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bvpT3

 

 

 

bvpT3
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{eqnarray*} 							\lambda z'' = - (2 + \cos(\pi t)) z' + z -(1 +\lambda \pi^{2}) \cos(\pi t) - (2 + \cos(\pi t)) \pi \sin(\pi t), \\ z(-1)=-1 \;\;\; z(1)=-1, 							\end{eqnarray*}

with

    \[ 							z \in \mathbb{R}, \;\;\; t\in [-1,1]. 							\]

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*} 							\left(\begin{array}{c} 							y_1\\ 							y_2 							\end{array}\right)'= 							\left(\begin{array}{c} 							y_2\\ 							\frac{1}{\lambda}f(t,y_1,y_2) 							\end{array}\right), 							\end{equation*}

where

    \begin{equation*} 							f(t,z,z') = - (2 + \cos(\pi t)) z' + z -(1 +\lambda \pi^{2}) \cos(\pi t) - (2 + \cos(\pi t)) \pi \sin(\pi t), 							\end{equation*}

and

    \[ 							(y_1,y_2)^T \in \mathbb{R}^{2}, \ \ \ t \in [-1,1]. 							\]

The boundary conditions are obtained from

    \begin{equation*} 							\left( 							\begin{array}{cc} 							1 & 0 \\ 							0 & 0 \\ 							\end{array} 							\right) 							\left(\begin{array}{c} 							y_{1}(-1)\\ 							y_{2}(-1) 							\end{array}\right) 							+ 							\left( 							\begin{array}{cc} 							0 & 0 \\ 							1 & 0 \\ 							\end{array} 							\right) 							\left(\begin{array}{c} 							y_{1}(1)\\ 							y_{2}(1) 							\end{array}\right)=\left(\begin{array}{c} 							-1 \\ 							-1 							\end{array}\right). 							\end{equation*}

Exact solution

    \[z(t) = \cos(\pi t).\]

 

 

 

 

The problem has no turning points and the solution is smooth.

 

 

 

 

 

 

 

 

 

 

 

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