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

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

A delay differential equation is an ODE which allows the use of previous values. In this case, the function
needs to be a JIT compiled Julia function. It looks just like the ODE, except in this case there is a function
`h(p,t)` which allows you to interpolate and grab previous values.

We must provide a history function `h(p,t)` that gives values for `u` before
`t0`. Here we assume that the solution was constant before the initial time point. Additionally, we pass
`constant_lags = c(20.0)` to tell the solver that only constant-time lags were used and what the lag length
was. This helps improve the solver accuracy by accurately stepping at the points of discontinuity. Together
this is:

```R
f <- JuliaCall::julia_eval("function f(du, u, h, p, t)
  du[1] = 1.1/(1 + sqrt(10)*(h(p, t-20)[1])^(5/4)) - 10*u[1]/(1 + 40*u[2])
  du[2] = 100*u[1]/(1 + 40*u[2]) - 2.43*u[2]
end")
h <- JuliaCall::julia_eval("function h(p, t)
  [1.05767027/3, 1.030713491/3]
end")
u0 <- c(1.05767027/3, 1.030713491/3)
tspan <- c(0.0, 100.0)
constant_lags <- c(20.0)
JuliaCall::julia_assign("u0", u0)
JuliaCall::julia_assign("tspan", tspan)
JuliaCall::julia_assign("constant_lags", tspan)
prob <- JuliaCall::julia_eval("DDEProblem(f, u0, h, tspan, constant_lags = constant_lags)")
sol <- de$solve(prob,de$MethodOfSteps(de$Tsit5()))
udf <- as.data.frame(t(sapply(sol$u,identity)))
plotly::plot_ly(udf, x = sol$t, y = ~V1, type = 'scatter', mode = 'lines') %>% plotly::add_trace(y = ~V2)
```

![delay](https://user-images.githubusercontent.com/1814174/39023532-10bdd750-43f0-11e8-837d-156d33ea2f99.png)

Notice that the solver accurately is able to simulate the kink (discontinuity) at `t=20` due to the discontinuity
of the derivative at the initial time point! This is why declaring discontinuities can enhance the solver accuracy.
