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