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metodos numericos
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Metodologa
Dada la siguiente ecuacin diferencial:
Primero despejamos y:
Si las condiciones iniciales son Y(0)=1, entonces x=0, Y=1
Evaluar para x=2
Por el mtodo de Euler se tiene:
Yi+1=Yi+ (H*dy/dx)
Donde H= (mximo valor de x mnimo valor de x) /n
Donde n es el nmero de iteraciones
Por el mtodo de Heun:
Yi+1=Yi+ (H/2)*(F(x, y) + F(x + h; yi+1))
Donde H= (mximo valor de x mnimo valor de x) /n
Donde n es el nmero de iteraciones
Resultados
1- Por el mtodo de Euler:
i
xi
yi
h
yi+1=yi+h*F(x,y)
0
0
1
0.1
9.333333333
1.933333333
1
0.1
1.93333333
0.1
2.67106787
2.20044012
2
0.2
2.20044012
0.1
1.948032285
2.395243349
3
0.3
2.39524335
0.1
1.581666023
2.553409951
4
0.4
2.55340995
0.1
1.370188672
2.690428818
5
0.5
2.69042882
0.1
1.244047759
2.814833594
6
0.6
2.81483359
0.1
1.170449591
2.931878553
7
0.7
2.93187855
0.1
1.130607739
3.044939327
8
0.8
3.04493933
0.1
1.11215108
3.156154435
9
0.9
3.15615444
0.1
1.106091793
3.266763615
10
1
3.26676361
0.1
1.10548475
3.37731209
11
1.1
3.37731209
0.1
1.104800753
3.487792165
12
1.2
3.48779216
0.1
1.099623049
3.59775447
13
1.3
3.59775447
0.1
1.086493681
3.706403838
14
1.4
3.70640384
0.1
1.062828567
3.812686695
15
1.5
3.81268669
0.1
1.026863334
3.915373028
16
1.6
3.91537303
0.1
0.977612448
4.013134273
17
1.7
4.01313427
0.1
0.914833347
4.104617607
18
1.8
4.10461761
0.1
0.838990663
4.188516674
19
1.9
4.18851667
0.1
0.751216423
4.263638316
20
2
4.26363832
0.1
0.653262232
4.328964539
2- Por el mtodo de Heun:
i
Xi
yi
h
F(xi+h;Y*i+1))
yEuler=yi+h*F(x,y)
yheun=yi+h*(F(x,y)+F(xi+h;Y*i+1))/2
0
0
1
0.1
9.333333333
2.67106787
1.933333333
1.60022006
1
0.1
1.60022006
0.1
4.082478106
2.469739342
2.008467871
1.927830933
2
0.2
1.92783093
0.1
2.729769773
2.003869128
2.20080791
2.164512878
3
0.3
2.16451288
0.1
2.094103137
1.695776515
2.373923191
2.35400686
4
0.4
2.35400686
0.1
1.736161091
1.498420963
2.527622969
2.515735963
5
0.5
2.51573596
0.1
1.518820646
1.37247684
2.667618027
2.660300837
6
0.6
2.66030084
0.1
1.38335742
1.293209395
2.798636579
2.794129178
7
0.7
2.79412918
0.1
1.29910503
1.244374473
2.924039681
2.921303153
8
0.8
2.92130315
0.1
1.247553728
1.214526385
3.046058526
3.044407159
9
0.9
3.04440716
0.1
1.216241403
1.195115872
3.166031299
3.164975023
10
1
3.16497502
0.1
1.196100852
1.179518241
3.284585108
3.283755977
11
1.1
3.28375598
0.1
1.18021486
1.162529199
3.401777463
3.40089318
12
1.2
3.40089318
0.1
1.163200737
1.140096163
3.517213254
3.516058025
13
1.3
3.51605803
0.1
1.140891874
1.109167965
3.630147212
3.628561017
14
1.4
3.62856102
0.1
1.110162981
1.067603566
3.739577315
3.737449344
15
1.5
3.73744934
0.1
1.068825307
1.014109614
3.844331875
3.84159609
16
1.6
3.84159609
0.1
1.015555736
0.948191011
3.943151664
3.939783428
17
1.7
3.93978343
0.1
0.949841487
0.870105383
4.034767576
4.030780771
18
1.8
4.03078077
0.1
0.871930284
0.780815251
4.1179738
4.113418048
19
1.9
4.11341805
0.1
0.782779368
0.68193297
4.191695985
4.186653665
20
2
4.18665366
0.1
0.683998212
0.575654349
4.255053486
4.249636293
2.1
4.24963629
Como los resultados por los diferentes mtodos solo varan en pequeas cantidades
Y Euler = 4.26363832 mientras que para Heun es:
Y Heun = 4.24963629
metodo de Euler
00.10.20.300000000000000040.40.50.60.70.799999999999999930.899999999999999910.999999999999999891.09999999999999991.21.31.40000000000000011.50000000000000021.60000000000000031.70000000000000041.80000000000000051.90000000000000062.000000000000000411.93333333333333362.20044012032626362.3952433488381712.55340995113041962.69042881832379482.81483359423522612.93187855333410543.044939327237023.15615443523713563.26676361457089783.37731208960527553.48779216486443343.59775446972322223.70640383778189893.8126866945125183.91537302794273684.01313427269655154.10461760740092264.18851667365419194.2636383159872864
Metodo de Heun
00.10.20.300000000000000040.40.50.60.70.799999999999999930.899999999999999910.999999999999999891.09999999999999991.21.31.40000000000000011.50000000000000021.60000000000000031.70000000000000041.80000000000000051.90000000000000062.000000000000000411.60022006016313181.92783093258850262.16451287761445292.35400686021364922.51573596288249312.66030083721362282.79412917797551822.92130315313339933.04440715878715423.16497502252816963.28375597716795833.40089318010773623.51605802511569683.62856101703209483.73744934435156223.84159609039743983.93978342774942644.03078077127015094.11341804803369284.1866536649177526