New fastest explicit non-stiff ODE solver? That's right, we now have something beating the pants off of the high order explicit RK methods! Check out the new symbolic-numeric optimized Taylor methods available in DifferentialEquations.jl! It uses a mix of Taylor-Mode AD, a symbolic post-processing trick, and a new order adaptivity algorithm to give a new level of performance.

See the paper: https://arxiv.org/abs/2602.04086

#julialang #diffeq #sciml

Your college professor teaches you "A-stable methods are required for stiff ODEs". But PSA, the most commonly used stiff ODE solvers (adaptive order BDF methods) are not A-stable. #sciml #numericalanalysis #diffeq

https://www.youtube.com/shorts/hmKVQ2B46i4

mathematical modeling of populations

https://www.youtube.com/watch?v=B5cMfyFqKmM

#rk4 #diffeq #simulation #demographics #systems

Postdoc in Single-Cell Multi-Omic Gene Regulatory Networks
University of Massachusetts Chan Medical School

Decode #GeneRegulatoryNetwork from #SingleCell multiomics with #CausalInference and #DiffEq as a #postdoc! No biomed bg needed.

See the full job description on jobRxiv: https://jobrxiv.org/job/university-of-massachusetts-chan-medical-...
https://jobrxiv.org/job/university-of-massachusetts-chan-medical-school-27778-postdoc-in-single-cell-multi-omic-gene-regulatory-networks-2/?fsp_sid=2615

WHY VARIATION OF PARAMETERS WORKS

This is more conceptual than a proof, but I find it comforting.

Consider the equation:

y' - y = xe^x

The solution will consist of a Homogeneous Solution and a Particular Solution; add the two for the complete solution.

The Homogeneous Solution is the solution that, when you run it through the left side of the equation, it always goes to zero. So you need the Homogeneous Solution for the same reason you need the "+ C" in antiderivatives: even if it's kind of the boring throwaway part of the solution, it is still part of the complete solution.

But Variation of Parameters finds another use for the Homogeneous Solution. Let us suppose that the Particular Solution is the Homogeneous Solution times a function "u". Well, if you feed the Particular Solution through the left side of the equation, the Homogeneous Solution part will tend to go away, leaving "u". So then you can multiply "u" by the Homogeneous Solution, and you've got your Particular Solution.

It kind of reminds me of how Taylor Series work. In a Taylor Series, you have a function that you can think of as secretly containing a multitude of polynomial terms, and the trick is finding a way to torture the function into confessing the coefficients on each polynomial term. In the case of Taylor it's done by iteratively differentiating and then setting "x" to zero, thus leaving a constant that is the coefficient for a given polynomial. (There's also that "n!" term but that's just details.)

Or Fourier Series: a periodic function secretly contains a multitude of sine and cosine terms, and again you find a way to torture it into confessing the coefficients on each sine / cosine. In that case the torture technique involves integration.

And in the case of Variation of Parameters, the torture technique is, we know what part of the particular solution gets burned away by the left side of the equation; the charred skeleton that remains is the other part of the particular solution.

#DifferentialEquations #VariationOfParameters #diffeq