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bvpT17

 

 

bvpT17
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*} z'' = -3\lambda z / (\lambda + t^{2})^{2}, \;\;\;z(-0.1) = -0.1 / \sqrt{\lambda + 0.01}, \;\;\; z(0.1) = 0.1 / \sqrt{\lambda + 0.01} \end{eqnarray*}

with

    \[ z \in \mathbb{R}, \;\;\; t\in [-0.1,0.1]. \]

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\\ f(t,y_1) \end{array}\right), \end{equation*}

where

    \begin{equation*} f(t,z) = -3\lambda z / (\lambda + t^{2})^{2}, \end{equation*}

with

    \[ (y_1,y_2)^T \in \mathbb{R}^{2} , \;\;\; t \in [-0.1,0.1]. \]

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}(-0.1)\\ y_{2}(-0.1) \end{array}\right) + \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array} \right) \left(\begin{array}{c} y_{1}(0.1)\\ y_{2}(0.1) \end{array}\right)=\left(\begin{array}{c} -0.1 / \sqrt{\lambda + 0.01}\\ 0.1 / \sqrt{\lambda + 0.01} \end{array}\right). \end{equation*}

 

Exact solution

    \[z(t) = t / \sqrt{\lambda + t^{2}}.\]

The solution has a boundary layer of width O(\sqrt\lambda) at t = 0.

 
 

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