Series and integrals of hypergeometric type have many applications in mathematical physics. Investigation of the corresponding special functions can therefore be relevant for various computations in theoretical models of reality. Bailey chains provide powerful tools for generating infinite sequences of summation or transformation formulas for hypergeometric type series. For a review of the corr...

Epstein-Hubbell 1963 elliptic-type integrals occur in radiation field problems. In this paper, we consider a generalization of the elliptic-type integrals introduced by Kalla and Tuan 1996 and developed by Salman 2006. We give some recurrence relations for this generalized special function. Some of these relations can be used for analytic continuation to different domains. Asymptotic relations ...

In this paper we obtain Meyers type regularity estimates for approximate solutions of nonlinear elliptic equations. These estimates are used in the analysis of a numerical scheme obtained from a numerical homogenization of nonlinear elliptic equations. Numerical homogenization of nonlinear elliptic equations results in discretization schemes that require additional integrability of the approxim...

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows to treat directly general elliptic operator which are not of Dirac type.

We discuss geometrical aspects of different dualities in the integrable systems of the Hitchin type and its various generalizations. It is shown that T duality known in the string theory context is related to the separation of variables procedure in dynamical system. We argue that there are analogues of S duality as well as mirror symmetry in the many-body systems of Hitchin type. The different...

Finite-difference approximations to an elliptic–hyperbolic system arising in vortex density models for type II superconductors are studied. The problem can be formulated as a non-local Hamilton–Jacobi equation on a bounded domain with zero Neumann boundary conditions. Monotone schemes are defined and shown to be stable. An L∞ error bound is proved for the approximations of the unique viscosity ...

Maxwell equations are the basis of the theory of electromagnetic fields. In the stationary case they split into independent problems for electric and magnetic fields. For the electric field in conductive media electrical conductivity problem is formulated, and in insulators the problem of electrostatics is formulated. Equations for stationary problems are of elliptic type, whereas in general th...

An optimal harvesting problem with a concave nonquadratic cost functional and a diffusive degenerate elliptic logistic state equation type is investigated. Under certain assumptions, we prove the existence and uniqueness of an optimal control. A characterization of the optimal control via the optimality system is also derived, which leads to approximating the optimal control.

We present a Lohner type algorithm for the computation of rigorous bounds for solutions of ordinary differential equations and its derivatives with respect to initial conditions up to arbitrary order. As an application we prove the existence of multiple invariant tori around some elliptic periodic orbits for the pendulum equation with periodic forcing and for Michelson system.

The aim of this paper is to demonstrate the mathematical problems of modelling the physical process of heat and moisture transfer in porous materials. The method of discretization in time is applied to derive a discretized (generally non{potential) system of PDEs of elliptic type from the original variational problem of evolution. The convergence of corresponding Rothe sequences is studied.