Related papers: The structure of general solutions and integrabili…
We describe a solving semi-decision method based on examination of the rational structures of the generalized integrating factors of first-order ODEs. We propose a conjecture that for some family of equations of the type…
In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can…
Given a polynomial matrix P(x) of grade g and a rational function $x(y) = n(y)/d(y)$, where $n(y)$ and $d(y)$ are coprime nonzero scalar polynomials, the polynomial matrix $Q(y) :=[d(y)]^gP(x(y))$ is defined. The complete eigenstructures of…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…
We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…
We design and analyse a new numerical method to solve ODE system based on the structural method. We compute approximations of solutions together with its derivatives up to order $K$ by solving an entire block corresponding to $R$ time…
In this paper we use the comparison method for investigation of first order polynomial differential equations. We prove two comparison criteria for these equations. The proved criteria we use to obtain some global solvability criteria for…
Many important systems across biology, engineering, physics, and economics are characterized by polynomial ordinary differential equations (ODEs), yet analytical solutions are rare. We develop a framework for identifying and solving a broad…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
We propose a sufficient condition of the convergence of a power-log series that formally satisfies an algebraic ordinary differential equation (ODE) of arbitrary order. A general form and properties of the functional coefficients of such a…