Integral equations on the half-line: a modified finite-section approximation

Abstract:

We consider the approximate solution of integral equations of the form $y(t) - \smallint _0^\infty k(t,s) y(s) ds = f(t)$, where the conditions on $k(t,s)$ are such that kernels of the Wiener-Hopf form $k(t,s) = \kappa (t - s)$ are included in the analysis. The finite-section approximation, in which the infinite integral is replaced by $\smallint _0^\beta$ for some $\beta > 0$, yields an approximate solution ${y_\beta }(t)$ that is known, under very general conditions, to converge to $y(t)$ as $\beta \to \infty$ with t fixed. However, the convergence is uniform only on finite intervals, and the approximation is typically very poor for $t > \beta$. Under the assumption that f 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 $\beta \to \infty$. A numerical example illustrates the improvement.