Transmission Boundary Value Problems Research Articles

A Study of The Scattering Properties of Eigenparameter- Dependent Matrix Difference Operator With Transmission Condition

In this paper, we set a transmission boundary value problem for a matrix valued difference equation on the semi axis. The main purpose of this study is to examine the properties of scattering solutions and scattering functions of this problem. Firstly, by giving the Jost solution and scattering solutions of this problem, we obtain the Jost function and the scattering function of the problem. We also investigate eigenvalues, spectral singularities, resolvent operator and continuous spectrum of this problem.

Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems

In this study, we developed a new modification of the well-known differential transform method (DTM) that differs from the classical one by the algorithm for calculating the coefficients of an approximate solution given as a series. The proposed new algorithm we will call α-parameterized DTM (α-p DTM,for short). By using the proposed α-p DTM, we solve the boundary value transmission problem for the third-order differential equation. For the same problem, an approximate solution is also found using also the classical DTM. The solutions obtained were compared graphically.

New scheme of the discrete sources method for two-dimensional scattering problems by penetrable obstacles

A new computational scheme of the Discrete Sources Method (DSM) is developed for the numerical solution of two-dimensional acoustic and electromagnetic transmission boundary-value problems (BVPs) pertaining to the Helmholtz equation. To establish the new DSM scheme, we show that the matrix integral operator, corresponding to the transmission boundary conditions, has a dense range. The approximate solution of the BVP is constructed according to the new DSM scheme and it is proven that it converges uniformly to the exact solution of the BVP. An analytic representation is derived for the calculation of the scattering cross section in terms of the determined DSs amplitudes avoiding an integration over the unit circle. The numerical implementation procedure of the DSM is described and numerical results for the application of the DSM are presented. The new scheme is shown to be accurate and efficient even for highly-elongated scatterers.

DISCONTINUOUS FRACTIONAL STURM-LIOUVILLE PROBLEMS WITH EIGEN-DEPENDENT BOUNDARY CONDITIONS

<abstract> In this paper, a fractional discontinuous Sturm-Liouville type boun-dary-value problem with eigenparameter-dependent boundary conditions and with two fractional transmission conditions is investigated. Using operator theory, a new inner product is defined by combining the parameters in the boundary and transmission conditions, then the boundary value transmission problem is transferred to an operator in a new Hilbert space such that the eigenvalues and eigenfunctions of the main problem coincide with those of this operator. Moreover, the fundamental solutions are constructed, and then the characteristic function whose zeros are the eigenvalues of the problem is established. </abstract>

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.

Third-order boundary value transmission problems

In this paper, we consider some third-order operators with transmission conditions. In particular, it is shown that such operators are formally symmetric in the corresponding Hilbert spaces and we introduce the resolvent operators associated with the differential operators. After showing that the eigenvalues of the problems are real and discrete we introduce some ordinary and Frechet derivatives of the eigenvalues with respect to some elements of data.

Higher dimensional transmission problems for Dirac operators on Lipschitz domains

Transmission boundary value problems for the time-harmonic Maxwell system, the Helmholtz operator, and the perturbed Dirac operator are formulated, using Clifford algebras, on bounded Lipschitz domains of Rm with m≥3. It is shown how the Dirac problem decouples into several Maxwell and Helmholtz problems. Necessary and sufficient conditions are provided for well-posedness in each case, and for the Dirac problem to be equivalent to one or several independent Maxwell problems.

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

We present a novel hybrid numerical–asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.

A New Method for Dissipative Dynamic Operator with Transmission Conditions

In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class \(C_{0}.\) With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction.

Dependence of eigenvalues of 2nth order boundary value transmission problems

The present paper deals with the dependence of eigenvalues of 2nth order boundary value transmission problems on the problem. The eigenvalues depend not only continuously but also smoothly on the problem. Some new differential expressions of eigenvalues with respect to an endpoint, a coefficient, the weight function, boundary conditions, and transmission conditions, are given.

Generalized Fourier Series as Green’s Function Expansion for Multi-interval Sturm–Liouville Systems

This study aims to investigate a class of boundary-value transmission problems consisting of Sturm–Liouville equation on finite number disjoint intervals together with eigenparameter-dependent boundary conditions and supplementary transmission conditions at the interior transmittal points. We introduce new Hilbert spaces for self-adjoint realization of the problem, and state the main spectral properties of eigenvalues and eigenfunctions of the considered problem. Then by suggesting own approaches, we have presented a formula for Green’s function and resolvent operator. Finally, we find the resolvent function for corresponding inhomogeneous problem and establish completeness relation of eigenfunctions as generalized Fourier series.

Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem

The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point. We introduce a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator. By using our own approaches we investigate such properties as uniform convergence of the eigenfunction expansions, the Parseval equality, the Rayleigh-Ritz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary value-transmission problem (BVTP).

Second-order differential operators with interior singularity

The purpose of this study is to investigate a new class of boundary value transmission problems (BVTPs) for a Sturm-Liouville equation on two separate intervals. We introduce a modified inner product in the direct sum space $L_{2}[a,c)\oplus L_{2}(c,b]\oplus C^{2}$ and define a symmetric linear operator in it in such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then, by suggesting own approaches, we construct the Green’s function for the BVTP under consideration and find the resolvent function for the corresponding inhomogeneous problem.

On the rate of the convergence of the characteristic values of an integral operator associated with a dissipative fourth order differential operator in lim-4 case with finite transmission conditions

In this paper, our main aim is to investigate the spectral properties of a singular dissipative fourth order boundary value problem in lim-4 case with finite transmission conditions. For this purpose we construct a suitable differential operator in an appropriate Hilbert space. After showing that this differential operator is a dissipative operator we pass to the resolvent operator with an explicit form. Using this resolvent operator and Krein’s theorem we prove a completeness theorem on the boundary value transmission problem.

Spectral analysis of eigenparameter dependent boundary value transmission problems

In this paper, the singular second order differential operators are considered defined on the multi-interval. Some boundary and transmission conditions are imposed on the maximal domain functions with the spectral parameter. After constructing the differential operators associated with the boundary value transmission problems on the suitable Hilbert spaces, it is proved that these operators are the maximal dissipative operators. Finally constructing the model operators which are established with the help of the scattering functions, it is proved that all root vectors of the maximal dissipative operators are complete in the Hilbert spaces.

Eigenparameter dependent Sturm–Liouville problems in boundary conditions with transmission conditions

In this paper, we investigate the nonselfadjoint (dissipative) boundary value transmission problems in Weyl’s limit-circle case. At first using the method of operator-theoretic formulation we pass to a new operator. After showing that this new operator is a maximal dissipative operator, we construct a selfadjoint dilation of the maximal dissipative operator. Using the equivalence of the Lax–Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function, we show that all eigenfunctions and associated functions are complete in the space Lw2(Ω).

The domain derivative of time‐harmonic electromagnetic waves at interfaces

The scattering of time‐harmonic electromagnetic waves by a penetrable obstacle is considered. In view of shape optimization or inverse reconstruction problems, the domain derivative of the scattering problem is investigated. Existence of the derivative in the sense of a Fréchet derivative and a characterization by a transmission boundary value problem are shown. Copyright © 2012 John Wiley &amp; Sons, Ltd.

On the characteristic values of the real component of a dissipative boundary value transmission problem

In this paper, using Krein’s theorem, we investigate the completeness of the system of root vectors of a Bessel-type singular dissipative boundary value transmission problem in the Weyl’s limit circle case.

Krein’s Theorems for a Dissipative Boundary Value Transmission Problem

Let us consider the differential expression $$\ell (y)=-y^{\prime \prime }+q(x)y,\quad x\in I:=[0,c)\cup (c,\infty ),$$ where c is a transmission point and is regular for the differential expression l(y). We assume that Weyl’s limit-circle case holds for the differential expression l(y) on I. In this paper, using Krein’s theorems, we investigate the completeness of the root vectors of a singular dissipative boundary value transmission problem generated by l(y).

Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach

We consider a bounded open subset 𝕀 o of ℝ n with 0 ∈ 𝕀 o , and a function f o of ∂𝕀 o to ℝ. Under reasonable assumptions, the Dirichlet problem Δu = 0 in 𝕀 o , u = f o on ∂𝕀 o , has one and only one solution ũ o . Then we consider another bounded open subset 𝕀 i of ℝ n with 0 ∈ 𝕀 i , and an increasing diffeomorphism F of ℝ onto itself, and a constant γ ∈]0, +∞[, and a function g of ∂𝕀 i to ℝ, and we consider the non-linear transmission boundary value problem for ε > 0 small, where νε𝕀 i is the outward unit normal to ε∂𝕀 i . Under suitable conditions on the data, we show that for sufficiently small, such a boundary value problem admits locally around (F (−1)(ũ o (0)), ũ o ) a family of solutions . Then we show that u i (ε, ε·) and (suitable restrictions of) u o (ε, ·) and u o (ε, ε·) can be continued real analytically in the parameter ε around ε = 0 for n ≥ 3, and can be represented in terms of real analytic functions of ε, log−1 ε, ε log2 ε for n = 2.