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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

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Source: ACM Digital Library