Uniform \(l^1\) behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations
In this paper, the time discrete scheme based on the first-order backward difference method for the following integro-differential problem is studied \[ u_t(t)+\int\limits_0^t\beta(t-s)Au(s)ds=0, \quad t>0, \quad u(0)=u_0. \] Here \(A\) is a positive self-adjoint linear operator defined on a dense subspace \(D(A)\) of the real Hilbert space \(H\), \(u_0\in H\), and \(\beta(t)\) is weakly singular kernel at \(t=0\) such that \(\beta(t)=t^{\alpha-1}/\Gamma(\alpha)\), \(0<\alpha<1\). The paper is organized as follows. Section 1 is an introduction. In this section abovementioned problem is stated and the main theorem of this work is formulated. A review of the suitable papers is also given in this section. The memory term is approximated by the interpolating quadrature in this paper. In Section 2, using the Laplace transform technique is shown that this interpolating quadrature scheme has a convergence rate of \(O(\Delta t)\), where \(\Delta t\) denotes the time step. Special attention is that the uniform \(\ell^1\) convergence property of the discretization in time is given.
scientific article; zbMATH DE number 7340773