#! /usr/bin/env python3
#
def flame_exact_test ( ):

#*****************************************************************************80
#
## flame_exact_test() tests flame_exact().
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    29 January 2022
#
#  Author:
#
#    John Burkardt
#
  import matplotlib.pyplot as plt
  import numpy as np
  import platform

  print ( '' )
  print ( 'flame_exact_test():' )
  print ( '  python version: ' + platform.python_version ( ) )
  print ( '  numpy version:  ' + np.version.version )
  print ( '  flame_exact() evaluates the exact solution of the flame ODE.' )
#
#  Get default parameters.
#
  t0, y0, tstop = flame_parameters ( )
#
#  Set final time to 300.
#
  tstop = 300.0
  flame_parameters ( t0, y0, tstop )

  print ( '' )
  print ( '  parameters:' )
  print ( '    t0 =    ', t0 )
  print ( '    tstop = ', tstop )
  print ( '' )

  t = np.linspace ( t0, tstop, 101 )

  plt.clf ( )

  for y0 in [ 0.005, 0.01, 0.02, 0.05 ]:

    print ( '  Evaluate for y0 = ', y0 )

    flame_parameters ( t0, y0, tstop )
    y = flame_exact ( t )

    plt.plot ( t, y, linewidth = 2 )

  s = ( 'flame_exact(): DELTA = 0.005, 0.01, 0.02, 0.05' )
  plt.title ( s )
  plt.grid ( True )
  plt.xlabel ( '<--- T --->' )
  plt.ylabel ( '<--- Y(T) --->' )
  plt.legend ( [ '0.005', '0.01', '0.02', '0.05' ] )
  filename = 'flame_exact.png'
  plt.savefig ( filename )
  print ( '  Graphics saved as "' + filename + '"' )
  plt.show ( block = False )
  plt.close ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'flame_exact_test():' )
  print ( '  Normal end of execution.' )
  return

def flame_exact ( t ):

#*****************************************************************************80
#
## flame_exact() evaluates the exact solution of flame_ode().
#
#  Discussion:
#
#    1 equation.
#
#    Moler attributes this problem to Lawrence Shampine.
#
#    The equation describes the radius of a ball of flame that
#    begins, at time 0, at DELTA.
#
#      Y(0) = DELTA
#
#    The rate of fuel consumption is proportional to the volume, and
#    the rate of fuel intake is proportional to the area of the ball.
#    We take the constant of proportionality to be 1.
#
#      Y' = Y^2 - Y^3
#
#    The data is normalized so that Y = 1 is the equilibrium solution.
#
#    The computation is to be made from T = 0 to T = 2/DELTA.
#
#    For values of DELTA close to 1, such as 0.01, the equation is
#    not stiff.  But for DELTA = 0.0001, the equation can become
#    stiff as the solution approaches the equilibrium solution Y = 1,
#    and computed solutions may be wildly inaccurate or cautious
#    solvers may take very small timesteps.
#
#    The exact solution involves the Lambert W function, defined by
#
#      W(z) * exp ( W(z) ) = z
#
#    and if we set
#
#      A = ( 1 / DELTA - 1 )
#
#    then
#
#      Y(T) = 1 / ( W(A*exp(A-T)) + 1 )
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    27 April 2024
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Cleve Moler,
#    Cleve's Corner: Stiff Differential Equations,
#    MATLAB News and Notes,
#    May 2003, pages 12-13.
#
#  Input:
#
#    real T(:): the times.
#
#  Output:
#
#    real Y(:), the exact solution values.
#
  from scipy.special import lambertw
  import numpy as np

  t0, y0, tstop = flame_parameters ( )

  a = ( 1.0 - y0 ) / y0
  y = 1.0 / np.real ( lambertw ( a * np.exp ( a - ( t - t0 ) ) ) + 1.0 )

  return y

def flame_parameters ( t0_user = None, y0_user = None, \
  tstop_user = None ):

#*****************************************************************************80
#
## flame_parameters() returns the parameters of flame_ode().
#
#  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:
#
#    28 January 2022
#
#  Author:
#
#    John Burkardt
#
#  Input:
#
#    real T0_USER: the initial time.
#
#    real Y0_USER: the initial condition.
#
#    real TSTOP_USER: the final time.
#
#  Output:
#
#    real T0: the initial time.
#
#    real Y0: the initial condition.
#
#    real TSTOP: the final time.
#
  import numpy as np
#
#  Initialize defaults.
#
  if not hasattr ( flame_parameters, "t0_default" ):
    flame_parameters.t0_default = 0.0

  if not hasattr ( flame_parameters, "y0_default" ):
    flame_parameters.y0_default = np.array ( [ 0.01 ] )

  if not hasattr ( flame_parameters, "tstop_default" ):
    flame_parameters.tstop_default = 2.0 / flame_parameters.y0_default[0]
#
#  Update defaults if input was supplied.
#
  if ( t0_user is not None ):
    flame_parameters.t0_default = t0_user

  if ( y0_user is not None ):
    flame_parameters.y0_default = y0_user

  if ( tstop_user is not None ):
    flame_parameters.tstop_default = tstop_user
#
#  Return values.
#
  t0 = flame_parameters.t0_default
  y0 = flame_parameters.y0_default
  tstop = flame_parameters.tstop_default
  
  return t0, y0, tstop

def timestamp ( ):

#*****************************************************************************80
#
## timestamp() prints the date as a timestamp.
#
#  Licensing:
#
#    This code is distributed under the MIT license. 
#
#  Modified:
#
#    21 August 2019
#
#  Author:
#
#    John Burkardt
#
  import time

  t = time.time ( )
  print ( time.ctime ( t ) )

  return

if ( __name__ == '__main__' ):
  timestamp ( )
  flame_exact_test ( )
  timestamp ( )