Chapter 3 Numerical Methods

3.1 Euler's Method

1
\(y_1=1.450000000,\; y_2=2.085625000,\; y_3= 3.079099746\)
2
\(y_1=1.200000000,\; y_2=1.440415946,\; y_3=1.729880994\)
3
\(y_1=1.900000000,\; y_2=1.781375000,\; y_3=1.646612970\)
4
\(y_1=2.962500000,\; y_2=2.922635828,\; y_3=2.880205639\)
5
\(y_1=2.513274123,\; y_2=1.814517822,\; y_3=1.216364496\)
6

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 48.298147362 51.492825643 53.076673685 54.647937102
7

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

2.0 1.390242009 1.370996758 1.361921132 1.353193719
8

\(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact

1.50 7.886170437 8.852463793 9.548039907 10.500000000
9

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)

3.0 1.469458241 1.462514486 1.459217010 0.3210 0.1537 0.0753

Approximate Solutions Residuals
10

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)

2.0 0.473456737 0.483227470 0.487986391 -0.3129 -0.1563 -0.0781

Approximate Solutions Residuals
11

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

1.0 0.691066797 0.676269516 0.668327471 0.659957689
12

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

2.0 -0.772381768 -0.761510960 -0.756179726 -0.750912371
13

Euler’s method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 0.538871178 0.593002325 0.620131525 0.647231889

Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 0.647231889 0.647231889 0.647231889 0.647231889

Applying variation of parameters to the given initial value problem yields

\(y=ue^{-3x}\), where (A) \(u'=7,\quad u(0)=6\). Since \(u''=0\), Euler’s method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem

.
14

Euler’s method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

3.0 12.804226135 13.912944662 14.559623055 15.282004826

Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

3.0 15.354122287 15.317257705 15.299429421 15.282004826
15

Euler’s method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact”

2.0 0.867565004 0.885719263 0.895024772 0.904276722

Euler semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact”

2.0 0.569670789 0.720861858 0.808438261 0.904276722
16

Euler’s method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact”

3.0 0.922094379 0.945604800 0.956752868 0.967523153

Euler semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact”

3.0 0.993954754 0.980751307 0.974140320 0.967523153
17

Euler’s method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact”

1.50 0.319892131 0.330797109 0.337020123 0.343780513

Euler semilinear method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact”

1.50 0.305596953 0.323340268 0.333204519 0.343780513
18

Euler’s method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact”

2.0 0.754572560 0.743869878 0.738303914 0.732638628

Euler semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact”

2.0 0.722610454 0.727742966 0.730220211 0.732638628
19

Euler’s method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact”

1.50 2.175959970 2.210259554 2.227207500 2.244023982

Euler semilinear method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact”

1.50 2.117953342 2.179844585 2.211647904 2.244023982
20

Euler’s method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

1.0 0.032105117 0.043997045 0.050159310 0.056415515

Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

1.0 0.056020154 0.056243980 0.056336491 0.056415515
21

Euler’s method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

1.0 28.987816656 38.426957516 45.367269688 54.729594761

Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

1.0 54.709134946 54.724150485 54.728228015 54.729594761
22

Euler’s method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

3.0 1.361427907 1.361320824 1.361332589 1.361383810

Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact”

3.0 1.291345518 1.326535737 1.344004102 1.361383810

3.2 The Improved Euler Method and Related Methods

1
\(y_1=1.542812500,\; y_2=2.421622101,\; y_3=4.208020541\)
2
\(y_1=1.220207973,\; y_2=1.489578775\; y_3=1.819337186\)
3
\(y_1=1.890687500,\; y_2=1.763784003,\; y_3=1.622698378\)
4
\(y_1=2.961317914\) \(y_2=2.920132727\) \(y_3=2.876213748\).
5
\(y_1=2.478055238,\; y_2=1.844042564,\; y_3=1.313882333\)
6

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 56.134480009 55.003390448 54.734674836 54.647937102
7

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

2.0 1.353501839 1.353288493 1.353219485 1.353193719
8

\(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact

1.50 10.141969585 10.396770409 10.472502111 10.500000000
9

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)

3.0 1.455674816 1.455935127 1.456001289 -0.00818 -0.00207 -0.000518

Approximate Solutions Residuals
10

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)

2.0 0.492862999 0.492709931 0.492674855 0.00335 0.000777 0.000187

Approximate Solutions Residuals
11

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.660268159 0.660028505 0.659974464 0.659957689
12

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

2.0 -0.749751364 -0.750637632 -0.750845571 -0.750912371
13
Applying variation of parameters to the given initial value problem

\(y=ue^{-3x}\), where (A) \(u'=1-2x,\quad u(0)=2\). Since \(u'''=0\), the improved Euler method yields the exact solution of (A). Therefore the improved Euler semilinear method produces the exact solution of the given problem.

Improved Euler method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 0.105660401 0.100924399 0.099893685 0.099574137

Improved Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 0.099574137 0.099574137 0.099574137 0.099574137
14

Improved Euler method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 15.107600968 15.234856000 15.269755072 15.282004826

Improved Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 15.285231726 15.282812424 15.282206780 15.282004826
15

Improved Euler method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.924335375 0.907866081 0.905058201 0.904276722

Improved Euler semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.969670789 0.920861858 0.908438261 0.904276722
16

Improved Euler method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

3.0 0.967473721 0.967510790 0.967520062 0.967523153

Improved Euler semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

3.0 0.967473721 0.967510790 0.967520062 0.967523153
17

Improved Euler method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact"

1.50 0.349176060 0.345171664 0.344131282 0.343780513

Improved Euler semilinear method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact"

1.50 0.349350206 0.345216894 0.344142832 0.343780513
18

Improved Euler method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.732679223 0.732721613 0.732667905 0.732638628

Improved Euler semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.732166678 0.732521078 0.732609267 0.732638628
19

Improved Euler method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact"

1.50 2.247880315 2.244975181 2.244260143 2.244023982

Improved Euler semilinear method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact"

1.50 2.248603585 2.245169707 2.244310465 2.244023982
20

Improved Euler method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.059071894 0.056999028 0.056553023 0.056415515

Improved Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.056295914 0.056385765 0.056408124 0.056415515
21

Improved Euler method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 50.534556346 53.483947013 54.391544440 54.729594761

Improved Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 54.709041434 54.724083572 54.728191366 54.729594761
22

Improved Euler method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 1.361395309 1.361379259 1.361382239 1.361383810

Improved Euler semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 1.375699933 1.364730937 1.362193997 1.361383810
23

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

2.0 1.349489056 1.352345900 1.352990822 1.353193719
24

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

2.0 1.350890736 1.352667599 1.353067951 1.353193719
25

\(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact

1.50 10.133021311 10.391655098 10.470731411 10.500000000
26

\(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact

1.50 10.136329642 10.393419681 10.470731411 10.500000000
27

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.660846835 0.660189749 0.660016904 0.659957689
28

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.660658411 0.660136630 0.660002840 0.659957689
29

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

2.0 -0.750626284 -0.750844513 -0.750895864 -0.751331499
30

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

2.0 -0.750335016 -0.750775571 -0.750879100 -0.751331499

3.3 The Runge-Kutta Method

1
\(y_1=1.550598190,\; y_2=2.469649729\)
2
\(y_1=1.221551366,\; y_2=1.492920208\)
3
\(y_1=1.890339767,\; y_2=1.763094323\)
4
\( y_1=2.961316248\) \( y_2=2.920128958\).
5
\(y_1=2.475605264,\; y_2=1.825992433\)
6

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

1.0 54.654509699 54.648344019 54.647962328 54.647937102
7

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) Exact

2.0 1.353191745 1.353193606 1.353193712 1.353193719
8

\(x\) \(h=0.05\) \(h=0.025\) \(h=0.0125\) Exact

1.50 10.498658198 10.499906266 10.499993820 10.500000000
9

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)

3.0 1.456023907 1.456023403 1.456023379 0.0000124 0.000000611 0.0000000333

Approximate Solutions Residuals
10

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) \(h=0.1\) \(h=0.05\) \(h=0.025\)

2.0 0.492663789 0.492663738 0.492663736 0.000000902 0.0000000508 0.00000000302

Approximate Solutions Residuals
11

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.659957046 0.659957646 0.659957686 0.659957689
12

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

2.0 -0.750911103 -0.750912294 -0.750912367 -0.750912371

Applying variation of parameters to the given initial value problem yields

\(y=ue^{-3x}\), where (A) \(u'=1-4x+3x^2-4x^3,\quad u(0)=-3\). Since \(u^{(5)}=0\), the Runge-Kutta method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem.

Runge-Kutta method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
0.0	-3.000000000	-3.000000000	-3.000000000	-3.000000000
0.1	-2.162598011	-2.162526572	-2.162522707	-2.162522468
0.2	-1.577172164	-1.577070939	-1.577065457	-1.577065117
0.3	-1.163350794	-1.163242678	-1.163236817	-1.163236453
0.4	-0.868030294	-0.867927182	-0.867921588	-0.867921241
0.5	-0.655542739	-0.655450183	-0.655445157	-0.655444845
0.6	-0.501535352	-0.501455325	-0.501450977	-0.501450707
0.7	-0.389127673	-0.389060213	-0.389056546	-0.389056318
0.8	-0.306468018	-0.306412184	-0.306409148	-0.306408959
0.9	-0.245153433	-0.245107859	-0.245105379	-0.245105226
1.0	-0.199187198	-0.199150401	-0.199148398	-0.199148273

Runge-Kutta semilinear method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
0.0	-3.000000000	-3.000000000	-3.000000000	-3.000000000
0.1	-2.162522468	-2.162522468	-2.162522468	-2.162522468
0.2	-1.577065117	-1.577065117	-1.577065117	-1.577065117
0.3	-1.163236453	-1.163236453	-1.163236453	-1.163236453
0.4	-0.867921241	-0.867921241	-0.867921241	-0.867921241
0.5	-0.655444845	-0.655444845	-0.655444845	-0.655444845
0.6	-0.501450707	-0.501450707	-0.501450707	-0.501450707
0.7	-0.389056318	-0.389056318	-0.389056318	-0.389056318
0.8	-0.306408959	-0.306408959	-0.306408959	-0.306408959
0.9	-0.245105226	-0.245105226	-0.245105226	-0.245105226
1.0	-0.199148273	-0.199148273	-0.199148273	-0.199148273

14

Runge-Kutta method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 15.281660036 15.281981407 15.282003300 15.282004826

Runge-Kutta semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 15.282005990 15.282004899 15.282004831 15.282004826
15

Runge-Kutta method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.904678156 0.904295772 0.904277759 0.904276722

Runge-Kutta semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.904592215 0.904297062 0.904278004 0.904276722
16

Runge-Kutta method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

3.0 0.967523147 0.967523152 0.967523153 0.967523153

Runge-Kutta semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

3.0 0.967523147 0.967523152 0.967523153 0.967523153

Runge-Kutta method
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	“Exact"
1.50	0.343839158	0.343784814	0.343780796	0.343780513

Runge-Kutta semilinear method
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	“Exact"
1.00	0.000000000	0.000000000	0.000000000	0.000000000
1.05	0.028121022	0.028121010	0.028121010	0.028121010
1.10	0.055393494	0.055393466	0.055393465	0.055393464
1.15	0.082164048	0.082163994	0.082163990	0.082163990
1.20	0.108862698	0.108862597	0.108862591	0.108862590
1.25	0.136058715	0.136058528	0.136058517	0.136058516
1.30	0.164564862	0.164564496	0.164564473	0.164564471
1.35	0.195651074	0.195650271	0.195650219	0.195650216
1.40	0.231542288	0.231540164	0.231540027	0.231540017
1.45	0.276818775	0.276811011	0.276810491	0.276810456
1.50	0.343839124	0.343784811	0.343780796	0.343780513

18

Runge-Kutta method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.732633229 0.732638318 0.732638609 0.732638628

Runge-Kutta semilinear method

\(x\) \(h=0.2\) \(h=0.1\) \(h=0.05\) “Exact"

2.0 0.732639212 0.732638663 0.732638630 0.732638628
19

Runge-Kutta method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact"

1.50 2.244025683 2.244024088 2.244023989 2.244023982

Runge-Kutta semilinear method

\(x\) \(h=0.0500\) \(h=0.0250\) \(h=0.0125\) “Exact"

1.50 2.244025081 2.244024051 2.244023987 2.244023982
20

Runge-Kutta method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.056426886 0.056416137 0.056415552 0.056415515

Runge-Kutta semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 0.056415185 0.056415495 0.056415514 0.056415515
21

Runge-Kutta method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 54.695901186 54.727111858 54.729426250 54.729594761

Runge-Kutta semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

1.0 54.729099966 54.729561720 54.729592658 54.729594761
22

Runge-Kutta method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 1.361384082 1.361383812 1.361383809 1.361383810

Runge-Kutta semilinear method

\(x\) \(h=0.1\) \(h=0.05\) \(h=0.025\) “Exact"

3.0 1.361456502 1.361388196 1.361384079 1.361383810
24

\(x\) \(h=.1\) \(h=.05\) \(h=.025\) Exact

2.00 -1.000000000 -1.000000000 -1.000000000 -1.000000000
25

\(x\) \(h=.1\) \(h=.05\) \(h=.025\) “Exact"

1.00 1.000000000 1.000000000 1.000000000 1.000000000
26

\(x\) \(h=.1\) \(h=.05\) \(h=.025\) Exact

1.50 4.142171279 4.142170553 4.142170508 4.142170505
27

\(x\) \(h=.1\) \(h=.05\) \(h=.025\) Exact

3.0 16.666666988 16.666666687 16.666666668 16.666666667

Euler’s method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
1.0	0.538871178	0.593002325	0.620131525	0.647231889

Euler semilinear method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
1.0	0.647231889	0.647231889	0.647231889	0.647231889

Euler’s method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exact”
2.0	0.867565004	0.885719263	0.895024772	0.904276722

Improved Euler method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
1.0	0.105660401	0.100924399	0.099893685	0.099574137

Improved Euler semilinear method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
1.0	0.099574137	0.099574137	0.099574137	0.099574137

\(x\)	\(h=.1\)	\(h=.05\)	\(h=.025\)	Exact
2.00	-1.000000000	-1.000000000	-1.000000000	-1.000000000

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
3.0	1.469458241	1.462514486	1.459217010	0.3210	0.1537	0.0753
	Approximate Solutions	Residuals

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
2.0	0.473456737	0.483227470	0.487986391	-0.3129	-0.1563	-0.0781
	Approximate Solutions	Residuals

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
3.0	1.455674816	1.455935127	1.456001289	-0.00818	-0.00207	-0.000518
	Approximate Solutions	Residuals

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
2.0	0.492862999	0.492709931	0.492674855	0.00335	0.000777	0.000187
	Approximate Solutions	Residuals