An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown

A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems

We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach

On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion

We propose an iterative procedure for the inverse problem of determining the displacement vector on the boundary of a bounded planar inclusion given the displacement and stress fields on an infinite (planar) line-segment. At each iteration step mixed boundary value problems in an elastostatic half-plane containing the bounded inclusion are solved. For efficient numerical implementation of the procedure these mixed problems are reduced to integral equations over the bounded inclusion. Well-posedness and numerical solution of these boundary integral equations are presented, and a proof of convergence of the procedure for the inverse problem to the original solution is given. Numerical investigations are presented both for the direct and inverse problems, and these results show in particular that the displacement vector on the boundary of the inclusion can be found in an accurate and stable way with small computational cost

An iterative method for the Cauchy problem for second-order elliptic equations

The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach

On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analyzed and made explicit, and standard quadrature rules are used for discretization. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data

On the Numerical Solution of a Hyperbolic Inverse Boundary Value Problem in Bounded Domains

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions

On the numerical solution of initial boundary value problems by the Laguerre transformation and boundary integral equations

this paper we wish to make a contribution to the solution of initial boundary value problems by studying the Laguerre transformation. The application of the Laguerre transformation for the approximate solution of nonstationary problems was suggested by Galazyuk[7] and used in [8, 16]. In this paper we will use the Laguerre transformation in combination with the boundary integral equation method for the numerical solution of an exterior initial boundary value problem for the wave equation with absorption in two space dimensions. The plan of the paper is as follows. In Section 2 we will introduce the initial boundary value problem which we want to consider and describe how it can be reduced to a sequence of exterior boundary value problems for the inhomogeneous Helmholtz equation with purely imaginary wave number by the Laguerre transformation. In Section 3 we will introduce appropriate singular solutions and reduce the sequence of boundary value problems to a sequence of integral equations of the first or second kind by a single- or a double-layer potential approach, respectively. Existence and uniqueness of solutions to these integral equations can be established by standard potential theoretic tools. The Section 4 is devoted to describing fully discrete numerical methods for the approximate solution both for the boundary in- 2 CHAPKO, KRESS tegral equation of the first and the second kind by using trigonometric interpolatory quadrature rules which explicitly acknowledge the logarithmic singularities of the kernels. For analytic boundaries and analytic boundary data this method is exponentially convergent. In the final Section 5 we illustrate by a numerical example that the method yields satisfactory approximations for the full time dependent problem. 2 THE LAGUERRE T..

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well

Recovering boundary data in planar heat conduction using a boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in ~\cite{Bast}, where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical experiments are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations

On the Numerical Solution of a Hypersingular Integral Equation for Elastic Scattering from a Planar Crack

In this paper we describe a fully discrete quadrature method for the numerical solution of a hypersingular integral equation of the first kind for the scattering of time-harmonic elastic waves by a cavity crack. We establish convergence of the method and prove error estimates in a Holder space setting. Numerical examples illustrate the convergence results