0080168
0.0128156
0.0130196
0.0142966
1.4223400
1.1896009
1.6033501
0.011284
0.024661
5
0.9
0.0081503
0.0153413
0.0156829
0.0173088
1.5525841
1.4362135
1.6300668
0.013058
0.029169
5
1
0.0083599
0.0184991
0.0190315
0.0211024
1.7092636
1.7279053
1.6719802
0.015297
0.034841
y" = (y'-xy')/x^2 ; x = 1, y = 0, y' = 1 ; x = 1,0(0,2)2,0
n
x
K1
K2
K3
K4
y
y'
y"
h=
0.1
p
q
0
1
-0.0050000
0.0052313
0.0057684
0.0066634
0.0000000
1.0000000
-1.0000000
0.002
0.007888
1
1.1
-0.0044825
0.0067199
0.0073640
0.0084432
0.1019999
1.0788755
-0.8964985
0.0032
0.01071
2
1.2
-0.0042017
0.0085064
0.0093006
0.0106228
0.2130879
1.1859708
-0.8403313
0.004535
0.014012
3
1.3
-0.0041056
0.0106815
0.0116796
0.0133203
0.3362201
1.3260883
-0.8211211
0.006085
0.017979
4
1.4
-0.0041666
0.0133614
0.0146321
0.0166896
0.4749141
1.5058776
-0.8333238
0.007942
0.022837
5
1.5
-0.0043732
0.0166968
0.0183297
0.0209327
0.6334442
1.7342442
-0.8746320
0.010218
0.028871
6
1.6
-0.0047259
0.0208847
0.0229976
0.0263166
0.8170864
2.0229526
-0.9451710
0.013052
0.036452
7
1.7
-0.0052358
0.0261847
0.0289340
0.0331959
1.0324338
2.3874707
-1.0471510
0.016628
0.046066
8
1.8
-0.0059241
0.0329409
0.0365359
0.0420448
1.2878085
2.8481297
-1.1848225
0.021184
0.058358
9
1.9
-0.0068233
0.0416123
0.0463348
0.0535006
1.5938057
3.4317108
-1.3646661
0.027041
0.074191
10
2
-0.0079790
0.0528152
0.0590466
0.0684249
1.9640181
4.1736160
-1.5958035
0.034628
0.094723