Acknowledgements: I would like to thank E. Sonnendrücker for the stimulating discussions we had together and from which the problem treated in this paper emerged. Abstract: we show that the solution to an oscillatory-singularly perturbed ordinary differential equation may be asymptotically expanded into a sum of oscillating terms. Each of those terms writes as an oscillating operator acting on ...

An extension of the finite element method–flux corrected transport stabilization for hyperbolic problems in the context of partial differential–algebraic equations is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ-scheme when applied to the time integration of the resulting differential–algebraic equation is sho...

This paper constructs extrapolated implicit-explicit time stepping methods that allow one to efficiently solve problems with both stiff and nonstiff components. The proposed methods are based on Euler steps and can provide very high order discretizations of ODEs, index-1 DAEs, and PDEs in the method of lines framework. Implicit-explicit schemes based on extrapolation are simple to construct, ea...

We study the ability of neural networks to calculate feedback control signals that steer trajectories continuous-time nonlinear dynamical systems on graphs, which we represent with ordinary differential equations (neural ODEs). To do so, present a ODE (NODEC) framework and find it can learn drive graph toward desired target states. While use loss functions not constrain energy, our results show...

An easy-to-use variant of the finite difference method (FDM), the polynomial finite difference method (PFDM), for the numerical solution of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs), is presented. Compared to the traditional implementation of the FDM, the PFDM approach has two major advantages: straightforward implementation, and easily adjustable accura...

Saccharomyces cerevisiae (baker’s yeast) can exhibit sustained oscillations during the operation in a continuous bioreactor that adversely affects its stability and productivity. Because of heterogeneous nature of cell populations, the cell population balance equation (PBE) can be used to capture the dynamic behavior of such cultures. In this work, an unstructured-segregated model is used f...

A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear differential equations have exact solutions which are general solution of the simplest integrable e...