Bounds of high quality for first kind Volterra integral equations

E-Methnds for .,aflving linear Volterra integral equatitms of the first kind with smot~th kernels are considered. E-MethotL~ are a new type of numerical algorithms computing numerical approximations together with mathematically guaranteed do.~ error I~amds, The basic concepts from verifi~ttion thet~ry are sketched and such ~lf-validating numerics derived. O~mputati~mal experiments show the effidency t)f these prtn:edures being an advance in numerical methods.