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<a name="ODE-Example-programs"></a>
<div class="header">
<p>
Next: <a href="ODE-References-and-Further-Reading.html#ODE-References-and-Further-Reading" accesskey="n" rel="next">ODE References and Further Reading</a>, Previous: <a href="Driver.html#Driver" accesskey="p" rel="previous">Driver</a>, Up: <a href="Ordinary-Differential-Equations.html#Ordinary-Differential-Equations" accesskey="u" rel="up">Ordinary Differential Equations</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-19"></a>
<h3 class="section">27.6 Examples</h3>
<a name="index-Van-der-Pol-oscillator_002c-example"></a>
<p>The following program solves the second-order nonlinear Van der Pol
oscillator equation,
</p>
<div class="example">
<pre class="example">u''(t) + \mu u'(t) (u(t)^2 - 1) + u(t) = 0
</pre></div>

<p>This can be converted into a first order system suitable for use with
the routines described in this chapter by introducing a separate
variable for the velocity, <em>v = u'(t)</em>,
</p>
<div class="example">
<pre class="example">u' = v
v' = -u + \mu v (1-u^2)
</pre></div>

<p>The program begins by defining functions for these derivatives and
their Jacobian. The main function uses driver level functions to solve
the problem. The program evolves the solution from <em>(u, v) = (1,
0)</em> at <em>t=0</em> to <em>t=100</em>.  The step-size <em>h</em> is
automatically adjusted by the controller to maintain an absolute
accuracy of <em>10^{-6}</em> in the function values <em>(u, v)</em>.  
The loop in the example prints the solution at the points
<em>t_i = 1, 2, \dots, 100</em>.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_errno.h&gt;
#include &lt;gsl/gsl_matrix.h&gt;
#include &lt;gsl/gsl_odeiv2.h&gt;

int
func (double t, const double y[], double f[],
      void *params)
{
  (void)(t); /* avoid unused parameter warning */
  double mu = *(double *)params;
  f[0] = y[1];
  f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
  return GSL_SUCCESS;
}

int
jac (double t, const double y[], double *dfdy, 
     double dfdt[], void *params)
{
  (void)(t); /* avoid unused parameter warning */
  double mu = *(double *)params;
  gsl_matrix_view dfdy_mat 
    = gsl_matrix_view_array (dfdy, 2, 2);
  gsl_matrix * m = &amp;dfdy_mat.matrix; 
  gsl_matrix_set (m, 0, 0, 0.0);
  gsl_matrix_set (m, 0, 1, 1.0);
  gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
  gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
  dfdt[0] = 0.0;
  dfdt[1] = 0.0;
  return GSL_SUCCESS;
}

int
main (void)
{
  double mu = 10;
  gsl_odeiv2_system sys = {func, jac, 2, &amp;mu};

  gsl_odeiv2_driver * d = 
    gsl_odeiv2_driver_alloc_y_new (&amp;sys, gsl_odeiv2_step_rk8pd,
				  1e-6, 1e-6, 0.0);
  int i;
  double t = 0.0, t1 = 100.0;
  double y[2] = { 1.0, 0.0 };

  for (i = 1; i &lt;= 100; i++)
    {
      double ti = i * t1 / 100.0;
      int status = gsl_odeiv2_driver_apply (d, &amp;t, ti, y);

      if (status != GSL_SUCCESS)
	{
	  printf (&quot;error, return value=%d\n&quot;, status);
	  break;
	}

      printf (&quot;%.5e %.5e %.5e\n&quot;, t, y[0], y[1]);
    }

  gsl_odeiv2_driver_free (d);
  return 0;
}
</pre></div>

<p>The user can work with the lower level functions directly, as in
the following example. In this case an intermediate result is printed
after each successful step instead of equidistant time points. 
</p>
<div class="example">
<pre class="verbatim">int
main (void)
{
  const gsl_odeiv2_step_type * T 
    = gsl_odeiv2_step_rk8pd;

  gsl_odeiv2_step * s 
    = gsl_odeiv2_step_alloc (T, 2);
  gsl_odeiv2_control * c 
    = gsl_odeiv2_control_y_new (1e-6, 0.0);
  gsl_odeiv2_evolve * e 
    = gsl_odeiv2_evolve_alloc (2);

  double mu = 10;
  gsl_odeiv2_system sys = {func, jac, 2, &amp;mu};

  double t = 0.0, t1 = 100.0;
  double h = 1e-6;
  double y[2] = { 1.0, 0.0 };

  while (t &lt; t1)
    {
      int status = gsl_odeiv2_evolve_apply (e, c, s,
                                           &amp;sys, 
                                           &amp;t, t1,
                                           &amp;h, y);

      if (status != GSL_SUCCESS)
          break;

      printf (&quot;%.5e %.5e %.5e\n&quot;, t, y[0], y[1]);
    }

  gsl_odeiv2_evolve_free (e);
  gsl_odeiv2_control_free (c);
  gsl_odeiv2_step_free (s);
  return 0;
}
</pre></div>

<p>For functions with multiple parameters, the appropriate information
can be passed in through the <var>params</var> argument in
<code>gsl_odeiv2_system</code> definition (<var>mu</var> in this example) by using
a pointer to a struct.
</p>

<p>It is also possible to work with a non-adaptive integrator, using only
the stepping function itself,
<code>gsl_odeiv2_driver_apply_fixed_step</code> or
<code>gsl_odeiv2_evolve_apply_fixed_step</code>. The following program uses
the driver level function, with fourth-order
Runge-Kutta stepping function with a fixed stepsize of
0.001.
</p>
<div class="example">
<pre class="verbatim">int
main (void)
{
  double mu = 10;
  gsl_odeiv2_system sys = { func, jac, 2, &amp;mu };

  gsl_odeiv2_driver *d =
    gsl_odeiv2_driver_alloc_y_new (&amp;sys, gsl_odeiv2_step_rk4,
                                   1e-3, 1e-8, 1e-8);

  double t = 0.0;
  double y[2] = { 1.0, 0.0 };
  int i, s;

  for (i = 0; i &lt; 100; i++)
    {
      s = gsl_odeiv2_driver_apply_fixed_step (d, &amp;t, 1e-3, 1000, y);

      if (s != GSL_SUCCESS)
        {
          printf (&quot;error: driver returned %d\n&quot;, s);
          break;
        }

      printf (&quot;%.5e %.5e %.5e\n&quot;, t, y[0], y[1]);
    }

  gsl_odeiv2_driver_free (d);
  return s;
}
</pre></div>

<hr>
<div class="header">
<p>
Next: <a href="ODE-References-and-Further-Reading.html#ODE-References-and-Further-Reading" accesskey="n" rel="next">ODE References and Further Reading</a>, Previous: <a href="Driver.html#Driver" accesskey="p" rel="previous">Driver</a>, Up: <a href="Ordinary-Differential-Equations.html#Ordinary-Differential-Equations" accesskey="u" rel="up">Ordinary Differential Equations</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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