ODE.ADA {mcODE}

Monte Carlo ODE Solver of Akhtar et al

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 rejection method.

Usage

ODE.ADA(G, initvalue, endpoint, X0 = 0, npoints = 1000, M, R, N = 1e+05)

Arguments

G

a function having two numeric vector arguments.

initvalue

a numeric initial value

endpoint

a numeric interval endpoint value.

X0

a numeric interval starting point value.

npoints

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

M

a numeric value specifying an upper bound for F(x, g).

R

a numeric value specifying a lower bound for F(x, g).

N

an integer value specifying the number of Monte Carlo simulations used for each subinterval.

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).

References

Akhtar, M. N., Durad, M. H., and Ahmed, A. (2015). Solving initial value ordinary differential equations by Monte Carlo method. Proc. IAM, 4:149-174.

Examples

G <- function(x, g) {
    -1000*g + sin(x)
}
gTrue <- function(x) (1000*sin(x) - cos(x))/1000001 
xF <- 1; x0 <- 0
g0 <- -1/1000001  # initial value
NAkhtar <- 800
nAkhtar <- 2500
MAkhtar <- 1e-3
RAkhtar <- 1e-6
gAkhtar <- ODE.ADA(G, initvalue = g0, endpoint = xF, X0 = x0, 
    npoints = nAkhtar, M=MAkhtar, R = RAkhtar, N = NAkhtar)
plot(gAkhtar, type = "l")