A.3.3: Section 3.3 Answers
- Page ID
- 43759
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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" |
\(2.0\) | \(-0.750911103\) | \(-0.750912294\) | \(-0.750912367\) | \(-0.750912371\) |
13. Applying variation of parameters to the given initial value problem yields \(y = ue^{−3x}\), where \((A) u' = 1 − 4x + 3x^{2} − 4x^{3}, u(0) = −3\). 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.
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" |
\(3.0\) | \(0.967523147\) | \(0.967523152\) | \(0.967523153\) | \(0.967523153\) |
17.
Runge-Kutta method | ||||
\(x\) | \(h=0.0500\) | \(h=0.0250\) | \(h=0.0125\) | "Exact" |
\(1.50\) | \(0.343839158\) | \(0.343784814\) | \(0.343780796\) | \(0.343780513\) |
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" |
\(3.0\) | \(16.666666988\) | \(16.666666687\) | \(16.666666668\) | \(16.666666667\) |