Author: Chris Rackauckas
Date: 2024-12-04

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

#### Chris Rackauckas

#### 2024-12-04

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:

    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

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

    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)