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

# include "kdv_exact.h"

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

void kdv_exact_rational ( double x, double t, double *u, double *ut, double *ux, 
  double *uxx, double *uxxx )

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

    kdv_exact_rational(): exact solution of KDV PDE using a rational function.

  Discussion:

    This solution u(x,t) satisfies the Korteweg-Devries PDE:
      u' - 6 u ux + uxxx = 0

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    01 May 2024

  Author:

    John Burkardt

  Reference:

    John D Cook,
    Rational solution to the Korteweg-De Vries equation,
    13 November 2023,
    https://www.johndcook.com/blog/2023/11/13/rational-kdv/

  Input:

    double x: the position.

    double t: the time.

  Output:

    double *u, *ut, *ux, *uxx, *uxxx:
    the values of the solution and its derivatives.
*/
{
  double t2;
  double t3;
  double x2;
  double x3;
  double x5;
  double x6;
  double x8;
  double x9;

  t2 = pow ( t, 2 );
  t3 = pow ( t, 3 );
  x2 = pow ( x, 2 );
  x3 = pow ( x, 3 );
  x5 = pow ( x, 5 );
  x6 = pow ( x, 6 );
  x8 = pow ( x, 8 );
  x9 = pow ( x, 9 );

  *u = 6.0 * x * ( x3 - 24.0 * t ) / pow ( x3 + 12.0 * t, 2 );

  *ut = - 288.0 * x * ( x3 - 6.0 * t ) / pow ( x3 + 12.0 * t, 3 );

  *ux = - 12.0 * ( x6 - 84.0 * t * x3 + 144.0 * t2 ) / pow ( x3 + 12.0 * t, 3 );

  *uxx =  36.0 * ( x8 - 192.0 * t * x5 + 1440.0 * t2 * x2 ) 
    / pow ( x3 + 12.0 * t, 4 );

  *uxxx = - 144.0 * x * ( x9 - 360.0 * t * x6 + 6480.0 * t2 * x3 - 8640.0 * t3 ) 
    / pow ( x3 + 12.0 * t, 5 );

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

void kdv_exact_sech ( double x, double t, double *u, double *ut, double *ux, 
  double *uxx, double *uxxx )

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

    kdv_exact_sech(): exact solution of KDV PDE using sech().

  Discussion:

    This solution u(x,t) satisfies the Korteweg-Devries PDE:
      u' - 6 u ux + uxxx = 0

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    01 May 2024

  Author:

    John Burkardt

  Reference:

    John D Cook,
    Solitons and the KdV equation,
    03 November 2023,
    https://www.johndcook.com/blog/2023/11/03/solitons-and-the-kdv-equation/

  Input:

    double x: the position.

    double t: the time.

  Output:

    double *u, *ut, *ux, *uxx, *uxxx:
    the values of the solution and its derivatives.
*/
{
  double a;
  double argument;
  double sa;
  double ta;
  double v;
/*
  Retrieve parameters a and v.
*/
  kdv_parameters ( NULL, NULL, NULL, NULL, &a, &v, NULL, NULL );

  argument = 0.5 * sqrt ( v ) * ( x - v * t - a );
  sa = 1.0 / cosh ( argument );
  ta = tanh ( argument );

  *u =    - 0.5  *       v        * pow ( sa, 2 );

  *ut =   - 0.5  * pow ( v, 2.5 ) * pow ( sa, 2 ) * ta;

  *ux =     0.5  * pow ( v, 1.5 ) * pow ( sa, 2 ) * ta;

  *uxx =  + 0.25 * pow ( v, 2 )   * pow ( sa, 4 ) 
          - 0.5  * pow ( v, 2 )   * pow ( sa, 2 ) * pow ( ta, 2 );

  *uxxx =        - pow ( v, 2.5 ) * pow ( sa, 4 ) * ta 
          + 0.5  * pow ( v, 2.5 ) * pow ( sa, 2 ) * pow ( ta, 3 );

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

void kdv_parameters ( double *a_in, double *v_in, double *t0_in, 
  double *tstop_in, double *a_out, double *v_out, double *t0_out, 
  double *tstop_out )

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

    kdv_parameters(): parameters for the Korteweg-Devries PDE.

  Discussion:

    If input values are specified, this resets the default parameters.
    Otherwise, the output will be the current defaults.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    01 May 2024

  Author:

    John Burkardt

  Input:

    double *a_user: the phase

    double *v_user: the velocity

    double *t0_user: the initial time

    double *tstop_user: the final time.

  Output:

    double *a_out: the phase.

    double *v_out: the velocity

    double *t0_out: the initial time

    double *tstop_out: the final time.
*/
{
  static double a_default = 0.0;
  static double t0_default = 0.0;
  static double tstop_default = 10.0;
  static double v_default = 1.0;
/*
  New values, if supplied on input, overwrite the current values.
*/
  if ( a_in )
  {
    a_default = *a_in;
  }

  if ( v_in )
  {
    v_default = *v_in;
  }

  if ( t0_in )
  {
    t0_default = *t0_in;
  }

  if ( tstop_in )
  {
    tstop_default = *tstop_in;
  }
/*
  The current values are copied to the output.
*/
  if ( a_out )
  {
    *a_out = a_default;
  }

  if ( v_out )
  {
    *v_out = v_default;
  }

  if ( t0_out )
  {
    *t0_out = t0_default;
  }

  if ( tstop_out )
  {
    *tstop_out = tstop_default;
  }
  
  return;
}
/******************************************************************************/

double kdv_residual ( double u, double ut, double ux, double uxxx )

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

    kdv_residual(): evaluate the residual of KDV PDE.

  Discussion:

    The residual of the Korteweg-Devries PDE:
      u' - 6 u ux + uxxx

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    30 April 2024

  Author:

    John Burkardt

  Reference:

    John D Cook,
    Solitons and the KdV equation,
    03 November 2023,
    https://www.johndcook.com/blog/2023/11/03/solitons-and-the-kdv-equation/

  Input:

    double u, ut, ux, uxxx:
    the values of the solution and its derivatives.

  Output:

    double r: the residual.
*/
{
  double r;

  r = ut - 6.0 * u * ux + uxxx;

  return r;
}