Justification of a Numerical Method for Solving Systems of Singular Integral Equations in Diffraction Grating Problems

Gandel, Y.V.; Polyanskaya, Tatyana

doi:10.1023/b:dieq.0000012697.36651.0d

Cited by 20 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…From the Propositions 3-4 and the existence of bounded operators Appealing to the Theorem 1 and (63) it concludes the uniqueness and the existence of the solution of the problems (16)- (17).…”

Section: Proof Of Theoremmentioning

confidence: 53%

“…The scheme for numerical solution of this problem and the results of numerical experiments had been proposed in the article [13]. A rigorous mathematical justification for the numerical solution of various electromagnetic problems by the method of discrete singularities [6,17,18] had existed at the time of publication of the articles [12,13]. But the systems of boundary integral equations of the problem of electromagnetic waves scattering by a system of superconducting band are different from the systems of integral equations of other problems that were previously solved numerically by the method of discrete singularities.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

V.¹,

Dushkin²

2014

AJAMS

View full text Add to dashboard Cite

show abstract

Section: Proof Of Theoremmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

V.¹,

Dushkin²

2014

AJAMS

View full text Add to dashboard Cite

show abstract

“…where [⋅] denotes the , -th entry of the matrix. Substituting (3.10) into (3.4), we obtain the second necessary condition (3.11) in the form 11) where, according to Remark 2.13 and (3.10), the right hand side of (3.11) is independent of the choice of Γ −1 + or Γ −1 − . 1.…”

Section: Necessary and Sufficient Conditions For The Existence Of The ( ) Solution To Equation (14)mentioning

confidence: 98%

Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L²

Katsevich

Bertola

Tovbis

2021

Mathematische Nachrichten

View full text Add to dashboard Cite

Given n disjoint intervals I-0.16emj on double-struckR together with n functions ψj∈L2(Ij), j=1,⋯n, and an n×n matrix Θ=(θjk), the problem is to find an L2 solution φ⃗=Col(φ1,⋯,φn), φj∈L2(Ij), to the linear system χΘHφ⃗=ψ⃗, where ψ⃗=Col(ψ1,⋯,ψn), H=diag(scriptH1,⋯,scriptHn) is a matrix of finite Hilbert transforms with H-0.16emj defined on L2(Ij), and χ=diag(χ1,⋯,χn) is a matrix of the corresponding characteristic functions on I-0.16emj. Since we can interpret χΘHφ⃗, as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of double-struckR and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.

show abstract

“…A rigorous mathematical justification for the numerical solution of various electromagnetic problems by the method of discrete singularities [6,17,18] had existed at the time of publication of the articles [12,13]. But the systems of boundary integral equations of the problem of electromagnetic waves scattering by a system of superconducting band are different from the systems of integral equations of other problems that were previously solved numerically by the method of discrete singularities.…”

Section: Introductionmentioning

confidence: 99%

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

V.N.¹,

V.²,

Dushkin³

2014

AJAMS

View full text Add to dashboard Cite

In this article the method for numerical solution of boundary integral equations of the original problem is proposed. This method is one of the modifications of Nystrom-type methods; particularly the method of discrete vortices. The convergence of the numerical solutions to the exact solution of the problem is guaranteed by propositions proved in this article. Also, the rate of convergence of the approximate solutions to the exact solution had been found. Keywords: singular integral equation, modification of method of discrete vortices, existence of approximate solution, the rate of convergence of the approximate solutionsCite This Article: Maurya V.N., Gandel Yu. V., and Dushkin V.D., "The Approximate Method for Solving the Boundary Integral

show abstract

Justification of a Numerical Method for Solving Systems of Singular Integral Equations in Diffraction Grating Problems

Cited by 20 publications

References 0 publications

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L²

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

Contact Info

Product

Resources

About

Justification of a Numerical Method for Solving Systems of Singular Integral Equations in Diffraction Grating Problems

Cited by 20 publications

References 0 publications

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L2

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

Contact Info

Product

Resources

About

Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L²