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Recycling solutions of boundary value problems: the Wiener--Hopf perspective on embedding formula

A. I. Korolkov, A. V. Kisil

Abstract

Embedding formula allows to recycle solution of a family boundary value problems by expressing all the solutions in terms of a small number of solutions. Such formulas have been previously derived in the context of diffraction by applying a cleverly chosen operator to the solution and the construction of edge Green's functions which are introduced in an elaborate manner specific for each problem. We demonstrate that embedding formula naturally appears from a matrix Wiener--Hopf equation, and the embedding formula is derived from the canonical solution to this matrix Wiener--Hopf problem. This allows to drive the embedding formula in any context where the problem can be formulated as a Wiener--Hopf equation. We illustrate the effectiveness of this approach by revisiting known problems, such as the problem of diffraction by half-line, a strip and the problem of diffraction by a wedge. Additionally, a new matrix Wiener--Hopf formulation is derived for wedge problems.

Recycling solutions of boundary value problems: the Wiener--Hopf perspective on embedding formula

Abstract

Embedding formula allows to recycle solution of a family boundary value problems by expressing all the solutions in terms of a small number of solutions. Such formulas have been previously derived in the context of diffraction by applying a cleverly chosen operator to the solution and the construction of edge Green's functions which are introduced in an elaborate manner specific for each problem. We demonstrate that embedding formula naturally appears from a matrix Wiener--Hopf equation, and the embedding formula is derived from the canonical solution to this matrix Wiener--Hopf problem. This allows to drive the embedding formula in any context where the problem can be formulated as a Wiener--Hopf equation. We illustrate the effectiveness of this approach by revisiting known problems, such as the problem of diffraction by half-line, a strip and the problem of diffraction by a wedge. Additionally, a new matrix Wiener--Hopf formulation is derived for wedge problems.

Paper Structure

This paper contains 20 sections, 174 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. Normal solutions to the Wiener--Hopf equations
  5. Edge Green's functions as normal solutions
  6. General procedure
  7. Canonical embedding
  8. Plane wave embedding
  9. Examples
  10. Diffraction by a half-plane
  11. Diffraction by a strip
  12. Diffraction by a right-angled wedge
  13. Wiener--Hopf equations derived via additional mappings
  14. Wiener--Hopf equation for the right-angled wedge via additional mapping
  15. Conclusions
  16. ...and 5 more sections

Figures (11)

  • Figure 1: Geometry of the problem of diffraction of a plane wave incident at angle $\theta_i$ by a finite strip.
  • Figure 2: Geometry of problems for edge Green's functions. There is a point source located either on the left or the right of the strip.
  • Figure 3: Directivities for the strip problem, with $ka=10$ (left: for the plane wave incidence problem, right: for the edge Green's function).
  • Figure 4: Geometry of the problem of diffraction by a half-plane, with a plane wave incidence at angle $\theta_i$.
  • Figure 5: Geometry of the problem of diffraction by a right-angled wedge.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Example 1
  • Remark 2
  • Remark 3
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9