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bvpT33

 

 

bvpT33
Contributor:  testset of J.R. Cash
Discipline:  academic test
Accession:  2013

 

 

 

 

 

 

Short description:

A syetm of two coupled  differential equations that is reduced to a first order system of 6 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 z''' - y y',\,\, \lambda y'' = y z' - z y',\\ y(0) =-1,\, y(1) = 1, \\ z(0) =z'(0) = z(1) =z'(1) = 0, \end{eqnarray*}

with

    \[ z,y \in \mathbb{R}, \;\;\; t\in [0,1]. \]

We write this problem in first order form by defining y_1=y,\,y_2 = y',\,y_3 = z,\,y_4 = z',\, y_5 = z'' and y_6 = z''', yielding a system of differential equations of the form

    \begin{equation*} \left(\begin{array}{c} y_1\\ y_2\\ y_3\\ y_4\\ y_5\\ y_6 \end{array}\right)'= \left(\begin{array}{c} y_2\\ \frac{1}{\lambda}y_1 y_4 - y_3 y_2\\ y_4\\ y_5\\ y_6\\ \frac{1}{\lambda}(-y_3 y_6 - y_1 y_2 ) \end{array}\right), \end{equation*}

with

    \[ (y_1,y_2,y_3,y_4,y_5,y_6)^T \in \mathbb{R}^{6}, \;\;\; t \in [0,1]. \]

The boundary conditions are obtained from

    \begin{equation*} \left( \begin{array}{cccccc} 1 & 0 & 0 & 0&0 & 0\\ 0 & 0 & 1 & 0&0 & 0\\ 0 & 0 & 0 & 1&0 & 0\\ 0 & 0 & 0 & 0&0 & 0\\ 0 & 0 & 0 & 0&0 & 0\\ 0 & 0 & 0 & 0&0 & 0 \end{array} \right) \left(\begin{array}{c} y_{1}(0)\\ y_{2}(0)\\ y_{3}(0)\\ y_{4}(0)\\ y_{5}(0)\\ y_{6}(0) \end{array}\right) + \left( \begin{array}{cccccc} 0 & 0 & 0 & 0&0 & 0\\ 0 & 0 & 0 & 0&0 & 0\\ 0 & 0 & 0 & 0&0 & 0\\ 1 & 0 & 0 & 0&0 & 0\\ 0 & 0 & 1 & 0&0 & 0\\ 0 & 0 & 0 & 1&0 & 0 \end{array} \right) \left(\begin{array}{c} y_{1}(1)\\ y_{2}(1)\\ y_{3}(1)\\ y_{4}(1) \\ y_{5}(1) \\ y_{6}(1) \end{array}\right)=\left(\begin{array}{c} -1 \\ 0 \\ 0\\ 1\\ 0\\ 0 \end{array}\right). \end{equation*}

 
 

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