# include <math.h>
# include <stdlib.h>

# include "sine_gordon_exact.h"

/******************************************************************************/

void sine_gordon_exact ( double a, double x, double y, double *u, double *uxy )

/******************************************************************************/
/*
  Purpose:

    sine_gordon_exact() evaluates an exact solution of the Sine-Gordeon PDE.

  Discussion:

    uxy = sin ( u )

    This is a one-soliton solution.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    02 May 2024

  Author:

    John Burkardt

  Reference:

    Daniel Arrigo,
    Analytical Techniques for Solving Nonlinear Partial Differential Equations,
    Morgan and Clayfoot, 2019,
    ISBN: 978 168 173 5351.

  Input:

    double A: a parameter.

    double X, Y: the X and Y coordinates of a point.

  Output:

    double *U, *UXY: the values of the solution, and its first 
    mixed derivative at (X,Y).
*/
{
  *u = 4.0 * atan ( exp ( a * x + y / a ) );

  *uxy = 4.0 * ( exp ( a * x + y / a ) - exp ( 3 * a * x + 3 * y / a ) ) 
    / pow ( 1.0 + exp ( 2 * a * x + 2 * y / a ), 2 );

  return;
}
/******************************************************************************/

double sine_gordon_residual ( double u, double uxy )

/******************************************************************************/
/*
  Purpose:

    sine_gordon_residual() evaluates the residual for the Sine-Gordeon PDE.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    02 May 2024

  Author:

    John Burkardt

  Reference:

    Daniel Arrigo,
    Analytical Techniques for Solving Nonlinear Partial Differential Equations,
    Morgan and Clayfoot, 2019,
    ISBN: 978 168 173 5351.

  Input:

    double U, UXY: the values of the solution, and its
    first mixed derivative.

  Output:

    double R, the residual.
*/
{
  double r;

  r = uxy - sin ( u );

  return r;
}