Integral Equations on the Half-Line: A Modified Finite-Section Approximation

We consider the approximate solution of integral equations of the form y(t) /if k(t,s)y(s)ds = f(t), where the conditions on k(t,s) are such that kernels of the Wiener-Hopf form k(t, s) = n(t s) are included in the analysis. The finite-section approximation, in which the infinite integral is replaced by }§ for some ß > 0, yields an approximate solution _Vß(t) that is known, under very general conditions, to converge to y(t) as ß -» oo with t fixed. However, the convergence is uniform only on finite intervals, and the approximation is typically very poor for t > ß. Under the assumption that / has a limit at infinity, we here introduce a modified finite-section approximation with improved approximation properties, and prove that the new approximate solution converges uniformly to y as ß -» oo. A numerical example illustrates the improvement.