NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

THE WELL-POSEDNESS OF THE ABSTRACT KINETIC EQUATION BOUNDARY VALUE PROBLEMS

In this paper the abstract boundary value problem on Hilbert space H is studied: where Q /Q- are the maximum positive/negative spectral projections of self-adjoint operator T. Under suitable conditions, it is shown that the boundary value problem of abstract kinetic equation is well-posed and can be applied to transport theory.     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

PROBABILISTIC APPROACH TO THE NEUMANN PROBLEM

In this paper, we consider the Neumann boundary value problem of Schrodinger operator withmeasure potentia1 μ. First, a martingale formulation of the Neumann problem and an analyticcharacterization of the martingale formulation are given. Then, by using the Dirichlet forms andStochastic analysis we obtain an explicit formula for the unique weak solotion of this problem interms of reflecting Brownian motion and it's boundary local time.     

Multivariate refinement equation with nonnegative masks

This paper is concerned with multivariate refinement equations of the type where (?) is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported nonnegative sequence on Zs, and M is an s×s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schrodinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

EXISTENCE OF SOLUTIONS TO A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

In this paper,we study the existence of solutions to a third-order three-point boundary value problem.By imposing certain restrictions on the nonlinear term,we prove the existence of at least one solution to the boundary value problem by the method of lower and upper solutions.We are interested in the construction of lower and upper solutions.     

THE COUPLING OF NATURAL BOUNDARY ELEMENT AND FINITE ELEMENT METHOD FOR 2D HYPERBOLIC EQUATIONS

In this paper, we investigate the coupling of natural boundary element and finite element methods of exterior initial boundary value problems for hyperbolic equations. The governing equation is first discretized in time, leading to a time-step scheme, where an exterior elliptic problem has to be solved in each time step. Second, a circular artificial boundary FR consisting of a circle of radius R is introduced, the original problem in an unbounded domain is transformed into the nonlocal boundary value problem in abounded subdomain. And the natural integral equation and the Poisson integral formula are obtained in the infinite domainΩ2 outside circle of radius R. The coupled variational formulation is given. Only the function itself, not its normal derivative at artificial boundary ΓR, appears in the variational equation, so that the unknown numbers are reducedand the boundary element stiffness matrix has a few different elements. Such a coupled method is superior to the one based on direct boundary element method. This paper discusses finite element discretization for variational problem and its corresponding numerical technique, and the convergence for the numerical solutions. Finally, the numerical example is presented to illustrate feasibility and efficiency of this method.     

Riemann–Hilbert-type Boundary Value Problems on a Half Hexagon

In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann–Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed.Finally, we investigate the Schwarz problem and the Riemann–Hilbert problem for inhomogeneous Cauchy–Riemann equations.     

Long-time asymptotics for the initial-boundary value problem of coupled Hirota equation on the half-line

The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem formulated in the complex k-plane.The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k)and S(k)that depend on the initial data and boundary values,respectively.Then,applying nonlinear steepest descent techniques to the associated 3×3 matrix-valued Riemann-Hilbert problem,we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation.     

BOUNDARY VALUE PROBLEM FOR NONLINEAR IMPULSIVE INTEGRO-DIFFERENTIAL EQUATION IN BANACH SPACE

In this paper, by establishing a new comparison result and using the monotone iterative technique, the existence of maximal and minimal solutions of the boundary value problem for second-order impulsive differential equation which depends on x' in Banach space is obtained.     

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

We consider a kind of scattering problem by a crack Γ that is buried in a bounded domain D,and we put a point source inside the domain D.This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Γ.Both sides of the crack Γ are given Dirichlet-impedance boundary conditions,and different boundary condition(Dirichlet,Neumann or Impedance boundary condition) is set on the boundary of D.Applying potential theory,the problem can be reformulated as a system of boundary integral equations.We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

ON HASEMAN BOUNDARY VALUE PROBLEM FOR A CLASS OF META-ANALYTIC FUNCTIONS

In this article, Haseman boundary value problem for a class of meta-analytic functions is studied. The expression of solution and the condition of solvability for Haseman boundary value problem are obtained by changing the problem discussed into the equivalent Haseman boundary value problem of bi-analytic function. And the expression of solution and the condition of solvability depend on the canonical matrix.     

POSITIVE SOLUTIONS TO FOURTH-ORDER NEUMANN BOUNDARY VALUE PROBLEM

In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtained under the assumption that the nonlinearity satisfies sublinear or superlinear conditions, which are relevant to the first eigenvalue of the corresponding linear operator.     

POISSON PRECONDITIONING FOR SELF-ADJOINT ELLIPTIC PROBLEMS

In this paper, we formulate interface problem and Neumann elliptic boundary value problem into a form of linear operator equations with self-adjoint positive definite op- erators. We prove that in the discrete level the condition number of these operators is independent of the mesh size. Therefore, given a prescribed error tolerance, the classical conjugate gradient algorithm converges within a fixed number of iterations. The main computation task at each iteration is to solve a Dirichlet Poisson boundary value problem in a rectangular domain, which can be furnished with fast Poisson solver. The overall computational complexity is essentially of linear scaling.     

THE GLOBAL SOLUTION AND BLOWUP OF A SPATIOTEMPORAL EIT PROBLEM WITH A DYNAMICAL BOUNDARY CONDITION

We study a spatiotemporal EIT problem with a dynamical boundary condition for the fractional Dirichlet-to-Neumann operator with a critical exponent.There are three major ingredients in this paper.The first is the finite time blowup and the decay estimate of the global solution with a lower-energy initial value.The second ingredient is the L^q(2 ≤q <∞) estimate of the global solution applying the Moser iteration,which allows us to show that any global solution is a classical solution...     

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.     

On Boundary Value Problems with the Shift for Nonlinear Elliptic Equations of Second Order

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain,For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincar boundary value problem.