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

  from sympy import cos
  from sympy import diff
  from sympy import pi
  from sympy import pprint
  from sympy import symbols
  import sympy

  print ( '' )
  print ( 'bvp_example():' )
  print ( '  sympy version: ' + sympy.__version__ )
  print ( '  We define a boundary value problem of the form:' )
  print ( '    -u" + u u\' = f(x).' )
  print ( '  We want the exact solution to be' )
  print ( '    u(x) = (2/pi) * (1 - cos (pi*x/2) ).' )
  print ( '  We need to manufacture the right hand side f(x).' )

  x = symbols ( 'x' )

  u0 = 2 * ( 1 - cos ( pi * x / 2 ) ) / pi
  print ( '' )
  print ( '  u(x):' )
  print ( '' )
  pprint ( u0 )

  u1 = diff ( u0, x )
  print ( '' )
  print ( '  dudx(x):' )
  print ( '' )
  pprint ( u1 )

  u2 = diff ( u0, x, 2 )
  print ( '' )
  print ( '  d2udx2(x):' )
  print ( '' )
  pprint ( u2 )

  f = - u2 + u0 * u1
  print ( '' )
  print ( '  f(x) = -u" + u u\'' )
  print ( '' )
  pprint ( f )

  return

if ( __name__ == "__main__" ):
  bvp_example ( )