Related papers: Second order difference equations and discrete ort…
We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z)) dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient $\kappa_n$…
Starting with the asymptotic expansion of the error equation of the shifted Gr\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted Gr\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction terms. We then…
The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods…
Under general conditions we show an a priori probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form d_tu = div (A\nabla u) + f (t, x, u;w) + g_i(t, x,…
In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schr\"odinger equation. The technique is based on the use of first order operators intertwining two difference…
In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order $\alpha\in(2,3)$, involving a general form of fractional derivative. First, we prove an…
Solutions which are quasimodular forms to a second order differential equation attached to a triangular group are explicitly described in terms of certain orthogonal polynomials.
Some solutions for one class of nonlinear fourth-order partial differential equations \[u_{tt} = ({\kappa u + \gamma u^2})_{xx} + \nu uu_{xxxx} + \mu u_{xxtt} + \alpha u_x u_{xxx} + \beta u_{xx}^2 \] where $\alpha ,\;\beta ,\;\gamma ,\;\mu…
The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…
For a discrete function $f\left( x\right) $ on a discrete set, the finite difference can be either forward and backward. If $f\left( x\right) $ is a sum of two such functions $f\left( x\right) =f_{1}\left( x\right) +f_{2}\left( x\right) $,…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
In Dunkl theory on $\mathbb{R}^{n}$ which generalizes classical Fourier analysis, we study the solution of the Klein-Gordon-equation defined by: \begin{eqnarray} \nonumber \partial_{t}^{2}u-\Delta_{k}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \…
The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…
An analysis of a fractional cubic differential equation is presented, which is a generalization of different versions of fractional logistic equations, in order to obtain simpler numerical methods that globalize and extend the results…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
This article analysis differential equations which represents damped and fractional oscillators. First, it is shown that prior to using physical quantities in fractional calculus, it is imperative that they are turned dimensionless.…
The second-order differential equation for the Uehling potential is derived explicitly. The right side of this differential equation is a linear combination of the two Macdonald's functions $K_{0}(b r)$ and $K_{1}(b r)$. This central…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
In this article, firstly we develop a method for a type of difference equations, applicable to solve approximately a class of first order ordinary differential equation systems. In a second step, we apply the results obtained to solve a…