In this paper, we investigate a singular perturbation problem including a fourth order O.D.E. with general linear boundary conditions. Firstly, we obtain the necessary conditions of solution of O.D.E. by making use of fundamental solution, then by compatibility of these conditions with boundary conditions, we determine that, for given perturbation problem, whether boundary layer is formed or not.

The lack of a general maximum principle for biharmonic equations suggests to study under which boundary conditions the positivity preserving property holds. We show that this property holds in general domains for suitable linear combinations of Dirichlet and Navier boundary conditions. The spectrum of this operator exhibits some unexpected features: radial data may generate nonradial solutions....

Given linear elliptic partial differential equations with mixed boundary conditions, with uncertain parameters constrained by inequalities, we show how to use finite element approximations to compute worst case a posteriori error bounds for linear response functionals determined by the solution. All discretization errors are taken into account. Our bounds are based on the dual weighted residual...

In this paper, an investigation is initiated of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts, i.e., where the secondorder derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary-value problems are associated with expected first-exit times of the membrane potential in ...

The finite analytic numerical method is developed to solve two-point boundary value problems of ordinary differential equations. The basic idea of the finite analytic method is the incorporation of the local analytic solution of the governing equation in the numerical solution of the boundary value problem. In this study, the finite analytic solution is developed for both linear and nonlinear s...

At the microand nanoscale, the standard continuity boundary conditions at fluid–solid interfaces of classical transport phenomena do not apply and must be replaced by boundary conditions that allow discontinuities. In this study, the classical thermal laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition for tangential velocity and continuous temp...

In this paper linear elliptic boundary value problems of second order with non-smooth data (L∞-coefficients, Lipschitz domains, regular sets, non-homogeneous mixed boundary conditions) are considered. It is shown that such boundary value problems generate Fredholm operators between appropriate Sobolev–Campanato spaces, that the weak solutions are Hölder continuous up to the boundary and that th...

We reduce the classification of all supersymmetric backgrounds in eleven dimensions to the evaluation of the supercovariant derivative and of an integrability condition, which contains the field equations, on six types of spinors. We determine the expression of the supercovariant derivative on all six types of spinors and give in each case the field equations that do not arise as the integrabil...