knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
A delay differential equation is an ODE which allows the use of previous values. In this case, the function
needs to be a JIT compiled Julia function. It looks just like the ODE, except in this case there is a function
h(p,t)
which allows you to interpolate and grab previous values.
We must provide a history function h(p,t)
that gives values for u
before
t0
. Here we assume that the solution was constant before the initial time point. Additionally, we pass
constant_lags = c(20.0)
to tell the solver that only constant-time lags were used and what the lag length
was. This helps improve the solver accuracy by accurately stepping at the points of discontinuity. Together
this is:
f <- JuliaCall::julia_eval("function f(du, u, h, p, t) du[1] = 1.1/(1 + sqrt(10)*(h(p, t-20)[1])^(5/4)) - 10*u[1]/(1 + 40*u[2]) du[2] = 100*u[1]/(1 + 40*u[2]) - 2.43*u[2] end") h <- JuliaCall::julia_eval("function h(p, t) [1.05767027/3, 1.030713491/3] end") u0 <- c(1.05767027/3, 1.030713491/3) tspan <- c(0.0, 100.0) constant_lags <- c(20.0) JuliaCall::julia_assign("u0", u0) JuliaCall::julia_assign("tspan", tspan) JuliaCall::julia_assign("constant_lags", tspan) prob <- JuliaCall::julia_eval("DDEProblem(f, u0, h, tspan, constant_lags = constant_lags)") sol <- de$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)
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.
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