mcODE (version 1.1)

ODE.MVT: Monte Carlo ODE Solver Based on Mean Value Theorem

Description

Given g' = G(x, g) and g(x0) = g0 satisfying conditions for existence and uniqueness of solution, a Monte Carlo approximation to the solution is found. The method is essentially a Monte Carlo version of Euler and Runge-Kutta type methods.

Usage

ODE.MVT(G, initvalue, endpoint, initpoint = 0, Niter = 2, npoints = 1000)

Value

A list consisting of

x

a vector of length npoints, consisting of the x-coordinate of the solution.

y

a vector of length npoints, consisting of the y-coordinate of the solution, i.e. g(x).

Arguments

G

a function having two arguments.

initvalue

a numeric initial value

endpoint

a numeric interval endpoint value.

initpoint

a numeric interval starting point value.

Niter

an integer value specifying the number of iterations at each step.

npoints

an integer value specifying the number of subintervals to build the approximation on.

Author

Braun, W.J.

References

Braun, W.J. (2024) Preprint.

Examples

# Initial Value Problem:
G <- function(x, g) {
    -1000*g + sin(x)
}
g0 <- -1/1000001; x0 <- 0 # initial condition
xF <- 1 # interval endpoint
nMVT <- 1000 # number of subintervals used
# Monte Carlo solution based on one iteration:
ghat1 <- ODE.MVT(G, initvalue = g0, endpoint = xF, initpoint = x0, Niter = 1, npoints = nMVT) 
# Monte Carlo solution based on five iterations:
ghat5 <- ODE.MVT(G, initvalue = g0, endpoint = xF, initpoint = x0, Niter = 5, npoints = nMVT) 
gTrue <- function(x) (1000*sin(x) - cos(x))/1000001 # true solution
oldpar <- par(mfrow=c(1,3))
curve(gTrue(x)) # 
lines(ghat1, col = 2, lty = 2, ylab="g(x)")
plot(abs(gTrue(ghat1$x) - ghat1$y) ~ ghat1$x, xlab = "x", 
    ylab = "Absolute Error", type = "l", ylim = c(0, 2e-6), main="1 Iteration")
plot(abs(gTrue(ghat5$x) - ghat5$y) ~ ghat5$x, xlab = "x", 
    ylab = "Absolute Error", type = "l", ylim = c(0, 2e-6), main="5 Iterations")
par(oldpar)