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bvpT31

 

 

bvpT31
Contributor:  testset of J.R. Cash
Discipline:  academic test
Accession:  2013

 

 

 

Short description:

A system of four coupled differential equations that is reduced to a first order system of 4 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*} \begin{array}{ll} z' = \sin \theta,\, \theta' = M,\, \lambda M' = -Q,\, \lambda Q' = (z-1) \cos \theta - MT,\, T = \sec \theta + \lambda Q \tan \theta, &\\ z(0) = z(1) = 0, \, M(0) = M(1) = 0. \\ \end{array} \end{eqnarray*}

with

    \[ z \in \mathbb{R}^{2} , \;\;\; t\in [0,1]. \]

We write this problem in first order form by defining y_1=z,\,y_2 = \theta,\,y_3 = M and y_4 = Q, yielding a system of differential equations of the form

    \begin{equation*} \left(\begin{array}{c} y_1\\ y_2\\ y_3\\ y_4 \end{array}\right)'= \left(\begin{array}{c} \sin y_2 \\ y_3\\ - y_4/ \lambda\\ f(y_1,y_2,y_3,y_4) \end{array}\right), \end{equation*}

where

    \begin{equation*} f(z,\theta,M,Q ) = \frac{1}{\lambda}((z - 1) \cos \theta - M \sec \theta) + \lambda Q \tan \theta) \end{equation*}

with

    \[ (y_1,y_2,y_3,y_4)^T \in \mathbb{R}^{4} , \;\;\; t \in [0,1]. \]

The boundary conditions are obtained from

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

This equation models nonlinear elastic beams.

 
 

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