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.

Value

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.

References

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

See also

Other integrals: integral()

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)