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<title>ODE Example programs - GNU Scientific Library -- Reference Manual</title>
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<p>
<a name="ODE-Example-programs"></a>
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<h3 class="section">25.5 Examples</h3>
<p><a name="index-Van-der-Pol-oscillator_002c-example-2076"></a>The following program solves the second-order nonlinear Van der Pol
oscillator equation,
<pre class="example"> x''(t) + \mu x'(t) (x(t)^2 - 1) + x(t) = 0
</pre>
<p class="noindent">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, y = x'(t),
<pre class="example"> x' = y
y' = -x + \mu y (1-x^2)
</pre>
<p class="noindent">The program begins by defining functions for these derivatives and
their Jacobian,
<pre class="example"><pre class="verbatim"> #include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv.h>
int
func (double t, const double y[], double f[],
void *params)
{
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)
{
double mu = *(double *)params;
gsl_matrix_view dfdy_mat
= gsl_matrix_view_array (dfdy, 2, 2);
gsl_matrix * m = &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)
{
const gsl_odeiv_step_type * T
= gsl_odeiv_step_rk8pd;
gsl_odeiv_step * s
= gsl_odeiv_step_alloc (T, 2);
gsl_odeiv_control * c
= gsl_odeiv_control_y_new (1e-6, 0.0);
gsl_odeiv_evolve * e
= gsl_odeiv_evolve_alloc (2);
double mu = 10;
gsl_odeiv_system sys = {func, jac, 2, &mu};
double t = 0.0, t1 = 100.0;
double h = 1e-6;
double y[2] = { 1.0, 0.0 };
while (t < t1)
{
int status = gsl_odeiv_evolve_apply (e, c, s,
&sys,
&t, t1,
&h, y);
if (status != GSL_SUCCESS)
break;
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv_evolve_free (e);
gsl_odeiv_control_free (c);
gsl_odeiv_step_free (s);
return 0;
}
</pre></pre>
<p class="noindent">For functions with multiple parameters, the appropriate information
can be passed in through the <var>params</var> argument using a pointer to
a struct.
<p>The main loop of the program evolves the solution from (y, y') =
(1, 0) at t=0 to t=100. The step-size h is
automatically adjusted by the controller to maintain an absolute
accuracy of <!-- {$10^{-6}$} -->
10^{-6} in the function values <var>y</var>.
<p class="noindent">To obtain the values at user-specified positions, rather than those
chosen automatically by the control function, the main loop can be modified
to advance the solution from one chosen point to the next. For example, the
following main loop prints the solution at the points t_i = 0,
1, 2, \dots, 100,
<pre class="example"> for (i = 1; i <= 100; i++)
{
double ti = i * t1 / 100.0;
while (t < ti)
{
gsl_odeiv_evolve_apply (e, c, s,
&sys,
&t, ti, &h,
y);
}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
</pre>
<p class="noindent">Note that arbitrary values of t_i can be used for each stage of
the integration. The equally spaced points in this example are just
used as an illustration.
<p>It is also possible to work with a non-adaptive integrator, using only
the stepping function itself. The following program uses the <code>rk4</code>
fourth-order Runge-Kutta stepping function with a fixed stepsize of
0.01,
<pre class="example"><pre class="verbatim"> int
main (void)
{
const gsl_odeiv_step_type * T
= gsl_odeiv_step_rk4;
gsl_odeiv_step * s
= gsl_odeiv_step_alloc (T, 2);
double mu = 10;
gsl_odeiv_system sys = {func, jac, 2, &mu};
double t = 0.0, t1 = 100.0;
double h = 1e-2;
double y[2] = { 1.0, 0.0 }, y_err[2];
double dydt_in[2], dydt_out[2];
/* initialise dydt_in from system parameters */
GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);
while (t < t1)
{
int status = gsl_odeiv_step_apply (s, t, h,
y, y_err,
dydt_in,
dydt_out,
&sys);
if (status != GSL_SUCCESS)
break;
dydt_in[0] = dydt_out[0];
dydt_in[1] = dydt_out[1];
t += h;
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv_step_free (s);
return 0;
}
</pre></pre>
<p class="noindent">The derivatives must be initialized for the starting point t=0
before the first step is taken. Subsequent steps use the output
derivatives <var>dydt_out</var> as inputs to the next step by copying their
values into <var>dydt_in</var>.
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