Related papers: Division of Differential operators, intertwine rel…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
In [1], a generalized type of Darboux transformations defined in terms of a twisted derivation was constructed in a unified form. Such twisted derivations include regular derivations, difference operators, superderivatives and…
Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential…
We consider optimal control problems for a wide class of bilateral obstacle problems where the control appears in a possibly nonlinear source term. The non-differentiability of the solution operator poses the main challenge for the…
The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary…
We propose a generalization of Laplace transformations to the case of linear partial differential operators (LPDOs) of arbitrary order in R^n. Practically all previously proposed differential transformations of LPDOs are particular cases of…
Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the…
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see…
In this paper we continue studying of matrix $n\times n$ linear differential intertwining operators. The problems of minimization and of reducibility of matrix intertwining operators are considered and criterions of weak minimizability and…
We describe the use of confluent Darboux transformations for Schr\"odinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlev\'e equations. As a preliminary illustration, we briefly…
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…
By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Sch\"odinger equation (DNLS). These transformations are shown to lead to two solution formulas for DNLS in…
Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…
In this paper we prove inversion formulas for the Dunkl intertwining operator $V_k$ and for its dual ${}^tV_k$ and we deduce the expression of the representing distributions of the inverse operators $V_k^{-1}$ and ${}^tV_k^{-1}$, and we…
In this article we study generalizations of the inhomogeneous Burgers equation. First at the operator level, in the sense that we replace classical differential derivations by operators with certain properties, and then we increase the…
Using bidifferential calculus, we derive a vectorial binary Darboux transformation for an integrable matrix version of the first negative flow of the Kaup-Newell hierarchy. A reduction from the latter system to an integrable matrix version…
Quantum calculus based on the right invertible divided difference operator $D_{\sigma}^{\tau}$ is proposed here in context of algebraic analysis \cite{DPR}. The linear operator $D_{\sigma}^{\tau}$, specified with the help of two fixed maps…
Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…
The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some…