R/bvp.R

In howardjp/cmna: Computational Methods for Numerical Analysis

## Copyright (c) 2016, James P. Howard, II <jh@jameshoward.us>
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#' @title Boundary value problems
#'
#' @name bvp
#' @rdname bvp
#'
#' @description
#' solve boundary value problems for ordinary differential equations
#'
#' @param x proposed initial \code{x}-value
#'
#' @details
#' The \code{euler} method implements the Euler method for solving
#' differential equations.  The code{midptivp} method solves initial
#' value problems using the second-order Runge-Kutta method.  The
#' \code{rungekutta4} method is the fourth-order Runge-Kutta method.
#'
#' @return a data frame of \code{x} and \code{y} values
#'
#' @examples
#' bvpexample(-2)
#' bvpexample(-1)
#' bvpexample(0)
#' bvpexample(1)
#' bvpexample(2)
#' ## (bvp.b <- bisection(bvpexample, 0, 1))
#' ## (bvp.s <- secant(bvpexample, 0))
#'
#' @importFrom utils tail
#'
#' @rdname bvp
#' @export
bvpexample <- function(x) {
    x0 <- 0
    y0 <- c(y1 = 1, y2 = x)
    yn <- 1

    odesystem <- function(x, y) {
        y1 <- y[2]
        y2 <- y[1]^2 - 2

        return(c(y1 = y1, y2 = y2))
    }

    z <- eulersys(odesystem, x0, y0, 1 / 1000, 1000)
    tail(z$y1, 1) - yn
}

#' @rdname bvp
#' @export
bvpexample10 <- function(x) {
  x0 <- 0
  y0 <- c(y1 = 1, y2 = x)
  yn <- 1

  odesystem <- function(x, y) {
      y1 <- y[2]
      y2 <- y[1]^2 - 2

      return(c(y1 = y1, y2 = y2))
  }

  z <- eulersys(odesystem, x0, y0, 1/ 10, 10)
  tail(z$y1, 1) - yn
}