# Exact solution of the DDE of Willé and Baker

Post date: Mar 8, 2015 8:55:19 PM

In their paper [1], Willé and Baker benchmark their DELSOL numerical solver for delay-differential equations on a number of test systems. Their *Example 3* is a system of three first-order differential equations with constant delays and was popularized by **a test case** in Matlab for the `dde23`

function. Matlab does indeed a good job at integrating the system - not the same can be said e.g. for OpenModelica.

Interestingly, nobody seems to have remarked that the system admits an explicit solution, which can be used to check the results of the numerical integration. The form of the solution is obviously piecewise polynomial and can be obtained mechanically with the method of steps. However, the coefficients are somewhat tedious to compute - the *monstre* solution up to *t*=5 takes about 20 printed pages and involves rational coefficients of over 40 digits!

For your enjoyment, here’s is the components of the solution **y** for *t*=1,2,3,4,5 computed symbolically (sorry, Google sites strips MathML):

i.e. approximately

Full code is here.

[1] Willé DR, Baker CTH. DELSOL—a numerical code for the solution of systems of delay-differential equations. Applied Numerical Mathematics. 1992 Apr;9(3–5):223–34.