# Solving Differential-Algebraic Equations (DAE) in R with diffeqr

#### 2019-09-22

A differential-algebraic equation is defined by an implicit function `f(du,u,p,t)=0`. All of the controls are the same as the other examples, except here you define a function which returns the residuals for each part of the equation to define the DAE. The initial value `u0` and the initial derivative `du0` are required, though they do not necessarily have to satisfy `f` (known as inconsistant initial conditions). The methods will automatically find consistant initial conditions. In order for this to occur, `differential_vars` must be set. This vector states which of the variables are differential (have a derivative term), with `false` meaning that the variable is purely algebraic.

This example shows how to solve the Robertson equation:

``````f <- function (du,u,p,t) {
resid1 = - 0.04*u[1]              + 1e4*u[2]*u[3] - du[1]
resid2 = + 0.04*u[1] - 3e7*u[2]^2 - 1e4*u[2]*u[3] - du[2]
resid3 = u[1] + u[2] + u[3] - 1.0
c(resid1,resid2,resid3)
}
u0 = c(1.0, 0, 0)
du0 = c(-0.04, 0.04, 0.0)
tspan = list(0.0,100000.0)
differential_vars = c(TRUE,TRUE,FALSE)
sol = diffeqr::dae.solve(f,du0,u0,tspan,differential_vars=differential_vars)
udf = as.data.frame(sol\$u)
plotly::plot_ly(udf, x = sol\$t, y = ~V1, type = 'scatter', mode = 'lines') %>%