The approximate solution of Wiener-Hopf integral equations

This paper develops an explicit approximate method of solving the integral equation (¹) f(x) = ∫₀^∞ h₁(x − t)f(t) dt + g(x), x > 0, where g(x), h₁(x) ϵ L¹(R) ∩ L²(R),f(x) = g(x) = 0 if x < 0. The approximate solution depends upon 2 parameters, h > 0 and k ϵ (0, 1). It is shown that if (¹) has a unique solution, then as h → 0⁺, and k → 1⁻, the approximate solution converges to the unique solution whenever a unique solution f ϵ L¹(R) ∩ L²(R) exists for every given g ϵ L¹(R) ∩ L²(R), provided that h∑i|H(ih +12h)z.sfnc;2→ ∫R| H(x)|2dx role=presentation style=font-size: 90%; display: inline-block; position: relative;>$\text{h}\text{∑}\text{i}\text{|H(ih +}\text{1}\text{2h}\text{)z.sfnc;}{}^2\text{→ ∫}{}^{\mathrm{R}}\text{| H(x)|}{}^2\text{dx}$$\text{h}\text{∑}\text{i}\text{|H(ih +}\text{1}\text{2h}\text{)z.sfnc;}{}^2\text{→ ∫}{}^{\mathrm{R}}\text{| H(x)|}{}^2\text{dx}$ as h → 0⁺, where H(x) is the Fourier transform of h₁(t). An example is given which illustrates the application of the method.