Uniform l~1 behaviour for time discretization of a Volterra equation with completely monotonic kernel: I. stability

摘要

This paper is the first of two papers on the time discretization of the equation u_t + ∫_0~tβ(t - s)Au(s)ds = 0 , t > 0 , u(0) = u_0, where A is a self-adjoint densely defined linear operator on a Hilbert space H with a complete eigensystem {λ_m, φ_m}_(m=1)~∞, and β(t) is completely monotonic and locally integrable, but not constant. The equation is discretized in time using first-order differences in combination with order-one convolution quadrature. The stability properties of the time discretization are derived in the l_t~1 (0, ∞; H) norm.