相关论文: Second order difference equations and discrete ort…
Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems,…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction…
We investigate the solutions of the second-order difference equation $u_{n+2}=(au_n)/(1+bu_nu_{n+1})$ using a group of transformations (Lie symmetries) that leaves the solutions invariant.
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
In this paper we obtain a set of polynomials which are orthogonal with respect to the classical discrete weight function of the Charlier polynomials at which an extra point mass at x=0 is added. We construct a difference operator of…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second…
In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of…
The bivariate difference filed $(\mathbb{F}(\alpha, \beta), \sigma)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of…
The mathematical theory of a novel variational approximation scheme for general second and fourth order partial differential equations \begin{equation}\label{eq: A} \partial_t u - \nabla\cdot\Big(u\nabla\frac{\delta\phi}{\delta…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…