We have suggested the numerical schemes of collocation methods for approximative solution of singular integrodifferential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejér points. Theoretical background of collocation methods has been obtained in Generalized Hölder spaces. Singular...

For a second order nonlinear ordinary differential equation (ODE), a singular boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble type (droplet type) solutions are sought. Being defined on a semi infinite interval 0 < r <∞, this ODE, with a polynomial nonlinearity of the third order with respect to a desir...

We study singular perturbations of a class two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. construct effective Hamiltonian and initial data prove the convergence value function solution limit (effective) Cauchy problem parabolic equation HJB type. use methods probability, viscosity solutions homog...

We study certain generalized Cauchy integral formulas for gradients of solutions to second order divergence form elliptic systems, which appeared in recent work by P. Auscher and A. Rosén. These are constructed through functional calculus and are in general beyond the scope of singular integrals. More precisely, we establish such Cauchy formulas for solutions u with gradient in weighted L2(R 1+...

We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically Cauchy–Leray Cauchy–Szegő projection associated to various classes bounded domains in \(\mathbb C^n\) with n ≥ 2.

We extend Krupnik’s criterion of self-adjointness of the Cauchy singular integral operator to the case of finitely connected domains. The main aim of the paper is to present a new approach for proof of the criterion. Let G+ be a finitely connected domain bounded by the rectifiable curve C = ∂G+, G− = C \ clos G+ and ∞ ∈ G−. Suppose also that w(z), z ∈ C is a nonnegative weight such that w(z) 6≡...

We study the well posedness of nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two three. Behind autonomous interest problem, this is model evolution so called singular solutions that are known analysis semilinear elliptic equations. show Cauchy problem for NLS considered enjoys local existence uniqueness strong (operator domain) solutions, depe...