Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

Blow-up analysis for solutions to Neumann boundary value problem

In this paper, we will analyze the blow-up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2-dimensional surface (M,g)$(M,g)$ with boundary. We will provide some accurate point-wise estimates for the profile of these sequences.     

Solvability of the Neumann problem for a Sobolev pseudoparabolic equation

The dynamic potential constructed in this paper is used to analyze the existence of a classical solution to the Neumann problem for a Sobolev equation.     

Small-amplitude solutions of the sine-Gordon equation on an interval under dirichlet or Neumann boundary conditions

Summary We give a complete classification of the small-amplitude finite-gap solutions of the sine-Gordon (SG) equation on an interval under Dirichlet or Neumann boundary conditions. Our classification is based on an analysis of the finite-gap solutions of the boundary problems for the SG equation by means of the Schottky uniformization approach.On leave from IPPI, Moscow, Russia     

Characteristic boundary problems for the Liouville equation

We obtain formulas for solutions of problems with boundary conditions of the 1st, 2nd, and 3rd kind.     

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A boundary value problem for Bitsadze equation in the unit disc

A boundary value problem for the Bitsadze equation

$\frac{{\partial ^2 }}{{\partial \bar z^2 }}u(x,y) \equiv \frac{1}{4}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right)^2 u(x,y) = 0$

in the interior of the unit disc is considered. It is proved that the problem is Noetherian and its index is calculated, and solvability conditions for the non-homogeneous problem are proposed. Some solutions of the homogeneous problem are explicitely found.

     

Bernstein Ritz‐Galerkin method for solving an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Solvability of discrete Neumann boundary value problems

In this article we gain solvability to a nonlinear, second-order difference equation with discrete Neumann boundary conditions. Our methods involve new inequalities on the right-hand side of the difference equation and Schaefer's Theorem in the finite-dimensional space setting.     

On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners

The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.     

Compact difference schemes for heat equation with Neumann boundary conditions

In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949–959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(τ² + h^2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(τ² + h³). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600–616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

On a degenerate variational inequality with Neumann boundary conditions

A stopping time problem for degenerate reflected diffusions is studied in this paper. We give a characterization of the optimal cost as the maximum solution of a degenerate elliptic variational inequality with Neumann boundary conditions.The author would like to thank Professor L. C. Evans for very helpful suggestions on this topic.     

On fourth-order boundary value problem with singular data

We consider a simply supported beam with restoring and external forces given as a sum of a continuous function and a Dirac delta distribution. We present sufficient conditions on these data in order to guarantee a unique positive or negative solution, respectively.     

奇异非线性二阶微分方程Neumann边值问题

研究了一个奇异非线性二阶微分方程的Neumann边值问题,通过摄动技巧和比较原理得到了所论问题解的存在唯一性。