The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

On global regular solutions of third order partial differential equations

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. Diffusion in a media with two distinct families of diffusion paths is modelled by two coupled linear partial differential equations of parabolic type with diffusivities D₁ and D₂. Physically the situation D₂ ? D₁ is of some considerable interest and previously established results, for D₂ non-zero, for the solution of boundary value problems, are not applicable to the idealized theory characterized by D₂ vanishing. An integral equation, which arises in the solution of boundary value problems for this idealized theory, is formally solved.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

The unbounded solution of a periodic mixed Sturm–Liouville problem in an infinite strip for the Laplacian

The unbounded solution, at the points where the boundary conditions change, for a mixed Sturm–Liouville problem of the Dirichlet–Neumann type can be obtained using the method of the integral equation formulation. Since this formulation is usually reduced to an infinite algebraic system in which the unknowns are the Fourier coefficients of the unknown unbounded entity, a study of ?_p-solutions imposes itself concerning the influence of the truncation on such systems. This study is achieved and the well-known theorem on the ?₂-solutions of the infinite algebraic systems is generalized.     

The Plemelj Formula of Higher Order Partial Derivatives of the Bochner-Martinelli Type Integral

In this paper, by using the technique of integral transformation, we obtain the Plemelj formulas with the Cauchy principal value and the Hadamard principal value of mixed higher order partial derivatives for integral of the Bochner-Martinelli type on a closed smooth manifold ∂D in Cⁿ. From the Plemelj formulas and using the theory of complex partial differential equation, we prove that the problem of higher order boundary value D^κΦ⁺(t) = D^κΦ⁻(t) + f(t) is equivalent to a complex linear higher order partial differential equation. Moreover, given a proper condition of the Cauchy boundary value problem, the problem of higher order boundary value has a unique branch complex harmonic solution satisfying Φ⁻(∞) = 0 in Cⁿ\∂D.     

Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the relationship of some Fredholm integral equations of the first kind to a family of boundary value problems

It is shown that the solution of a class of forced convection heat transfer problems can be used as a vehicle for exhibiting a correspondence between certain boundary value problems and their associated integral equations. If the solution of the boundary value problem is known then the solution of the integral equation can be found by a simple calculation. This generalizes the relationship of certain solutions of Laplace's equation to integral equations having logarithmic or Cauchy kernels. When the boundary value problems have separable solutions, the solution of the integral equation can be reduced to the verification of a set of identities μ_n ∫_EP(t) φ_n(t) k(x ? t) dt = φ_n(x), x?E, where the {φ_n} form an orthonormal set on E.Once the method of solution has been derived from the physical approach it can be put on a firm mathematical basis.     

Solution of problems on the anti-plane deformation of elastic bodies with corner points by the method of integral equations

The problem of the antiplane deformation of an elastic cylinder with a multiconnected finite or infinite section, bounded by a system of closed curves that can have corner points, is examined. Forces or displacements are given on the whole boundary of the body. The problem is reduced to an integral equation whose kernel has strong stationary singularities at the corner points. Results of an investigation of the solvability of this equation and the smoothness of its solution are presented. A procedure for the numerical solution of the integral equation is described. A space with a prismatic hole of rectangular section or a rigid inclusion subjected to a uniform tangential force at infinity is considered as an example. The generalized stress intensity factors are calculated.     

Diffraction of water waves due to a presence of two headlands

This paper is concerned with a boundary value problem for the Helmholtz equation on a horizontal infinite strip with obstacles. The derivation of Helmholtz equation from shallow water equations is given and the boundary value problem with an arbitrary shap of headland is stated. The boundary conditions are of the general Neumann type, and thus we use the finite difference method in numerical solution. Helholtz equation is replaced by the five-points formula and for the points close to the boundary, Taylors expansions are made useful with non-uniform spacing. For solving the resulting system of linear equations, the “Mathematica” package is used. The graphs show the velocity potential contours in the cases, of semielliptic, semicircular and narrow headland. Also, we discuss the problem in the presence of two headlands.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.     

Transient elastodynamic three-dimensional problems in cracked bodies

The general formulation of the transient elastodynamic second boundary value problem in an isotropic linear elastic body with a crack of arbitrary shape by combining the boundary integral equation method and the Laplace transform with respect to time is presented in this paper. Both finite and infinite elastic bodies are considered. A numerical solution of the transformed boundary integral equations is proposed.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

Interactions in an infinite medium of planar cracks with completely rigid planar inclusions

The problem of determining the interactions in an infinite medium of planar cracks with absolutely rigid inclusions leads to a system of integral equations, the regular kernals of which represent the interaction. The system of integral equations is completely determined under boundary conditions for the equilibrium of the inclusions as a rigid body. An approximate solution for the system of integral equations is used. The dependence of the magnitude of the external load on parameters characterizing the distribution in the medium of disc-shaped cracks and inclusions is graphically presented.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 63–68, 1989.     

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R ³. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.     

The analytic continuation of solutions to elliptic boundary value problems in two independent variables

An explicit representation is derived for the continuation across an analytic boundary of the solution to a boundary value problem for an analytic elliptic equation of second order in two independent variables. The representation is in terms of Cauchy data on the boundary and the complex Riemann function. This is equivalent to a representation for the solution to Cauchy's problem given by Henrici in 1957. It is confirmed that the method of complex characteristics is satisfactory for locating real singularities in the solution provided that the Riemann function is entire in its four arguments. Applications to Laplace's and Helmholtz's equations are discussed. By inserting known, simple solutions to the latter equation into the representation formula, several nontrivial integral relations involving the Bessel function J₀, and a possibly new series expansion for J_μ(x), are found.