Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.     

Classical solution of a one-dimensional parabolic boundary value problem with nonlinear boundary conditions and moving boundary

We consider a nonlinear parabolic boundary value problem of the Stefan type with one space variable, which generalizes the model of hydride formation under constant conditions. We suggest a grid method for constructing approximations to the unknown boundary and to the concentration distribution. We prove the uniform convergence of the interpolation approximations to a classical solution of the boundary value problem. (The boundary is smooth, and the concentration distribution has the necessary derivatives.) Thus, we prove the theorem on the existence of a solution, and the proof is given in constructive form: the suggested convergent grid method can be used for numerical experiments.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.     

Impulsive boundary value problems with nonlinear boundary conditions

In this paper, the upper and lower solution method and Schauder’s fixed point theorem are employed in the study of boundary value problems for a class of second-order impulsive ordinary differential equations with nonlinear boundary conditions. We prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

Singularly perturbed boundary value problems with a power-law boundary layer

We consider boundary value problems for second-order singularly perturbed equations whose solution has a power-law boundary layer that occurs because the degenerate equation has multiple roots.     

On boundary value problem with singular inhomogeneity concentrated on the boundary

We study the asymptotic behavior of solutions to a boundary value problem for the Poisson equation with a singular right-hand side, singular potential and with alternating type of the boundary condition. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem by means of the unfolding method. The proof requires that the dimension be larger than two.     

The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

Spectral smoothed boundary methods: The role of external boundary conditions

The spectral smoothed boundary method (SSBM) is a recently proposed numerical method to approximate the solution of partial differential equations in irregular domains with no‐flux boundary conditions by means of Fourier spectral methods. In this article we explore the robustness and accuracy of the scheme under variations of the artificial boundary conditions that must be imposed on the boundary of the enlarged domain in which the problem is solved. As a test model, we present quantitative numerical results based on a problem of propagation of waves of electrical activity in cardiac tissue for which the method is relevant. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions

By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Characterizations of boundary pluripolar hulls

We present some basic properties of the so-called boundary relative extremal function and discuss boundary pluripolar sets and boundary pluripolar hulls. We show that for B-regular domains the boundary pluripolar hull is always trivial on the boundary of the domain and present a “boundary version” of Zeriahi’s theorem on the completeness of pluripolar sets.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers

A boundary value method for solving a class of nonlinear singularly perturbed two point boundary value problems with a boundary layer at one end is proposed. Using singular perturbation analysis the method consists of solving two problems; namely, a reduced problem and a boundary layer correction problem. We use Pade’ approximation to obtain the solution of the latter problem and to satisfy the condition at infinity. Numerical examples will be given to illustrate the method.     

Non-reflecting boundary condition on ellipsoidal boundary

The modeling of wave propagation problems using finite element methods usually requires the truncation of the computation domain around the scatterer of interest. Absorbing boundary conditions are classically considered in order to avoid spurious reflections. In this paper, we investigate some properties of the Dirichlet to Neumann map posed on a spheroidal boundary in the context of the Helmholtz equation.     

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relation-ship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman's criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.     

On an optimal boundary control problem with a dynamic boundary condition

We are the first solve the optimization problem for a boundary displacement control of string vibrations at one endpoint in the case of a nonstationary boundary condition containing a directional derivative at the other endpoint.     

A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals

A free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in Fazio (SIAM J. Numer. Anal. 33 (1996) 1473). We consider here a survey on recent developments related to the free boundary identification of the truncated boundary. The goals of this survey are: to recall the reasoning for a free boundary identification of the truncated boundary, to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature, and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line.     

An augmented Galerkin procedure for the boundary integral method applied to mixed boundary value problems

Here we present the asymptotic error analysis for the boundary element approximation of the direct boundary integral equations for the plane mixed boundary value problem of the Laplacian. The boundary elements are defined by B-splines for the smooth parts of the boundary charges and additional singular functions at the collision points. The asymptotic error estimates include estimates for the stress intensity factors which occur as additional unknowns to be computed within the Galerkin scheme. The numerical analysis is based on the uniqueness of the problem, a coerciveness inequality, the triangular principal part and an extended shift theorem of the boundary integral operators.     

Regularity of boundary wavelets

The conventional way of constructing boundary functions for wavelets on a finite interval is by forming linear combinations of boundary-crossing scaling functions. Desirable properties such as regularity (i.e. continuity and approximation order) are easy to derive from corresponding properties of the interior scaling functions. In this article we focus instead on boundary functions defined by recursion relations. We show that the number of boundary functions is uniquely determined, and derive conditions for determining regularity from the recursion coefficients. We show that there are regular boundary functions which are not linear combinations of shifts of the underlying scaling functions.