Stability estimate for an inverse boundary coefficient problem in thermal imaging

We establish a stability estimate for an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition (we note here that the initial condition is assumed to be a priori unknown). Our stability estimate is of logarithmic type and it is essentially based on a logarithmic estimate for a Cauchy problem for the Laplace equation.     

Some Generalized Problems for Polyharmonic Functions

In the first section of this paper, some non-local boundary value problem for the polyharmonic equation in the plane is considered. This problem consists in determining solution of the polyharmonic equation satisfying some special non-local-type boundary condition on two curves. The existence theorem is proved. In the second section, an example for the case of the biharmonic equation is considered. In the third section, some non-local, non-linear problem of Riquier type is examined. The Riquier-type problem consists in determining the polyharmonic function in the plane whose value together with its successive Laplacians are prescribed on the boundary. The existence theorem is proved and an example for the case of the biharmonic equation is considered.     

On an inhomogeneous problem for parabolic-hyperbolic equation

We study a boundary value problem for an inhomogeneous parabolic-hyperbolic equation with a noncharacteristic type change line. Boundary conditions of the first kind are posed on characteristics in the parabolic and hyperbolic parts of the domain where the equation is given, and a condition of the third kind is posed on the noncharacteristic part of the boundary in the parabolic part. First, we study the solvability of an inhomogeneous initial–boundary value problem for a parabolic equation.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

The method of lines for Poisson's equation with nonlinear or free boundary conditions

Summary The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.This work was supported by the U.S. Army Research Office under Grant DA-AG29-76-G-0261     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

Finding the unknown boundary in the initial boundary-value problem for the heat equation

The article considers the determination of the boundary of a two-dimensional region in which an initial boundary-value problem for the heat equation is defined, given the solution of the problem for all time instants at some points of the region. The direct problem is reduced to an integral equation, and numerical solutions of the inverse problem are obtained for the case when the boundary is an ellipse. We investigate the sensitivity of the observed variables to the location (relative to the boundary) of the point where the right-hand side of the equation is specified. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 18–24.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

Nonclassical boundary value problem for the transport equation

We consider a nonclassical boundary value problem for the transport equation. The particle transport process is described by a stationary linear integro-differential equation, and the outgoing particle flux density on some part of the boundary is specified as the boundary condition. We find the particle flux density for a given outgoing flux and known coefficients of the equation. We show that, under some restrictions on the medium, there exists a unique solution of the problem, which can be represented by an infinite convergent series.     

Numerical solution of the inverse problem for the polarized-radiation transfer equation

An inverse problem for the steady vector transfer equation for polarized radiation in an isotropic medium is studied. For this problem, an attenuation factor is found from a given solution of the equation at a medium boundary. An approach is propounded to solve the inverse problem by using special external radiative sources. A formula is derived which relates the Radon transform of an attenuation factor to a radiation-flux density at the boundary. Numerical experiments show that the algorithm for the polarized-radiation transfer equation has an advantage over the method used in the scalar case.     

Liouville Property for Solutions of the Linearized Degenerate Thin Film Equation of Fourth Order in a Halfspace

We consider a boundary value problem in a half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to the normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.     

Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Boundary Identification for a Blast Furnace

In this paper,the authors discuss an inverse boundary problem for the axi- symmetric steady-state heat equation,which arises in monitoring the boundary corrosion for the blast-furnace.Measure temperature at some locations are used to identify the shape of the corrosion boundary. The numerical inversion is complicated and consuming since the wear-line varies during the process and the boundary in the heat problem is not fixed.The authors suggest a method that the unknown boundary can be represented by a given curve plus a small perturbation,then the equation can be solved with fixed boundary,and a lot of computing time will be saved. A method is given to solve the inverse problem by minimizing the sum of the squared residual at the measuring locations,in which the direct problems are solved by axi- symmetric fundamental solution method. The numerical results are in good agreement with test model data as well as industrial data,even in severe corrosion case.     

On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion

A two-point Neumann boundary value problem for a two-ion electro-diffusion model reducible to the Painlevé II equation is investigated. The problem is unconventional in that the model equation involves yet-to-be-determined boundary values of the solution. In prior work by Thompson, the existence of a solution was established subject to an inequality on the physical parameters. Here, a two-dimensional shooting method is used to show that this restriction may be removed. A practical algorithm for the solution of the boundary value problem is presented in an appendix.     

Initial-boundary value problem for an equation of internal gravity waves in a 3-D multiply connected domain with Dirichlet boundary condition

Method of boundary integral equations is applied to the initial-boundary value problem for an equation of fourth order and composite type in 3-D multiply connected domain with Dirichlet boundary condition. The problem controls nonsteady internal gravity waves in a stratified fluid. The problem is reduced to the time-dependent integral equation. It is shown that the integral equation is solvable. The solution of the problem is obtained in the form of dynamic potentials. The density in potentials obeys this integral equation. Therefore, the existence theorem is proved. Besides, the uniqueness of the solution is studied. All results hold for both interior and exterior domains with appropriate conditions at infinity.     

Scattering by Rough Surfaces: the Dirichlet Problem for the Helmholtz Equation in a Non-locally Perturbed Half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.     

Existence and uniqueness of solutions for acoustic scattering over infinite obstacles

The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.