void traveling_wave_exact ( double x, double t, double *u, double *ut, 
  double *utt, double *ux, double *uxx );
void traveling_wave_parameters ( double *a_in, double *k_in, double *w_in, 
  double *phi_in, double *xmin_in, double *xmax_in, double *t0_in, 
  double *tstop_in, double *a_out, double *k_out, double *w_out, 
  double *phi_out, double *xmin_out, double *xmax_out, double *t0_out, 
  double *tstop_out );
void traveling_wave_residual ( double t, double x, double *r );