Related papers: Explicitly Solvable Systems of First-Order Differe…
In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.
We obtain explicit formulas for the solutions of the system of second-order difference equations of the form $x_{n+ 1} = \frac{x_n y_{n-1}}{y_n (a_n + b_n x_n y_{n - 1})}, \quad y_{n+1} = \frac{x_{n - 1} y_n}{x_n (c_n+d_n x_{n-1} y_n)}$,…
The motivation for this paper are computer calculations of complete lists of weight systems of quasihomogeneous polynomials with isolated singularity at 0 up to rather large Milnor numbers. We review combinatorial characterizations of such…
In this article, we prove an existence of solutions for a non-local boundary value problem with nonlinearity in a nonlocal condition. Our method is based upon the Mawhin's coincidence theory.
The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference…
One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both…
In this paper two related simplification problems for systems of linear inequalities describing precedence relation systems are considered. Given a precedence relation system, the first problem seeks a minimum subset of the precedence…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on Bour-Mayer method to determine compatibility conditions via Jacobi-Mayer brackets. We solve…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
A primary branch solution (PBS) is defined as a solution with $n$ independent $m-1$ dimensional arbitrary functions for an $n$ order $m$ dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
We introduce two classes of homogeneous polynomials and show their role in constructing of integrable hierarchies for some integrable lattices.
We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
We present a method for solving a class of initial valued, coupled, non-linear differential equations with `moving singularities' subject to some subsidiary conditions. We show that this type of singularities can be adequately treated by…
This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function…