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

TL;DR

This paper shows that embedding formulas for boundary value problems can be derived from matrix Wiener--Hopf equations, enabling solution recycling across various diffraction problems, including new formulations for wedge problems.

Contribution

It reveals that embedding formulas naturally emerge from matrix Wiener--Hopf equations, broadening their applicability to diverse boundary value problems.

Findings

Embedding formulas are derived from canonical solutions to matrix Wiener--Hopf problems.

The approach is demonstrated on classical diffraction problems like half-line, strip, and wedge.

A new matrix Wiener--Hopf formulation for wedge diffraction problems is introduced.

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.