bvpexample <- function ( yp0 )

#*****************************************************************************80
#
## bvpexample attempts to match a boundary value problem.
#
#  Discussion:
#
#    The BVP is:
#      y''(x,y) = y^2 - 2.0
#      y(0.0) = 1.0
#      y(1.0) = 1.0
#    Instead, we solve the IVP
#      z''(x,z) = z^2 - 2.0
#      z(0.0) = 1.0
#      z'(0.0) = yp0
#    and return
#     value = z(1.0) - y(1.0)
#    
#  Licensing:
#
#    Copyright 2016 James P. Howard, II
#
#    The computer code and data files on this web page are distributed under
#    https://opensource.org/licenses/BSD-2-Clause, the BSD-2-Clause license.
#
#  Modified:
#
#    28 March 2020
#
#  Author:
#
#    Original R code by James Howard;
#    Modifications by John Burkardt.
#
#  Reference:
#
#    James Howard,
#    Computational Methods for Numerical Analysis with R,
#    CRC Press, 2017
#    ISBN13: 978-1-4987-2363-3.
#
{
  source ( "/home/burkardt/public_html/r_src/eulersys/eulersys.R" )  

  x0 <- 0.0
  y0 <- c ( y1 = 1.0, y2 = yp0 )
  yn <- 1.0

  odesystem <- function ( x, y )
  {
    y1 <- y[2]
    y2 <- y[1]^2 - 2.0
    return ( c ( y1 = y1, y2 = y2 ) )
  }

  n = 1000
  h <- 1.0 / n

  z <- eulersys ( odesystem, x0, y0, h, n )

  value <- tail ( z$y1, 1 ) - yn

  return ( value )
}