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The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
The paper deals with the following system of nonlinear difference equations \begin{equation*} x_{n+1}=ax_{n}^{2}y_{n}+bx_{n}y_{n}^{2},\ y_{n+1}=cx_{n}^{2}y_{n}+dx_{n}y_{n}^{2},\ n\in \mathbb{N}_{0}, \end{equation*} where the initial values…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
For first order differential equations of the form $y'=\sum_{p=0}^P F_p(x)y^p$ and second order homogeneous linear differential equations $y''+a(x)y'+b(x)y=0$ with locally integrable coefficients having asymptotic (possibly divergent) power…
The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev\'{e} test, the calculation of the integrals relies on a variety of…
In this paper we report a few examples of algebraically solvable dynamical systems characterized by 2 coupled Ordinary Differential Equations which read as follows: x_n = P(n) (x1, x2) , n = 1, 2 , with P(n) (x1, x2) specific polynomials of…
This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$…
We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by…
Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however,…
We prove that any globally periodic rational discrete system in K^k(where K denotes either R or C), has unconfined singularities, zero algebraic entropy and it is complete integrable (that is, it has as many functionally independent first…
A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational…
Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in…
We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
We find a criterion for correct solvability in L_p(R) of a linear differential equation of a first order with non-negative locally integrated coefficient and study the asymptotic properties of its solutions.
Based on an original classification of differential equations by types of regular Lie group actions, we offer a systematic procedure for describing partial differential equations with prescribed symmetry groups. Using a new powerful…
The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of…