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We show the direct applicability of the Brouwer fixed point theorem for the existence of equilibrium points and periodic solutions for differential systems on general domains satisfying geometric conditions at the boundary. We develop a…
In this paper, we present a numerical homogenization scheme for indefinite, time-harmonic Maxwell's equations involving potentially rough (rapidly oscillating) coefficients. The method involves an $\mathbf{H}(\mathrm{curl})$-stable,…
Continuing the research initiated in \cite{Fr-Ki2}, we study the existence of solutions and their regularity for the cohomological equations $X u=f$ for locally Hamiltonian flows (determined by the vector field $X$) on a compact surface $M$…
In this paper we construct a family of commuting multidimensional differential operators of order 3, which is closely related to the KdV hierarchy. We find a common eigenfunction of this family and an algebraic relation between these…
In this paper we study the construction of a discrete solution for a hyperbolic system of partial differentials of the strongly coupled type. In its construction, the discrete separation of matricial variable method was followed. Two…
We prove that for generic geometry, the curl-eigenfield solutions to the steady Euler equations on the three torus are all hydrodynamically unstable (linear, L^2 norm). The proof involves a marriage of contact topological methods with the…
Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants…
We solve the convection-diffusion equation using a coupling of cell-centered finite volume (FV) and discontinuous Galerkin (DG) methods. The domain is divided into disjoint regions assigned to FV or DG, and the two methods are coupled…
The paper investigates solutions of the fractional hyperbolic diffusion equation in its most general form with two fractional derivatives of distinct orders. The solutions are given as spatial-temporal homogeneous and isotropic random…
In this paper we study the Dirichlet problem for the Kobayashi--Warren--Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain…
We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on…
A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods…
This article investigates uniform well-posedness and inviscid limit behavior for the periodic Korteweg-de Vries-Burgers (KdV-B) and modified Korteweg-de Vries-Burgers (mKdV-B) equations: \[ \partial_t u + \partial_x^3 u - \varepsilon…
Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing…
We construct divergence-free Sobolev vector fields in C([0,1];W^{1,r}(T^d;R^d)) with r < d and d >=2 which simultaneously admit any finite number of distinct positive solutions to the continuity equation. We then show that the vector fields…
In the 80's D. Eisenbud and J. Harris considered the following problem: ``What are the limits of Weierstrass points in families of curves degenerating to stable curves?'' But for the case of stable curves of compact type, treated by them,…
In a bounded domain $\mathcal{O}\subset\mathbb{R}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability…
We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) full von Karman shallow shell equations that accounts for both…
Using continuation methods, we study the global solution structure of periodic solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of…
In the paper obtained equivalent system of Fredholm integral equations in the study of the Dirichlet problem for the generalized Manjeron equation with non-smooth coefficients in non-classical treatment (1), (4). When non-smooth conditions…