knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) knitr::opts_knit$set( global.par = TRUE )
library(calculus)
The function ode
provides solvers for systems of ordinary differential equations of the type:
$$ \frac{dy}{dt} = f(t,y), \quad y(t_0)=y_0 $$
where $y$ is the vector of state variables. Two solvers are available: the simpler and faster Euler scheme^[https://en.wikipedia.org/wiki/Euler_method] or the more accurate 4-th order Runge-Kutta method^[https://en.wikipedia.org/wiki/Runge-Kutta_methods].
Although many packages already exist to solve ordinary differential equations in R^[https://CRAN.R-project.org/view=DifferentialEquations], they usually represent the function $f$ either with an R function
or with characters
. While the representation via R functions
is usually more efficient, the symbolic representation is easier to adopt for beginners and more flexible for advanced users to handle systems that might have been generated via symbolic programming. The function ode
supports both the representations and uses hashed environments
to improve symbolic evaluations.
The vector-valued function $f$ representing the system can be specified as a vector of characters
, or a function
returning a numeric vector, giving the values of the derivatives at time $t$. The initial conditions are set with the argument var
and the time variable can be specified with timevar
.
par(mar = c(4, 4, 1, 1))
$$ \frac{dx}{dt}=x, \quad x_0 = 1 $$
f <- "x" var <- c(x=1) times <- seq(0, 2*pi, by=0.001) x <- ode(f = f, var = var, times = times) plot(times, x, type = "l")
$$ \frac{dx}{dt}=\cos(t), \quad x_0 = 0 $$
f <- "cos(t)" var <- c(x=0) times <- seq(0, 2*pi, by=0.001) x <- ode(f = f, var = var, times = times, timevar = "t") plot(times, x, type = "l")
$$ \frac{d}{dt} \begin{bmatrix} x\ y \end{bmatrix}= \begin{bmatrix} x\ x(1+\cos(10t)) \end{bmatrix}, \quad \begin{bmatrix} x_0\y_0 \end{bmatrix}= \begin{bmatrix} 1\1 \end{bmatrix} $$
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 = f, var = var, times = times, timevar = "t") matplot(times, x, type = "l", lty = 1, col = 1:2)
$$ \frac{d}{dt} \begin{bmatrix} x\ y \end{bmatrix}= \begin{bmatrix} x\ y \end{bmatrix}, \quad \begin{bmatrix} x_0\y_0 \end{bmatrix}= \begin{bmatrix} 1\2 \end{bmatrix} $$
f <- function(x, y) c(x, y) var <- c(x=1, y=2) times <- seq(0, 2*pi, by=0.001) x <- ode(f = f, var = var, times = times) matplot(times, x, type = "l", lty = 1, col = 1:2)
$$ \frac{d}{dt} \begin{bmatrix} x\ y\ z \end{bmatrix}= \begin{bmatrix} x\ y\ y(1+cos(10t)) \end{bmatrix}, \quad \begin{bmatrix} x_0\y_0\z_0 \end{bmatrix}= \begin{bmatrix} 1\2\2 \end{bmatrix} $$
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 = f, var = var, times = times, timevar = "t") matplot(times, x, type = "l", lty = 1, col = 1:3)
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
A BibTeX entry for LaTeX users is
@Article{calculus, title = {{calculus}: High-Dimensional Numerical and Symbolic Calculus in {R}}, author = {Emanuele Guidotti}, journal = {Journal of Statistical Software}, year = {2022}, volume = {104}, number = {5}, pages = {1--37}, doi = {10.18637/jss.v104.i05}, }
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