Specify Ordinary Differential Equation (ODE)

Define compiled code for ordinary differential equation.

Usage

specify_ode(code, fname = NULL, pname = c("dy", "x", "y", "p"))

Arguments

code: string with the body of the function definition (see details)
fname: Optional name of the exported C++ function
pname: Vector of variable names (results, inputs, states, parameters)

Value

pointer (externalptr) to C++ function

Details

The model (code) should be specified as the body of of C++ function. The following variables are defined bye default (see the argument pname)

dy: Vector with derivatives, i.e. the rhs of the ODE (the result).
x: Vector with the first element being the time, and the following elements additional exogenous input variables,
y: Vector with the dependent variable
p: Parameter vector

\(y'(t) = f_{p}(x(t), y(t))\) All variables are treated as Armadillo (http://arma.sourceforge.net/) vectors/matrices.

As an example consider the Lorenz Equations \(\frac{dx_{t}}{dt} = \sigma(y_{t}-x_{t})\) \(\frac{dy_{t}}{dt} = x_{t}(\rho-z_{t})-y_{t}\) \(\frac{dz_{t}}{dt} = x_{t}y_{t}-\beta z_{t}\)

We can specify this model as ode <- 'dy(0) = p(0)*(y(1)-y(0)); dy(1) = y(0)*(p(1)-y(2)); dy(2) = y(0)*y(1)-p(2)*y(2);' dy <- specify_ode(ode)

As an example of model with exogenous inputs consider the following ODE: \(y'(t) = \beta_{0} + \beta_{1}y(t) + \beta_{2}y(t)x(t) + \beta_{3}x(t)\cdot t\) This could be specified as mod <- 'double t = x(0); dy = p(0) + p(1)*y + p(2)*x(1)*y + p(3)*x(1)*t;' dy <- specify_ode(mod)

Author

Klaus Kähler Holst

Examples

ode <- paste0(
  "dy(0) = p(0)*(y(1)-y(0));",
  "dy(1) = y(0)*(p(1)-y(2));",
  "dy(2) = y(0)*y(1)-p(2)*y(2);", collapse="\n"
)
 # Reduce test time
dy <- specify_ode(ode)
tt <- seq(0, 100, length.out=2e4)
yy <- solve_ode(dy, input=tt, init=c(1, 1, 1), par=c(10, 28, 8/3))