### An algorithm for the approximate solution of Wiener-Hopf integral equationsAn algorithm for the approximate solution of Wiener-Hopf integral equations Access Restriction
Subscribed

 Author Stenger, Frank Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Language English
 Subject Keyword Wiener-hopf ♦ Convolution ♦ Integral equations ♦ Approximate solution Abstract An explicit approximate solution ƒ(h)α is given for the equation ƒ(t) = ∫∞0 k(t - τ)ƒ(τ) dτ + 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)α of (*) depends on two positive parameters, h and α. 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)α(t) | ≤ c1 exp (-πd/h) + c2 exp (-πd/α) where c1 and c2 are constants. As an example, we compute ƒ(h)α(t), t = 0.2(0.2)1, h = π/10, α = π/50, for the case of k(t) = exp(-| t |)/(2π), g(t) = t4 exp (-3t). The resulting solution is correct to five decimals. Description Affiliation: Univ. of Utah, Salt Lake City (Stenger, Frank) Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2005-08-01 Publisher Place New York Journal Communications of the ACM (CACM) Volume Number 16 Issue Number 11 Page Count 3 Starting Page 708 Ending Page 710

#### Open content in new tab

Source: ACM Digital Library