The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic stability

AbstractWe study discretization in classes of integro-differential equations $u(t)+0t(λ1a1(tτ)+λ2a2(tτ)++λnan(tτ))u(τ)dτ=0,t>0,u(0)=1,λj1,j=1,2,...,n, $, where the functions aj (t), 1 ⩽ j ⩽ n, are completely monotonic on (0,∞) and locally integrable, but not constant. The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term. The stability properties of the discretization are derived in the weighted l1(ρ; 0,∞) norm, where ρ is a given weight function. Applications to the weighted l1 stability of the numerical solutions of a related equation in Hilbert space are given.

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