Related papers: Elliptic Hypergeometric Solutions to Elliptic Diff…
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a…
Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlev\'e equation.
We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.
In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.
Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for…
This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very…
We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the…
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…
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…
General structure of the multivariate plain and q-hypergeometric terms and univariate elliptic hypergeometric terms is described. Some explicit examples of the totally elliptic hypergeometric terms leading to multidimensional integrals on…
A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g.…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
This book aims to provide a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. Such a class of equations often arises in analysis, probability theory,…
An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are…
The solutions of the discrete Painlev\'e equation I were constructed in terms of elliptic and hyperelliptic $\psi$ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference…
An interpolation problem related to the elliptic Painlev\'e equation is formulated and solved. A simple form of the elliptic Painlev\'e equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
In one of his work, appeared in 1969, John A. Baker initiated the systematic investigation of some partial difference equations. The main purpose of this paper is to continue and to extend these investigations. Firstly, we present how such…