Related papers: An Axiomatic Approach to Liveness for Differential…
In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions…
We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to…
The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE…
A discrete-time linear dynamical system (LDS) is given by an update matrix $M \in \mathbb{R}^{d\times d}$, and has the trajectories $\langle s, Ms, M^2s, \ldots \rangle$ for $s \in \mathbb{R}^d$. Reachability-type decision problems of…
Mechanistic dynamic models of biochemical networks such as Ordinary Differential Equations (ODEs) contain unknown parameters like the reaction rate constants and the initial concentrations of the compounds. The large number of parameters as…
This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform…
Neural Ordinary Differential Equations (ODE) are a promising approach to learn dynamic models from time-series data in science and engineering applications. This work aims at learning Neural ODE for stiff systems, which are usually raised…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
In [1], we have presented the theoretical background for finding the Elementary Invariants for a 3D system of first order rational differential equations (1ODEs). We have also provided an algorithm to find such Invariants. Here we introduce…
A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…
Pointwise-in-time stability notions for Ordinary Differential Equations (ODEs) provide quantitative metrics for system performance by establishing bounds on the rate of decay of the system state in terms of initial condition -- allowing…
Formal verification has been successfully developed in computer science for verifying combinatorial classes of models and specifications. In like manner, formal verification methods have been developed for dynamical systems. However, the…
A random walk-based method is proposed to efficiently compute the solution of a large class of fractional in time linear systems of differential equations (linear F-ODE systems), along with the derivatives with respect to the system…
In this paper, ordinary and exponential dichotomies are defined in differential equations with equations with piecewise constant argument of general type. We prove the asymptotic equivalence between the bounded solutions of a linear system…
We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive…
When mathematical/computational problems reach infinity, extending analysis and/or numerical computation beyond it becomes a notorious challenge. We suggest that, upon suitable singular transformations (that can in principle be…
Recent works on optical flow estimation use neural networks to predict the flow field that maps positions of one image to positions of the other. These networks consist of a feature extractor, a correlation volume, and finally several…
We consider state-dependent delay equations (SDDE) obtained by adding delays to a planar ordinary differential equation with a limit cycle. These situations appear in models of several physical processes, where small delay effects are…
We discuss the non-uniqueness of continuous solutions to differential equations with a {\it discrete } state-dependent delay and continuous initial functions. We are interested not only in the fact (conditions) of non-uniqueness, but in…
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different…