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.
\(x\) |
\(h=0.5\) |
\(h=0.025\) |
\(h=0.0125\) |
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 \((A) u' = 1 − 2x, u(0) = 2\). 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 semilinar 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 semilinar 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" |
\(2.0\) |
\(-0.750335016\) |
\(-0.750775571\) |
\(-0.750879100\) |
\(-0.751331499\) |