Numerical modelling of qualitative behaviour of solutions to convolution integral equations

We consider the qualitative behaviour of solutions to linear integral equations of the form(1) y ( t ) = g ( t ) + ∫ 0 t k ( t - s ) y ( s ) d s , where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley–Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.