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

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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 ∈ L 1 (- ∞, ∞) ∩ L 2 (-∞, ∞), and where it is assumed that the classical Wiener-Hopf technique may be applied to (*) to yield a solution ƒ ∈ L 1 (0, ∞) ∩ L 2 (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 ) | ≤ c 1 exp (- πd / h ) + c 2 exp (- πd / α ) where c 1 and c 2 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 |)/(2 π ), g ( t ) = t 4 exp (-3 t ). The resulting solution is correct to five decimals.

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