pracma (version 0.5-3)

ode23: Runge-Kutta

Description

Runge-Kutta (2, 3)-method with variable step size.

Usage

ode23(f, t0, tfinal, y0, ..., rtol = 1e-3, atol = 1e-6)

Arguments

f

function in the differential equation $y' = f(x, y)$; defined as a function $R \times R^m \ R^m$, where $m$ is the number of equations.

t0, tfinal

start and end points of the interval.

y0

starting values as column vector; for $m$ equations u0 needs to be a vector of length m.

rtol, atol

relative and absolute tolerance.

...

Additional parameters to be passed to the function.

Value

• List with components t for grid (or `time') points between t0 and tfinal, and y an n-by-m matrix with solution variables in columns, i.e. each row contains one time stamp.

Details

ode23 is an integration method for systems of ordinary differential equations using second and third order Runge-Kutta-Fehlberg formulas with automatic step-size.

References

Moler, C. (2004). Numerical Computing with Matlab. Revised Reprint, SIAM. http://www.mathworks.com/moler/chapters.html.

See Also

rk4sys, deval

Examples

##  Example1: Three-body problem
f <- function(t, y)
		as.matrix(c(y[2]*y[3], -y[1]*y[3], 0.51*y[1]*y[2]))
y0 <- as.matrix(c(0, 1, 1))
t0 <- 0; tf <- 20
sol <- ode23(f, t0, tf, y0, rtol=1e-5, atol=1e-10)
matplot(sol$t, sol$y, type = "l", lty = 1, lwd = c(2, 1, 1),
        col = c("darkred", "darkblue", "darkgreen"),
        xlab = "Time [min]", ylab= "",
        main = "Three-body Problem")
grid()

##  Example2: Van der Pol Equation
#   x'' + (x^2 - 1) x' + x = 0
f <- function(t, x)
        as.matrix(c(x[1] * (1 - x[2]^2) -x[2], x[1]))
t0 <- 0; tf <- 20
x0 <- as.matrix(c(0, 0.25))
sol <- ode23(f, t0, tf, x0)
plot(c(0, 20), c(-3, 3), type = "n",
     xlab = "Time", ylab = "", main = "Van der Pol Equation")
lines(sol$t, sol$y[, 1], col = "blue")
lines(sol$t, sol$y[, 2], col = "darkgreen")
grid()