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bab 5
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BAB VPENYELESAIAN PERSAMAAN DIFERENSIAL ORDINER SIMULTANDENGAN RUNGE KUTTA
A. TUJUANAgar mahasiswa dapat menyelesaikan bentuk penyelesaian differensial ordiner simultan menggunakan penyelesaian numerik.B. DASAR TEORIJika dijumpai bentuk :
I.c : x = x0 ; y = y0 ; z = z0 Maka cara Range Kutta untuk mencari xi+1, yi+1,zi+1 berdasar harga xi,yi,zi adalah :K1 = f1 ( xi , yi, zi ) . L1 = = f2 ( xi , yi, zi ) .
K2 = f1 ( xi + / 2 , yi + k1 / 2 + l1/2 ) .L2 = f2 ( xi + / 2 , yi + k1 / 2 + l1/2 ) .
K3 = f1 (xi + / 2 , yi + k2 / 2 + l1/2 ) .L3 = f2 (xi + / 2 , yi + k2 / 2 + l1/2 ) .
K4 = f1 ( xi + , yi + k3 ,z1+l3) . L4 = f2 ( xi + , yi + k3 ,z1+l3) .
Dapat diperoleh :Xi+1= xi + Yi+1= yi + ( k1 + 2k2 + 2k3 + k4 ) / 6}Zi+1= zi + ( l1 + 2l2 + 2l3 + l4 ) / 6}
Algoritma1. Menentukan persamaan yang akan diselesaikan
2. Menentukan nilai X0,Y0,Z0,Xn,n,x3. Menghitung nilai K1 L1,K2 L2,K3 L3,K4 L4.4. Menghitung harga x,y,z baru
BAB 2x00,5
y01
z01,5
xn1,5
n10
x0,1
A. LATIHAN1.
TENTUKAN NILAI y,z PADA SAAT x = 1.5 DENGAN MENGGUNAKAN METODE RANGE KUTTA ?nx0y0z0k1l1k2l2k3l3k4l4yz
00,511,50,12500,24750,13790,25890,13830,25920,15200,27131,13821,7592
10,61,13821,75920,15200,27130,16640,28390,16690,28420,18210,29741,30502,0433
20,71,30502,04330,18210,29740,19820,31120,19880,31160,21580,32601,50372,3548
30,81,50372,35480,21580,32600,23380,34090,23450,34130,25350,35691,73802,6960
40,91,73802,69600,25340,35690,27340,37310,27420,37340,29530,39022,01203,0694
512,01203,06940,29530,39020,31750,40760,31840,40790,34180,42592,33013,4772
61,12,33013,47720,34180,42590,36640,44450,36730,44480,39320,46402,69723,9220
71,22,69723,92200,39320,46400,42030,48370,42130,48410,44990,50443,11834,4060
81,33,11834,40600,44990,50440,47970,52540,48080,52570,51220,54733,59884,9316
91,43,59884,93160,51220,54730,54500,56940,54610,56970,58050,59244,14465,5013
101,54,14465,50130,58040,59240,616290,61570,61750,61610,65500,64004,76186,1173
Maka untuk x =2 harga y =5.1344 dan z = 4.0399x01
y01
z00,1
xn2
n20
x0,05
2.
nx0y0z0k1l1k2l2k3l3k4l4yz
0110,10,06580,02660,06820,02820,06830,02820,07080,02991,06830,1282
11,051,06830,12820,07080,02990,07340,03170,07340,03180,07610,03371,14170,1600
21,11,14170,16000,07610,03370,07890,03570,07900,03570,08190,03781,22060,1957
31,151,22060,19570,08190,03780,08490,04010,08500,04010,08820,04251,30560,2359
41,21,30560,23590,08820,04250,09150,04510,09160,04520,09510,04791,39720,2810
51,251,39720,28100,09510,04790,09870,05080,09880,05080,10260,05391,49600,3318
61,31,49600,33180,10260,05390,10670,05730,10680,05740,11110,06091,60270,3892
71,351,60270,38920,11110,06090,11550,06480,11570,06490,12050,06911,71840,4541
81,41,71840,45410,12050,06910,12550,07350,12570,07370,13100,07861,84410,5278
91,451,84410,52780,13100,07860,13670,08390,13700,08410,14300,08991,98100,6119
101,51,98100,61190,14300,08990,14940,09630,14970,09660,15660,10362,13060,7084
111,552,13060,70840,15660,10360,16400,11130,16440,11170,17230,12032,29490,8200
121,62,29490,82000,17230,12020,18080,12970,18130,13030,19050,14102,47610,9502
131,652,47610,95020,19050,14090,20040,15280,20100,15360,21190,16712,67701,1037
141,72,67701,10370,21180,16710,22350,18230,22430,18350,23720,20092,90111,2870
151,752,90111,28700,23720,20090,25110,22080,25220,22250,26770,24573,15301,5092
161,83,15301,50920,26770,24560,28460,27220,28600,27480,30500,30653,43871,7835
171,853,43871,78350,30500,30640,32590,34330,32780,34730,35160,39213,76602,1301
181,93,76602,13010,35150,39190,37800,44480,38070,45140,41120,51744,14602,5804
191,954,14602,58040,41100,51710,44530,59640,44930,60790,48960,71024,59433,1865
2024,59433,18650,48940,70960,53530,83550,54150,85690,59701,02635,13444,0399
Maka untuk x =2 harga y =5.1344 dan z = 4.0399