Abstract
This paper reviews the modern state of the Wiener–Hopf factorization method and its generalizations. The main constructive results for matrix Wiener–Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener–Hopf technique.
Highlights
The Wiener–Hopf method has been motivated by interdisciplinary interests ever since its inception
It resulted from a collaboration between Norbert Wiener, who worked on stochastic processes, and Eberhard Hopf, who worked on partial differential equations (PDEs)
This review offers a short introduction to the application, theory and numerical implementation of the Wiener–Hopf method; each element forms an important part of the whole
Summary
The Wiener–Hopf method has been motivated by interdisciplinary interests ever since its inception. We begin by formulating the basic Wiener–Hopf equation (resulting from applying the Fourier transform to equation of type (1.2)) and discussing the standard method of its solution (for details of its derivation, see [17]) This will be compared and contrasted with the Riemann–Hilbert boundary value problem at the end of the section. (ii) The sum of the orders κj at infinity of the columns of the matrix X(z) (i.e. solutions Xj(z)) is equal to the Cauchy index of the determinant det X(z) They are called partial indices of the Riemann–Hilbert boundary value problem (2.15). As an illustration of the concept, we consider a class of partial differential equations (PDEs) that can be reduced to a Wiener–Hopf equations or a Riemann–Hilbert boundary value problem ([3], §20.3) This method is suitable for linear PDEs of the form. We list a number of approaches to particular constructive exact or approximate factorization methods in the sections below
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