Wiener-Hopf Method

The Wiener-Hopf method is a powerful technique which enables certain linear partial differential equations subject to boundary conditions on semi-infinite domains to be solved explicitly. The method is sometimes referred to as the Wiener-Hopf technique or the Wiener-Hopf factorization.

The Wiener-Hopf method begins by applying the generalized upper and lower Fourier transforms to obtain an identity


on a strip

 S={z=sigma+itau:tau_i<tau<tau_+ and -infty<sigma<infty}

of the complex alpha-plane where alpha=sigma+itau is a complex variable. Note that identity () is in terms of the unknown functions Phi_+=Phi_+(alpha) and Psi_-=Psi_-(alpha) which are analytic in the half-planes tau>tau_- and tau<tau_+, respectively, while A(alpha), B(alpha), and C(alpha) are "parameter functions" in the alpha-plane which are analytic on all of S.

For simplicity, assume that A and B are non-zero in S. The most fundamental step of the Wiener-Hopf process is to find a solution for Phi_+ and Psi_- in () by finding functions K_+(alpha) and K_-(alpha)--analytic and nonzero in tau>tau_- and in tau<tau_+, respectively--so that


Upon doing so, the factorization () can be used to rewrite () as


whereby the last summand K_-(alpha)C(alpha)/B(alpha) can be decomposed as


for C_+, respectively C_-, analytic in the region of S satisfying tau>tau_-, respectively tau<tau_+.

Substituting () into () and rewriting induces a function J=J(alpha) of the form


which, despite being defined only in the strip S, can be defined and made analytic on the entire complex alpha-plane by way of analytic continuation. The idea behind () is to next show the existence of positive integers p,q in Z^+ for which

 |K_+(alpha)Phi_+(alpha)+C_+(alpha)|<|alpha|^p as alpha->infty,tau>tau_-


 |K_-(alpha)Psi_-(alpha)C_-(alpha)|<|alpha|^q as alpha->infty,tau<tau_+,

whereby Liouville's theorem applies and requires that J=J(alpha) be a polynomial P(alpha) of degree less than or equal to min(p,q). In particular,




thus defining Phi_+ and Psi_- to within the arbitrary polynomial P, i.e., to within a finite number of arbitrary constants which must be determined using other methods.

While the Wiener-Hopf method itself is a useful tool for solving various types of partial differential equations, one of its most significant strengths is the vast array of other equation solving methods derived therefrom. Indeed, the techniques spawned from the Wiener-Hopf factorization have proven useful in a number of very different circumstances across a diverse array of disciplines including theoretical and applied physics (Noble 1958), diffraction theory (Linton and McIver 2001), and fluid dynamics (Ho 2007).

See also

Fourier Transform, Generalized Fourier Integral, Laplace Transform, Liouville's Boundedness Theorem, Partial Differential Equation

This entry contributed by Christopher Stover

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Ho, J. "The Wiener-Hopf Method and Its Applications in Fluids." 2007., C. M. and McIver, P. Handbook of Mathematical Techniques for Wave/Structure Interactions. Boca Raton, FL: CRC Press, 2001.Noble, B. Methods Based on the Wiener-Hopf Technique For the Solution of Partial Differential Equations. Belfast, Northern Ireland: Pergamon Press, 1958.

Cite this as:

Stover, Christopher. "Wiener-Hopf Method." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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