In this paper, we consider the following Neumann boundary value problem { −u′′(x) = u(x)− λ|u(x)|, x ∈ (0, 1), u′(0) = 0 = u′(1), where λ ∈ R is parameter. We study the positive and negative solutions of this problem with respect to a parameter ρ (i.e. u(0) = ρ) in all R∗. By using a quadrature method, we obtain our results. Also we provide some details about the solutions that are obtained.

Inspired by the penalization of the domain approach of Lions & Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in s...

Given an open bounded connected subset Ω of R, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u) = 1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A...

— We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz–Sobolev space. Résumé. — On étudie un problème aux limites de Neumann associé à un opérateur différe...

The mixed Dirichlet-Neumann problem for the Helmholtz equation in the exterior of several bodies (obstacles) is studied in 2 and 3 dimensions. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called the "method of interior boundaries", because additional boundaries are introduced inside scattering bodies, where the Neumann ...

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes espe...

We introduce a new fictitious domain method for the solution of second-order elliptic boundary-value problems with Dirichlet or Neumann boundary conditions on domains with C2 boundary. The main advantage of this method is that it extends the solutions smoothly, which leads to better performance by achieving higher accuracy with fewer degrees of freedom. The method is based on a least-squares in...

The non-local hyperbolic partial differential equations have many applications in sciences and engineering. A collocation finite element approach based on exponential cubic B-spline and quintic B-spline are presented for the numerical solution of the wave equation subject to nonlocal boundary condition. Von Neumann stability analysis is used to analyze the proposed methods. The efficiency, accu...