Related papers: Physical applications of second-order linear diffe…
In this paper, a quantum dot mathematical model based on a two-dimensional Schr\"odinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a…
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic…
We derive the general conditions for fully-nonlinear symmetry-integrable second-order evolution equations and their first-order recursion operators. We then apply the established Propositions to find links between a class of fully-nonlinear…
Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order…
In this article, we study about the solutions of second order linear differential equations by considering several conditions on the coefficients of homogenous linear differential equation and its associated non-homogenous linear…
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike…
An extension of the ideas of the Prelle-Singer procedure to second order differential equations is proposed. As in the original PS procedure, this version of our method deals with differential equations of the form…
I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.
We discuss the explicit construction of the Schroedinger equations admitting a representation through some family of general polynomials. Almost all solvable quantum potentials are shown to be generated by this approach. Some generalization…
A recently proposed algorithm to obtain global solutions of the double confluent Heun equation is applied to solve the quantum mechanical problem of finding the energies and wave functions of a particle bound in a potential sum of a…
We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a…
Second order linear non-autonomous differential equations with negative stiffness are considered. Using Chetaev-like (Lyapunov-like) functions, necessary (sufficient) conditions are found for the solutions to be bounded for all initial…
A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these…
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a…
A class of exact solutions is obtained for the Li\'{e}nard type ordinary non-linear differential equation. As a first step in our study the second order Li\'{e}nard type equation is transformed into a second kind Abel type first order…
We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and…
We prove the bispectrality of some class of matrix Schr\"odinger operators with polynomial potentials that satisfy a second-order matrix autonomous differential equation. The physical equation is constructed using the formal theory of the…
Our aim in this paper is to prove, under some growth conditions on the datas, the solvability in a Gevrey class of a polynomially nonlinear functional differential equation.