The free space Maxwell dielectric problem can be reduced to a system of surface integral equations (SIE). A numerical formulation for the Maxwell dielectric problem using an SIE system presents two key advantages: first, the radiation condition at infinity is exactly satisfied, and second, there is no need to artificially define a truncated domain. Consequently, these SIE systems have generated much interest in physics, electrical engineering, and mathematics, and many SIE formulations have been proposed over time. In this article we introduce a new SIE formulation which is in the desirable operator form identity plus compact, is well-posed, and remains well-conditioned as the frequency tends to zero. The unknowns in the formulation are three-dimensional vector fields on the boundary of the dielectric body. The SIE studied in this paper is derived from a formulation developed in earlier work by Ganesh, Hawkins, and Volkov. Our initial formulation utilized linear constraints to obtain a uniquely solvable system for all frequencies. The new SIE introduced and analyzed in this article combines the integral equations from that previous work with new constraints. We show that the new system is in the operator form identity plus compact in a particular functional space, and we prove well-posedness at all frequencies and low-frequency stability of the new SIE.

A continuous nonlinear Fornasini–Marchesini system of fractional order with nonzero initial conditions is considered. The existence, uniqueness and continuous dependence of solutions on functional parameters are studied. Some results concerning single and mixed fractional derivatives of functions of two variables are presented.

An optimized finite difference (FD) extrapolated scheme for the two-dimensional (2D) integro-differential equation of parabolic type is established by means of the proper orthogonal decomposition (POD) technique. The stability and error estimates of the optimized FD extrapolated solutions are provided and a numerical example is presented to explain the correction of the theoretical results and the superiority of the optimized FD extrapolated scheme.

We examine the existence of a weak solution to a class of variational-hemivariational inequalities governed by a fractional Laplace operator. Moreover, an existence result is proved for a bilateral obstacle problem for a nonlocal hemivariational inequality. Our approach is based on a surjectivity result for pseudomonotone mappings, properties of Clarke’s subgradient, and the Moreau–Yosida approximation.

A mixed boundary value problem for the partial differential equation of diffusion in an inhomogeneous medium in a Lipschitz domain is reduced to a system of direct segregated parametrix-based boundary-domain integral equations (BDIEs). We use a parametrix different from the one employed in previous papers by Mikhailov (2002, 2006) and Chkadua, Mikhailov and Natroshvili (2009). We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.

A boundary value problem consisting of an integrodifferential equation, together with boundary conditions dependent on the spectral parameter, is investigated. The asymptotic behavior of the eigenvalues is studied, and we prove the uniqueness and the stability theorems for the solution of the inverse problem.

A new generalization of the Browder’s degree for mappings of the type ${\left(S\right)}_{+}$ is presented. The main idea is rooted in the observation that the Browder’s degree remains unchanged for mappings of the form $A:Y\to X^{\ast }$, where $Y$ is a reflexive uniformly convex Banach space continuously embedded in the Banach space $X$. The advantage of the suggested degree lies in the simplicity it provides for the calculations of degree associated to nonlinear operators. An application from the theory of phase transition in liquid crystals is presented for which the suggested degree has been successfully applied.

We propose a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for linear Volterra integro-differential equations with vanishing delays. We derive an $h$-$p$ version a-priori error estimate in the $H^1$-norm that is completely explicit in the local time steps and the local polynomial degrees. Numerical experiments are provided to confirm the theoretical results.