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Contributor:  testset of J.R. Cash
Discipline:  academic test
Accession:  2013





Short description:

A second order differential equations that is reduced to a first order system of 2 equations.

Applicable solvers:

all the solvers supported by the Test Set.


Plots of the solution <- click to generate the plots of the solution and the textual output



Mathematical description:



The problem is

    \begin{equation*} 	\begin{matrix} 	\lambda z'' =z, \quad z(0)=1, \quad z(1)=0, 	\end{matrix} 	\end{equation*}


    \begin{equation*} 	\begin{matrix} 	z \in \mathbb{R}, \quad t\in [0,1]. 	\end{matrix} 	\end{equation*}

We write this problem in first order form by defining y_1=z, and y_2=z', yielding a system of differential equations of the form

    \begin{equation*} 	\begin{matrix} 	\left(\begin{matrix} 	y_1\\ 	y_2 	\end{matrix}\right)'= 	\left(\begin{matrix} 	y_2\\ 	\frac{1}{\lambda}f(y_1) 	\end{matrix}\right), 	\end{matrix} 	\end{equation*}


    \begin{equation*} 	\begin{matrix} 	f(z)= z, 	\end{matrix} 	\end{equation*}


    \begin{equation*} 	\begin{matrix} 	(y_1,y_2)^T \in \mathbb{R}^{2}, \quad t \in [0,1]. 	\end{matrix} 	\end{equation*}

The boundary conditions are obtained from

    \begin{equation*} 	\begin{matrix} 	\left( 	\begin{matrix} 	1 & 0 \\ 	0 & 0 \\ 	\end{matrix} 	\right) 	\left(\begin{matrix} 	y_{1}(0)\\ 	y_{2}(0) 	\end{matrix}\right) 	+ 	\left( 	\begin{matrix} 	0 & 0 \\ 	1 & 0 \\ 	\end{matrix} 	\right) 	\left(\begin{matrix} 	y_{1}(1)\\ 	y_{2}(1) 	\end{matrix}\right)=\left(\begin{matrix} 	1 \\ 	0 	\end{matrix}\right). 	\end{matrix} 	\end{equation*}

Exact solution

    \begin{equation*} 	\begin{matrix} 	z(t) = (\exp(-t/\sqrt \lambda) - \exp((t-2)/\sqrt \lambda)) / (1 - \exp(-2/\sqrt \lambda)). 	\end{matrix} 	\end{equation*}