The Mixed Problem for the Laplace Equation in an Exterior Domain with an Arbitrary Partition of the Boundary

In this paper we propose a method for solving the mixed boundary-value problem for the Laplace equation in a connected exterior domain with an arbitrary partition of the boundary. All simple closed curves making up the boundary are divided into three sets. On the elements of the first set the Dirichlet condition is given, on the elements of the second set the third boundary condition is prescribed, and the third set, in turn, is divided into two subsets of simple closed arcs, with the Dirichlet condition prescribed on the elements of one of these subsets and the third boundary condition on the elements of the other subset. The problem is reduced to a uniquely solvable Fredholm equation of the second kind in a Banach space. The third boundary-value problem and the mixed Dirichlet--Neumann problem are particular cases of the problem under study.     

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain. In many practice applications, we cannot know the initial temperature during heat process, therefore we consider a non-characteristic Cauchy problem for the heat equation. The method of fundamental solutions is applied to solve this problem. Due to ill-posedness of this problem, we first discretize the problem and then regularize it in the form of discrete equation. Numerical tests are conducted for showing the effectiveness of the proposed method.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.     

On a New Approach to Frequency Sounding of Layered Media

Frequency sounding of layered media is modeled by a hyperbolic problem. Within the framework of this model, we formulate an inverse problem. Applying the Laplace transform and introducing the impedance function, the latter is first reduced to the inverse boundary value problem for the Riccati equation and then to the Cauchy problem for a first-order quadratic equation. The advantage of such transformations is that the quadratic equation does not contain an unknown coefficient. For a specific class of data, it is shown that the Cauchy problem is uniquely solvable. Based on the asymptotic behavior of solutions to both the Riccati and quadratic equations, a stable reconstruction algorithm is constructed. Its feasibility is demonstrated in computational experiments.     

一类双重退化抛物方程局部解的存在性

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate p- Laplace equation(p＞2) u_t-div(b(x,t,u)|▽u|~(p-2)▽u)=f(x,u,t). The initial-boundary value problem in a bounded domain of R~N is considered under mixed boundary conditions.The existence of local-in-time weak solutions is obtained.     

Inverse shape and surface heat transfer coefficient identification

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin is considered. This problem which consists of determining an unknown inner boundary of an annular domain and possibly its surface heat transfer coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux) is solved numerically using the meshless method of fundamental solutions (MFS). A nonlinear unconstrained minimisation of the objective function is regularised when noise is added to the input boundary data. The stability of the numerical results is investigated for several test examples with respect to noise in the input data and various values of the regularisation parameters.     

Determination of an unknown function in a parabolic inverse problem by Sinc‐collocation method

In this article, an inverse problem of determining an unknown time‐dependent source term of a parabolic equation is considered. We change the inverse problem to a Volterra integral equation of convolution‐type. By using Sinc‐collocation method, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are given to demonstrate the computational efficiency of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1584–1598, 2010     

Determining of three unknown functions of a semilinear ultraparabolic equation

In this paper, we consider the inverse problem for second‐order semilinear ultraparabolic equation. The equation has unknown function of time variable in its minor coefficient and two unknown functions of time and spacial variables in its right‐hand side. Initial, boundary, and integral type overdetermination conditions are posed. By using the properties of the solutions of the corresponding initial‐boundary value problem and the method of successive approximations, the sufficient conditions of the existence, and the uniqueness of the solution for the inverse problem are obtained on some time interval that depends on the coefficients of the equation.     

一个抛物型方程侧边值问题的正则逼近解在一类Sobolev空间中的最优误差界

逆热传导问题(IHCP)是严重不适定问题,即问题的解(如果存在)不连续依赖于数据．但目前关于逆热传导问题的已有结果主要是针对标准逆热传导问题．文中给出了出现在实际问题中的一个抛物型方程侧边值问题,即一个含有对流项的非标准型逆热传导问题的正则逼近解一类Sobolev空间中的最优误差界．     

The direct and inverse problem for an inclusion within a heat-conducting layered medium

This paper is concerned with the problem of heat conduction from an inclusion in a heat transfer layered medium. Making use of the boundary integral equation method, the well-posedness of the forward problem is established by the Fredholm theory. Then an inverse boundary value problem, i.e. identifying the inclusion from the measurements of the temperature and heat flux on the accessible exterior boundary of the medium is considered in the framework of the linear sampling method. Based on a careful analysis of the Dirichlet-to-Neumann map, the mathematical fundamentals of the linear sampling method for reconstructing the inclusion are proved rigorously.     

一类Laplace双曲型方程广义Cauchy问题的反问题

本文讨论了确定Ｌａｐｌｉｔｃｅ双曲型方程u_xy(x,y)+a(x,y)u_x(x,y)+b(x,y)+u_y(x,y)+q(x)u(x,y)=f(x,y)的广义Ｃａｕｃｈｙ问题中系数ｑ（ｘ）的反问题。文中利用特征法线及不动点理论，导出了与反问题等价的非线性积分方程组，证明了反问题局部解的存在唯一性，最后给出了反问题整体用的唯一性定理。     

局部扰动半平面上的散射问题

该文讨论半平面上有局部扰动情况下的散射问题．通过位势理论,应用边界积分方程的方法研究了该问题解的存在与唯一性．主要方法是运用对称反射,使该无界区域上的散射问题变成一个有界区域上的散射问题,只是这一有界区域的边界不光滑．通过仔细分析相应的边界积分算子,作者得到了其解的存在与唯一性．     

On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition

We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 522–534.Original Russian Text Copyright © 2005 by V. L. Kamynin.This revised version was published online in April 2005 with a corrected issue number.     

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhge?m–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.     

ANALYSIS OF TWO-DIMENSIONAL ELECTROMAGNETIC SCATTERING BY A PERFECTLY CONDUCTING OBSTACLE IN A HOMOGENEOUS CHIRAL ENVIRONMENT

The time-harmonic electromagnetic plane waves incident on a perfectly conducting obstacle in a homogeneous chiral environment are considered.A two-dimensional direct scat- tering model is established and the existence and uniqueness of solutions to the problem are discussed by an integral equation approach.The inverse scattering problem to find the shape of scatterer with the given far-field data is formulated.Result on the uniqueness of the inverse problem is proved.     

On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution of heat equation with nonlocal boundary and overdetermination conditions is considered. The existence, uniqueness and continuous dependence upon the data are studied. Some considerations on the numerical solution for this inverse problem are presented with the examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier

The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions.