Discretization with variable time steps of an evolution equation with a positive-type memory term
The authors consider the problem \[ u'(t) + \int^t_0 \beta (t-s) Au(s)ds = f(t) \quad \text{for} \quad t>0,\;u(0)=v. \] Here \(A\) is a selfadjoint positive definite operator in a real Hilbert space \(H\), and \(\beta\) is a real-valued positive definite kernel. The equation is discretized in time using backward Euler and Crank-Nicolson methods, and incorporating appropriate quadratures for the memory term (integral term). Variable time steps are permitted. Error estimates of the methods are derived.
scientific article; zbMATH DE number 908402