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Kisil, Anastasia V.

A generalization of the Wiener-Hopf method for an equation in two variables with three unknown functions. (English) Zbl 07831791

SIAM J. Appl. Math. 84, No. 2, 464-476 (2024).
The purpose of the author is to study the following generalization of the Wiener-Hopf equation \[ (1):\ A(x,y)\Pi(x,y)+B(x,y)\Psi_1(x)+C(x,y)\Psi_2(y)+D(x,y)=0, \] for \((x,y)\in D_x\times D_y=\{x\in\mathbb C:|x|\le 1\}\times\{y\in\mathbb C:|y|\le 1\},\) where \(\Pi(x,y), \Psi_1(x),\Psi_2(y)\) are unknowns analytic functions in \(\overset{\circ}{D_x}\times\overset{\circ} D_y\), \(\overset{\circ}D_x\), \(\overset{\circ}D_y\), respectively, and \(B(x,y)\), \(C(x,y)\), \(D(x, y)\) are given polynomials in \(x\) and in \(y\), \(A(x,y)=xy(x+x^{-1}+y+y^{-1}+a)\) such that \(\operatorname{Im}{a}\) is a very small positive number. The method of solution is described in the fifth section.
Reviewer: Mohammed El Aïdi (Bogotá)

MSC:

47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
35Q15 Riemann-Hilbert problems in context of PDEs
78A45 Diffraction, scattering
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74J20 Wave scattering in solid mechanics
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane

Keywords:

Wiener-Hopf equation; discrete Helmholtz equation; Riemann-Hilbert problem; acoustics; Toeplitz operator
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References:

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