Related papers: A Universal Ordinary Differential Equation

When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can…

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…

In their 1968 paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the…

The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods…

A system of polynomial ordinary differential equations (ODEs) is specified via a vector of multivariate polynomials, or vector field, $F$. A safety assertion $\psi\rightarrow[F]\phi$ means that the trajectory of the system will lie in a…

In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions;…

We study a class of linear ordinary differential equations (ODE)s with distributional coefficients. These equations are defined using an {\it intrinsic} multiplicative product of Schwartz distributions which is an extension of the…

In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) $y^{\prime}=p(y)$ over \emph{unbounded time domains}, where $p$ is a vector of polynomials. Contrarily to the bounded (compact) time…

We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$. We prove an existence and uniqueness theorem for a linear differential equation of infinite…

I construct a new universal differential equation (B), in the sense of Rubel. That is, its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line.

Reachability analysis is a fundamental problem for safety verification and falsification of Cyber-Physical Systems (CPS) whose dynamics follow physical laws usually represented as differential equations. In the last two decades, numerous…

There exists a huge number of numerical methods that iteratively construct approximations to the solution $y(x)$ of an ordinary differential equation (ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a finite…

We consider a natural generalisation of the Painlev\'e property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with…

We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind $ \frac{d}{dt}\tilde{u}(t) = \tilde{f}(t) \tilde{u}(t)$, $\tilde{u}(-1)=1$, with $\tilde{f}(t)$ an analytic…

We present the first uniform XP exact algorithm for unconstrained binary optimization of quadratic, polynomial, fractional, and other objectives under a single parameter, the differentially affine (DA) rank $r$. An objective $f: \{0,1\}^n…

We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second…

In 1957, Hans Lewy constructed a counterexample showing that very simple and natural differential equations can fail to have local solutions. A geometric interpretation and a generalization of this counterexample were given in 1960 by…

A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…

The study of existence and uniqueness of solutions became important due to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years…

Recently in [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab. 43, 2 (2015), 468--527] and [A. Jentzen, T. M\"uller-Gronbach, and L. Yaroslavtseva, Commun. Math. Sci. 14, 6 (2016), 1477--1500] stochastic differential equations (SDEs)…