heat <- function ( u, alpha, xdelta, tdelta, n )

#*****************************************************************************80
#
## heat solves the 1D heat equation using the forward time centered space method.
#
#  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:
#
#    29 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.
#
{
  m <- length ( u )
  uarray <- matrix ( u, nrow = 1 )
  newu <- u

  h <- alpha * tdelta / xdelta^2

  for ( i in 1 : n )
  {
    for ( j in 2 : ( m - 1 ) )
    {
      ustep <- u[j-1] - 2.0 * u[j] + u[j+1]
      newu[j] <- u[j] + h * ustep
    }
    u <- newu
    uarray <- rbind ( uarray, u )
  }

  return ( uarray )
}