Integro-DDE that includes integral over delta function

Specific Domains Modelling & Simulations
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Hi everyone. I’ve solved some DDE systems in Julia, but I’m stumped on my latest problem. I’m not sure if this is a math question or a Julia question, but I’d appreciate any guidance you can provide.

How would you go about solving a DDE that includes a term like this?

\frac{dP\left(t\right)}{dt} = ... + \int_0^t dt' \frac{dP\left(t'\right)}{dt'} \delta\left(t-\tau\left(t'\right)\right)

Context: I’m developing a model of a quantity that is periodically refreshed (i.e. a product that must be periodically repurchased due to obsolescence). This DDE has a time/state-dependent delay, and the value of \frac{dP\left(t\right)}{dt} is a function of refreshed purchases from t' = t- \tau(t'), where \tau(t) changes over time as a function of macroeconomic conditions and other exogenous factors. It’s possible that purchases at multiple values of t' could all get refreshed at time t, hence the integral over t'.

I’ve looked at this thread, but I don’t think it answers my question: Integro-differential equations with DifferentialEquations.jl

Does DifferentialEquations.jl already anticipate situations like this?

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These are called Neutral delay differential equations (NDDEs) , there is a whole litterature on them.

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Yes, so we give you the whole interpolation function from which you can write the integral into the rhs of the equation. Of course there can be numerical issues that are tough around discontinuities, which we have some support for though with NDDEs there can be some extra propagation that needs to be handled through adaptivity.