## 1. Introduction

[2] The Wiener-Hopf (W-H) method is a general analytical technique which is able to deal with different kind of mathematical-physical and/or engineering problems [*Noble*, 1988; *Weinstein*, 1969; *Daniele*, 2003, 2004]. According to the authors' opinion the Wiener-Hopf method is one of the most important mathematical tool to obtain closed form solutions for a consistent number of fundamental (canonical) problems.

[3] In application problems, the W-H technique deals with the solution of the functional equation defined in the complex *α* plane:

This equation is called the Wiener-Hopf equation and it can be classified as scalar or vector, by respectively involving scalar quantities (*F*_{+}(*α*), *R*, *F*_{−}^{s}(*α*), *F*_{−}(*α*) and *G*(*α*)) or *n*-dimensional vector quantities (*F*_{+}(*α*), *R*, *F*_{−}^{s}(*α*) and *F*_{−}(*α*)) and matrix quantities of order *n* (*G*(*α*)).

[4] The unknowns of equation (1) are the plus and the minus functions *F*_{+}(*α*) and *F*_{−}^{s}(*α*). The functions *F*_{+}(*α*) and *F*_{−}(*α*) are generally Laplace transforms and they vanish as *α* → ∞.

The plus and the minus functions are called standard if they are regular respectively in the upper half plane *Im*[*α*] ≥ 0 and in the lower half plane *Im*[*α*] ≤ 0.

[5] The function *G*(*α*) is called kernel of the W-H equation. Note that the *G*(*α*) and its inverse (*G*^{−1}(*α*) = [*G*(*α*)]^{−1}) are regular function in the real axis *Im*[*α*] = 0. The kernel is defined by the physical problem; therefore it is known together with *R* and *α*_{o}. *R* and *α*_{o} are related to the source, it is always supposed that *Im*[*α*_{o}] ≠ 0. The properties of the problem are defined by the spectrum of *G*(*α*). We assume the absence of essential singularities in the spectrum, except at the infinity. The zeroes of det[G(*α*)] and det[G^{−1}(*α*)] define the structural singularities. We observe that the branch point singularities define the continuous spectrum and the poles define the discrete spectrum.

[6] Although (1) presents two unknowns, *F*_{+}(*α*) and *F*_{−}^{s}(*α*), the Wiener-Hopf equation constitute a closed mathematical problem which yields proper solution without any additional information, see proofs of the existence and the uniqueness of the solution in the classical mathematical literature [*Gohberg and Krein*, 1960].

[7] The W-H technique yields the following solution of the equation (1) [*Noble*, 1988; *Daniele*, 2004]:

where *G*_{+}^{−1}(*α*) = [*G*_{+}(*α*)]^{−1} and *G*_{−}^{−1}(*α*) = [*G*_{−}(*α*)]^{−1} arise from the factorization of the kernel *G*(*α*):

[8] In (3) and (4) the factorized function *G*_{+}(*α*) and *G*_{−}(*α*) and their inverses *G*_{+}^{−1}(*α*) and *G*_{−}^{−1}(*α*) are regular respectively in the half planes *Im*[*α*] ≥ 0 and *Im*[*α*] ≤ 0 and present algebraic behavior as *α* → ∞. The solution (3)–(4) shows that the central problem for the solution of Wiener-Hopf equation (1) is the factorization of the kernel (5). The simplest class of W-H equations are the scalar ones where closed form solutions are always possible [*Noble*, 1988; *Weinstein*, 1969; *Daniele*, 2004]. Conversely the general solution for vector W-H equations is not available in closed form and only some progress has been done to develop a general method of explicit solution [*Daniele*, 2004; *Hashimoto et al.*, 1993].

[9] Today we are able to factorize in closed form triangular matrices, rational matrices, i.e., matrices with entries that are rational functions of *α*, and matrices that commute with rational matrices [*Daniele*, 1984, 2004]. The need of developing an approximate technique of factorization arises from all the cases where no explicit factorization is possible.

[10] Several powerful mathematical methods of functional analysis help to efficiently solve W-H problems, for instance: iterative methods, moment methods, regularization methods and so on.

[11] Furthermore, since it is possible to factorize rational matrices in closed form, the vector factorization problem can be approximated by introducing rational approximants (for instance Padé type [*Abrahams*, 2000]) for the entries of the matrix kernel. We have investigated this technique in practical engineering applications [see *Daniele*, 2004].

[12] We are convinced that the reduction to Fredholm equations is possibly the best way to face the factorization problems.

[13] We confide in the Fredholm method since we experienced its efficacy and efficiency in different problems using different quadratures. Conversely the Padé approximants (or more in general rational approximants) do not assure accuracy/convergency when the order of the rational representation is increased.

[14] The aim of this paper is to present an efficient method to solve a general W-H problem which is based on the reduction of the factorization problem to the solution of a Fredholm equation of the second kind.

[15] This paper is organized as follow. In section 2 we derive the factorization of kernel using an homogeneous W-H formulation. Section 3 and 4 shows respectively the reduction of the W-H equations to Fredholm integral equations and their numerical solution. Section 4 includes some tools to improve the convergence of Fredholm integral equations and details on the analytical continuation of the numerical solution. Section 5 validates the proposed technique, presenting several numerical tests with kernel of practical applications. This section presents the efficiency and the convergence of the approximate Fredholm solution in particular in the framework of scattering/diffraction problems. See *Gohberg and Krupnik* [1992] and *Jones* [1979] for proofs on the compactness of kernel for Fredholm integral equations derived from W-H problems.