The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential po...
The paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we...
This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie alge...
This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein...
This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equat...
A high-order single-step implicit method, the Legendre–Darboux Method of order six (LDM6), is introduced for solving both linear and nonlinear initial value problems. Unlike classical Taylor expansions, LDM6 systematically constructs higher-ord...
We study monic Hermite–Sobolev polynomials for the Gaussian weight with discrete Sobolev masses placed at conjugate purely imaginary points
We present an adaptive certifying semi-algorithm for rational second-order ordinary differential equations (ODEs) in the Darboux-representative, non-degenerate subclass of the Liouvillian setting considered here. The method first computes a nonlocal...
In this paper, the linearizability of a 2:−3 resonant system with quadratic nonlinearities is studied. We provide a list of the conditions for this family of systems having a linearizable center. The conditions for linearizablity are obtained by comp...
The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrica...
Given a quasi-definite linear functional
We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of...
Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of appl...
In this paper, we extend some differential and structural results for monic Jacobi–Sobolev orthogonal polynomials, associated with a general discrete Sobolev inner product, with a Jacobi continuous part. We consider finitely many exterior mass...
In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phas...
We present different methods to construct solutions to the Kadomtsev–Petviashvili (KP) equation. In the first method, from the solutions to the NLS equation, we construct solutions to the KP equation in terms of Fredholm determinants. We deduce...