ode function

Ordinary Differential Equations

Solves a numerical or symbolic system of ordinary differential equations.

ode(
  f,
  var,
  times,
  timevar = NULL,
  params = list(),
  method = "rk4",
  drop = FALSE
)

Arguments

• f: vector of characters, or a function returning a numeric vector, giving the values of the derivatives in the ODE system at time timevar. See examples.
• var: vector giving the initial conditions. See examples.
• times: discretization sequence, the first value represents the initial time.
• timevar: the time variable used by f, if any.
• params: list of additional parameters passed to f.
• method: the solver to use. One of "rk4" (Runge-Kutta) or "euler" (Euler).
• drop: if TRUE, return only the final solution instead of the whole trajectory.

Returns

Vector of final solutions if drop=TRUE, otherwise a matrix with as many rows as elements in times and as many columns as elements in var.

Examples

## ==================================================
## Example: symbolic system 
## System:  dx = x dt
## Initial: x0 = 1
## ==================================================
f <- "x"
var <- c(x=1)
times <- seq(0, 2*pi, by=0.001)
x <- ode(f, var, times)
plot(times, x, type = "l")

## ==================================================
## Example: time dependent system
## System:  dx = cos(t) dt
## Initial: x0 = 0
## ==================================================
f <- "cos(t)"
var <- c(x=0)
times <- seq(0, 2*pi, by=0.001)
x <- ode(f, var, times, timevar = "t")
plot(times, x, type = "l")

## ==================================================
## Example: multivariate time dependent system
## System:  dx = x dt 
##          dy = x*(1+cos(10*t)) dt
## Initial: x0 = 1
##          y0 = 1
## ==================================================
f <- c("x", "x*(1+cos(10*t))")
var <- c(x=1, y=1)
times <- seq(0, 2*pi, by=0.001)
x <- ode(f, var, times, timevar = "t")
matplot(times, x, type = "l", lty = 1, col = 1:2)

## ==================================================
## Example: numerical system
## System:  dx = x dt 
##          dy = y dt 
## Initial: x0 = 1
##          y0 = 2
## ==================================================
f <- function(x, y) c(x, y)
var <- c(x=1, y=2)
times <- seq(0, 2*pi, by=0.001)
x <- ode(f, var, times)
matplot(times, x, type = "l", lty = 1, col = 1:2)

## ==================================================
## Example: vectorized interface
## System:  dx = x dt 
##          dy = y dt 
##          dz = y*(1+cos(10*t)) dt  
## Initial: x0 = 1
##          y0 = 2
##          z0 = 2
## ==================================================
f <- function(x, t) c(x[1], x[2], x[2]*(1+cos(10*t)))
var <- c(1,2,2)
times <- seq(0, 2*pi, by=0.001)
x <- ode(f, var, times, timevar = "t")
matplot(times, x, type = "l", lty = 1, col = 1:3)

References

Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. tools:::Rd_expr_doi("10.18637/jss.v104.i05")

See Also

Other integrals: integral()