---
title: "Solving Differential-Algebraic Equations (DAE) in R with diffeqr"
author: "Chris Rackauckas"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Solving Differential-Algebraic Equations (DAE) in R with diffeqr}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

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 inconsistent initial conditions). The methods will automatically find consistent 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:

```R
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 <- c(0.0,100000.0)
differential_vars <- c(TRUE,TRUE,FALSE)
prob <- de$DAEProblem(f,du0,u0,tspan,differential_vars=differential_vars)
sol <- de$solve(prob)
udf <- as.data.frame(t(sapply(sol$u,identity)))
plotly::plot_ly(udf, x = sol$t, y = ~V1, type = 'scatter', mode = 'lines') %>%
  plotly::add_trace(y = ~V2) %>%
  plotly::add_trace(y = ~V3)
```

![daes](https://user-images.githubusercontent.com/1814174/39022955-d600814c-43ec-11e8-91bb-e096ff3d3fb7.png)

Additionally, an in-place JIT compiled form for `f` can be used to enhance the speed:

```R
f = JuliaCall::julia_eval("function f(out,du,u,p,t)
  out[1] = - 0.04u[1]              + 1e4*u[2]*u[3] - du[1]
  out[2] = + 0.04u[1] - 3e7*u[2]^2 - 1e4*u[2]*u[3] - du[2]
  out[3] = u[1] + u[2] + u[3] - 1.0
end")
u0 <- c(1.0, 0, 0)
du0 <- c(-0.04, 0.04, 0.0)
tspan <- c(0.0,100000.0)
differential_vars <- c(TRUE,TRUE,FALSE)
JuliaCall::julia_assign("du0", du0)
JuliaCall::julia_assign("u0", u0)
JuliaCall::julia_assign("p", p)
JuliaCall::julia_assign("tspan", tspan)
JuliaCall::julia_assign("differential_vars", differential_vars)
prob = JuliaCall::julia_eval("DAEProblem(f, du0, u0, tspan, p, differential_vars=differential_vars)")
sol = de$solve(prob)
```
