From 12829572f0f51a7ab662af2e743715afc2b0e996 Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 2 Dec 2024 13:14:39 -0500 Subject: [PATCH] vault backup: 2024-12-02 13:14:39 --- .../2024-12-02 Delay Differential Equation.md | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-12-02 Delay Differential Equation.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-12-02 Delay Differential Equation.md index 18b10d1a9..2f143e574 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-12-02 Delay Differential Equation.md +++ b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-12-02 Delay Differential Equation.md @@ -6,3 +6,12 @@ First order DDE: $$\frac{dx}{dt} = -x(t-1)$$ This is a problem. We cannot use an initial value, we need **an initial history function (IHF)**. This is the behaviour of x(t) defined in an interval $[-\tau_0, 0]$, assuming solution time starts at $t=0$ +## Method of Steps +Think of DDE as being a mapping between the past interval and the present interval +$$x(t) = \phi_{i-1}(t)$$ on any interval $[t_i-1,t_i]$ +$$\int_{\phi_{i-1}(t)}^{x(t)}dx' = - \int_{t_i}^t \phi_{i-1}(t' - 1)dt'$$ +Then after some steps +$$\frac{dx}{dt} = -x(t-1) \rightarrow dx' = -x'(t'-1)dt'$$ +And then this can be solved at each interval. +- This gets annoying. +- Need to solve at each interval over and over. \ No newline at end of file