4: Linear Systems of Ordinary Differential Equations (LSODE)
- Page ID
- 15010
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definition of Linear Systems of Ordinary Differential Equations (LSODE)
\begin{eqnarray*}
y_{1}'(t)& =& \alpha_{11}(t)y_{1}(t)\ +\alpha_{12}(t)y_{2}+\ldots+\alpha_{1n}(t)y_{n}+f_{1}(t)\\
y_{2}'(t)& =& \alpha_{21}(t)y_{1}(t)\ +\alpha_{22}(t)y_{2}+\ldots+\alpha_{2n}(t)y_{n}+f_{2}(t)\\
y_{n}'(t)& =& \alpha_{n1}(t)y_{1}(t)\ +\alpha_{n2}(t)y_{2}+\ldots+\alpha_{nn}(t)y_{n}+f_{n}(t),\\
\end{eqnarray*}
where \(y_{i}'(t) :=\displaystyle \frac{dy_{i}(t)}{dt}\) denotes the first derivative of functions \(y_{i}(t),\ i= 1,2, \ldots,\ n\), with respect to \(t\).
Matrix Form of LSODE
\(\lfloor_{y_{n}'(t)}|^{y_{2}'(t)}\lceil 1\ldots]=\lfloor\alpha_{n1}(t)|^{\alpha_{21}.(t)}\lceil\alpha_{11}.(t)\alpha_{n2}(t)\alpha_{22}.(t)\alpha_{12}.(t).\cdot.\cdot.\ \alpha_{nn}(t)\alpha_{2n}.(t)\alpha_{1n}.(t)]\lfloor y_{n}(t)|^{y_{2}(t)}\lceil y_{1}.(t)]+\lfloor_{f_{n}(t)}|^{f_{2}(t)}\lceil f_{1}.(t)]\),
or in matrix form
\begin{eqnarray*}
y^{\prec}(t)=A(t)\vec{y}(t)+f^{\rightarrow}(t),
\end{eqnarray*}
where
\(\vec{y}(t)=|\lceil y_{2}'(ty_{1}'(t)\rceil_{1}, A(t) =|\lceil\alpha_{21}.(t\alpha_{11}.(t) \alpha_{22}.(t\alpha_{12}.(t) \alpha_{2n}.(t\alpha_{1n}.(t)\rceil_{1}, f^{\rightarrow}(t)=|\lceil f_{2}(tf_{1}(t)\rceil_{1} \lfloor y_{n}'(t) \rfloor \lfloor\alpha_{n1}(t) \alpha_{n2}(t) \alpha_{nn}(t) \rfloor \lfloor f_{n}(t) \rfloor\), (3), \(A(t)\)in (3) is called coefficient matrix of (2) and \(f^{\rightarrow}(t)\)