Related papers: Exactly solvable quadratic differential equation s…
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
Full set of autonomous completely solvable differential systems of equations in total differentials is built by basis of infinitesimal operators, universal invariant, and structure constants of admited multiparametric Lie group (abelian and…
Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…
We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems…
We consider in C^n the class of symmetric homogeneous quadratic dynamical systems. We introduce the notion of algebraic integrability for this class. We present a class of symmetric quadratic dynamical systems that are algebraically…
We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential…
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular…
Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that…
Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a…
We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically…
Consider systems of equations $q_i(x)=0$, where $q_i: {\Bbb R}^n \longrightarrow {\Bbb R}$, $i=1, \ldots, m$, are quadratic forms. Our goal is to tell efficiently systems with many non-trivial solutions or near-solutions $x \ne 0$ from…
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…
We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
A novel integrability condition for the Riccati equation, the simplest form of nonlinear ordinary differential equations, is obtained by using elementary quadrature method. Under this condition, the analytic general solution is presented,…
In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by…
Optimal decentralized controller design is notoriously difficult, but recent research has identified large subclasses of such problems that may be convexified and thus are amenable to solution via efficient numerical methods. One recently…
Preliminary results of our investigations on solving indefinite qua\-dra\-tic programs by dynamical systems are given. First, dynamical systems corresponding to two fundamental DC programming algorithms to deal with indefinite quadratic…