Helmholtz equation outside an open arc in a plane with a mixed boundary condition

A boundary value problem for the Helmholtz equation outside an open arc in a plane is studied with mixed boundary conditions. In doing so, the Dirichlet condition is specified on one side of the open arc and the boundary condition of the third kind is specified on the other side of the open arc. We consider non-propagative Helmholtz equation, real-valued solutions of which satisfy maximum principle. By using the potential theory the boundary value problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original boundary value problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

A new integral equation approach to the Neumann problem in acoustic scattering

We suggest a new approach of reduction of the Neumann problem in acoustic scattering to a uniquely solvable Fredholm integral equation of the second kind with weakly singular kernel. To derive this equation we placed an additional boundary with an appropriate boundary condition inside the scatterer. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies a uniquely solvable Fredholm integral equation of the second kind and can be computed by standard codes. Copyright © 2001 John Wiley & Sons, Ltd.     

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated.     

On a generalization of the Neumann problem for the Laplace equation outside cuts in a plane

A boundary value problem for harmonic functions outside cuts in a plane is considered. The jump of the normal derivative is specified on the cuts as well as a linear combination of the normal derivative on one side of the cut and the jump of the unknown function. The problem is studied with three different conditions at infinity, which lead to different results on existence and number of solutions. The integral representation for a solution is obtained in the form of potentials density in which satisfies the uniquely solvable Fredholm integral equation of the 2nd kind. Copyright © 2004 John Wiley & Sons, Ltd.     

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.     

Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.     

The axisymmetric mixed problem of elasticity theory for a cone clamped along its side surface with an attached spherical segment

The axisymmetric mixed problem of the stress state of an elastic cone, with a spherical segment attached to the base, is considered. The side surface of the cone is rigidly clamped, while the surface of the spherical segment is under a load. By using a new integral transformation over the meridial angle the problem is reduced in transformant space to a vector boundary value problem, the solution of which is constructed using the solution of a matrix boundary value problem. The unknown function (the derivative of the displacements), which occurs in the solution, is determined from the approximate solution of a singular integral equation, for which a preliminary investigation is carried out of the nature of the singularity of the function at the ends of the integration interval. Subsequent use of inverse integral transformations leads to the final solution of the initial problem. The values of the stresses obtained are compared with those that arise in the cone for a similar load, when sliding clamping conditions are specified on the side surface of the cone (for this case an exact solution of this problem is constructed, based on the known result).     

The modified jump problem for the Laplace equation and singularities at the tips

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified of the cuts. The unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The singularities at the ends of the cuts are investigated.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A problem for the Helmholtz equation outside cuts on the plane with the dirichlet condition and the oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the Helmholtz equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition with pure imaginary coefficient of the tangential derivative is posed on the other side. We prove the uniqueness of the solution. The solvability of the problem is proved for the case in which the above-mentioned pure imaginary coefficient is less than unity in absolute value. In this case, we obtain an integral representation of the solution of the problem in the form of potentials. The densities of the potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. The boundary value problem considered here generalizes the mixed Dirichlet-Neumann problem.     

The direct electromagnetic scattering problem by a mixed impedance screen in chiral media

In this article solvability results for the direct electromagnetic scattering problem for a mixed perfectly conducting-impedance screen in a chiral environment is studied. In particular, incident time-harmonic electromagnetic waves in a chiral medium upon a partially coated open surface Γ (the ‘screen’), that satisfies an impedance boundary condition on one side and a perfectly conducting boundary condition on the other side, are considered. We introduce the Beltrami fields, appropriate boundary integral relations for these fields are proved and via them a uniqueness result is established. A variational method in a suitable functional space setting is considered and using a Calderon type operator for the chiral case, existence for the scattering problem is established.     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Pasternak地基上简支板振动问题的准格林函数方法

提出一种新的数值方法——准格林函数方法．以Pasternak地基上简支多边形薄板的振动问题为例,详细阐明了准格林函数方法的思想．即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将Pasternak地基上薄板自由振动问题的振型控制微分方程化为两个耦合的第二类Fredholm积分方程．边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性,最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率．数值方法表明,该方法具有较高的精度．     

The Elementary Solution of Triple Integral Equations and The Solution of Triple Series Equations Involving Associated Legendre Polynomials and their Application

A method is developed for the formal solution of an important class of triple integral equations involving Bessel functions. The solution of the triple integral equations is reduced to two simultaneous Fredholm integral equations and the results obtained are simpler than those of other authors and also superior for the purposes of solution by iteration. In the same manner the formal solution of triple series equations involving associated Legendre polynomials is presented. The solution of the problem is reduced to that of solving a Fredholm integral equation of the first kind. Finally to illustrate the application of the results an electrostatic problem is discussed.     

求解三维第一类Fredholm积分方程的GMRES法

利用数值求积公式,将三维第一类Fredholm积分方程进行离散,通过引入正则化方法,将离散后的积分方程转化为一离散适定问题,通过广义极小残余算法得到了其数值解．数值模拟结果表明该方法的可行有效性．     

An improved boundary element formulation for the motion of spherical bubbles

We suggest a modified boundary element method for modeling the potential flow caused by the motion of many spherical bubbles. The problem for the fluid velocity potential is reduced to a Fredholm integral equation of the second kind. The kernel of that integral equation has no singularity; as a result, its numerical solution does not encounter any difficulties. The matrix representation of the integral equation is diagonally dominant and is well suited to handle multiple bubble system.     

Boundary control of a hyperbolic system with one space variable

We consider a mixed problem in a half-strip for a hyperbolic system with one space variable and with constant coefficients. The control problem is to find boundary conditions ensuring that the system has a given state vector at a given instant of time. We study whether the problem is asymptotically solvable, i.e., whether there exists a sequence of boundary conditions such that the corresponding sequence of final state vectors uniformly converges to the given vector. We reduce the construction of a family of such sequences of boundary conditions with a function parameter to the solution of a Fredholm integral equation of the second kind and prove a sufficient condition for its unique solvability in terms of the problem data.     

Preconditioning techniques for the iterative solution of scattering problems

We consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems. We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.     

Inverse problem for a nonlinear Benney–Luke type integro-differential equations with degenerate kernel

We consider the problem on the unique solvability of the inverse problem for a nonlinear partial Benney–Luke type integro-differential equation of the fourth order with a degenerate kernel. We modify the degenerate kernelmethod which has been designed for Fredholm integral equations of the second kind to apply to the case of the above-mentioned equation. We exploit the Fouriermethod of separation of variables. By means of designations, the Benney–Luke type integro-differential equation is reduced to a system of algebraic equations. Using an additional condition, we obtain the countable system of nonlinear integral equations with respect to the main unknown function. We employ the method of successive approximations together with the contraction mapping principle. Finally, the restore function is defined.