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bvpT4

 

 

 

bvpT4
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'' = - z' + (1 +\lambda) z, \;\;\; z(-1) = 1 + \exp(-2), \;\;\; z(1) = 1 + \exp(-2(1+\lambda)/\lambda), \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(y_1,y_2) \end{array}\right), \end{equation*}

where

    \begin{equation*} f(z,z') = -z' + (1 +\lambda) z, \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 + \exp(-2) \\ 1 + \exp(-2(1+\lambda)/\lambda) \end{array}\right). \end{equation*}

Exact solution

    \[z(t) = \exp(t -1) + \exp(-(1 +\lambda)(1 +t) /\lambda).\]

The solution has a boundary layer of width O(\epsilon) at t = -1.

 
 

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