An algorithm for the approximate solution of Wiener-Hopf integral equations

An explicit approximate solution ƒ(h)&agr; is given for the equation ƒ(t) = ∫∞0 k(t - &tgr;)ƒ(&tgr;) d&tgr; + g(t), t > 0, (*) where k, g ∈ L1(- ∞, ∞) ∩ L2(-∞, ∞), and where it is assumed that the classical Wiener-Hopf technique may be applied to (*) to yield a solution ƒ ∈ L1(0, ∞) ∩ L2(0, ∞) for every such given g. It is furthermore assumed that the Fourier transforms K and G+ of k and g respectively are known explicitly, where K(x) = ∫∞-∞ exp (ixt)k(t) dt, G+(x) = ∫∞0 exp (ixt)g(t) dt. The approximate solution ƒ(h)&agr; of (*) depends on two positive parameters, h and &agr;. If K(z) and G+(z) are analytic functions of z = x + iy in the region {x + iy : | y | ≤ d}, and if K is real on (-∞, ∞), then | ƒ(t) - ƒ(h)&agr;(t) | ≤ c1 exp (-&pgr;d/h) + c2 exp (-&pgr;d/&agr;) where c1 and c2 are constants. As an example, we compute ƒ(h)&agr;(t), t = 0.2(0.2)1, h = &pgr;/10, &agr; = &pgr;/50, for the case of k(t) = exp(-| t |)/(2&pgr;), g(t) = t4 exp (-3t). The resulting solution is correct to five decimals.

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