Related papers: Difference operators and difference equations on l…
It is shown how to define difference equations on particular lattices $\{x_n\}$, $n\in\mathbb{Z}$, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special…
A differential operator of weight $\lambda$ is the algebraic abstraction of the difference quotient $d_\lambda(f)(x):=\big(f(x+\lambda)-f(x)\big)/\lambda$, including both the derivation as $\lambda$ approaches to $0$ and the difference…
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with…
In this paper we develop the calculus of pseudo-differential operators on the lattice $\mathbb{Z}^n$, which we can call pseudo-difference operators. An interesting feature of this calculus is that the phase space is compact so the symbol…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. The objective…
We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special…
In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…
Lattice Green's Functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
In this paper, we obtain the ladder operators and associated compatibility conditions for the type I and the type II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to…
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…
This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually…
We develop an elliptic theory based in $L^2$ of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum. In particular, we do not require the indicial roots to be…
In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as…
We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…
This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…