Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials. (Received: September 27, 2005)     

Duality theorems in some overdetermined boundary value problems

We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

Boundary Integral Equations for Mixed Boundary Value Problems in R3

Both exterior and interior mixed Dirichlet-Neumann problems in R³ for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.     

Axisymmetric Stokes' flow in a spherical shell revisited via the Fokas method. Part I: Irrotational flow

The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiband sampling theorems for Mittag-Leffler transforms bandlimited on rays

We derive a biorthogonal Kramer analytic theorem, for integral transforms whose kernels generate biorthogonal bases in Hilbert spaces. The theorem is applied to various integral transforms associated with classes of fractional integro-differential eigenvalue problems, leading to Lagrange-type interpolation sampling theorems, derived by Djrabshian [Harmonic analysis and boundary value problems in the complex domain. Basel: Birkhäuser; 1993]. We work out some concrete examples, illustrating these sampling expansions.     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

The Heat Equation in the Interior of an Equilateral Triangle

We present the solution of the classical problem of the heat equation formulated in the interior of an equilateral triangle with Dirichlet boundary conditions. This solution is expressed as an integral in the complex Fourier space, i.e., the complex k₁ and k₂ planes, involving appropriate integral transforms of the Dirichlet boundary conditions. By choosing Dirichlet data so that their integral transforms can be computed explicitly, we show that the solution is expressed in terms of an integral whose integrand decays exponentially as . Hence, it is possible to evaluate this integral numerically in an efficient and straightforward manner. Other types of boundary value problems, including the Neumman and Robin problems, can be solved similarly.     

Integral representations and boundary value problems for a linear third-order elliptic system with an interior supersingular point

For a supersingular elliptic system, we find an integral representation of the solution and the corresponding inversion formula depending on the values of roots of the characteristic equation, which is of interest from both the theoretical and the practical viewpoint. All studies are carried out for the case in which the singular point is an interior point of the domain. Note that this case is most complicated. In the resulting integral representations, we clearly single out the singular part of the solutions, which permits analyzing their asymptotic behavior with respect to r. We study the influence of the supersingular point on the solvability of boundary value problems and find a well-posed statement of a number of Dirichlet and Riemann-Hilbert boundary value problems.     

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound‐soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.     

Sharp inequalities of singular values of smooth kernels

New inequalities of singular values of the integral operators with smoothL ² kernels are obtained and shown by examples to be sharp if the kernels satisfy also certain boundary conditions. These results are based on an idea of Gohberg-Krein by which the singular values of the integral operators are interrelated to the eigenvalues of some two point boundary value problems.Dedicate to Professor Ky Fan on the occasion of his 85th birthday     

Spectral theory and iterative methods for the Maxwell system in nonsmooth domains

We study spectral properties of boundary integral operators which naturally arise in the study of the Maxwell system of equations in a Lipschitz domain Ω ? ?³. By employing Rellich‐type identities we show that the spectrum of the magnetic dipole boundary integral operator (composed with an appropriate projection) acting on L²(?Ω) lies in the exterior of a hyperbola whose shape depends only on the Lipschitz constant of Ω. These spectral theory results are then used to construct generalized Neumann series solutions for boundary value problems associated with the Maxwell system and to study their rates of convergence (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

Boundary value problems in the theory of binary mixtures

In this paper, the boundary value problems of steady oscillation (vibration) of the linear theory of thermoelasticity for binary mixtures are investigated by means of the boundary integral equation method (potential method). The uniqueness and existence theorems of solutions of the exterior boundary value problems by means potential method and multidimensional singular integral equations are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radial symmetry and partially overdetermined problems in a convex cone

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

The exact order of information-based complexity of weakly singular integral equations with periodic analytic coefficients

The exact order of complexity of weakly singular integral equations including logarithmic singularities with periodic and analytic coefficients is found. This class of equations contains the boundary equations of exterior boundary value problems for the two-dimensional Helmholtz equation. Translated fromMatematicheskie Zarnetki, Vol. 62, No. 5, pp. 643–656, November, 1997. Translated by O. V. Sipacheva     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

An integral equation technique for scattering problems with mixed boundary conditions

This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a result, Calderón identities can be utilized to avoid the use of hypersingular integral operators. Numerical results illustrate the performance of the proposed solution technique.