相关论文: Solution to Rubel's question about differentially …
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
We show that if any four distinct solutions of a rational difference equation are algebraically independent, then any number of distinct solutions to the equation are independent. A nontrivial variant of this result is given for autonomous…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the…
Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of…
It is proved that the Halphen system of ordinary differential equations has no non-trivial rational first integrals.
Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in…
The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…
Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact…
We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed…
In present paper we propose an approach based on examination of the structure of the general solution of equations of the type dy/dx=P(x,y)/Q(x,y), with P and Q polynomials only in y. Under the term structure we mean the dependency…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
Partial ordinary Bell polynomials are used to formulate and prove a version of the Fa\`{a} di Bruno's formula which is convenient for handling nonlinear terms in the differential transformation. Applicability of the result is shown in two…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been…
We investigate global and local regularity of generalized solutions to parabolic initial-boundary value problem for Petrovskii system of second order differential equations. Results are formulated in terms of the belonging of right-hand…
The constraints proposed recently by Bershadsky to produce $W^l_n$ algebras are a mixture of first and second class constraints and are degenerate. We show that they admit a first-class subsystem from which they can be recovered by…
We prove a result on the convex dependence of solutions of ordinary differential equations on an ordered finite-dimensional real vector space with respect to the initial data.
The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a commutative ring (in which case the linear…