Compact alternating direction implicit scheme for integro-differential equations of parabolic type (Q1668735)

In this article the following two-dimensional parabolic integro-differential equation in the domain \((0,1)\times (0,1)\times (0,T]\) is considered: \[ u_{t} -\mu \Delta u=\int\limits_{0}^{t}\beta (t-s)\Delta u(s)ds. \] Here \(\Delta\) is the Laplace operator and the kernel \(\beta(t)=t^{\alpha -1} /\Gamma (\alpha )\), \((0<\alpha <1)\) is a weakly singular function. The corresponding initial boundary value problem with homogenous boundary conditions is stated. A compact alternating direction implicit scheme combined with a second-order fractional quadrature rule suggested by \textit{C. Lubich} [Numer. Math. 52, No. 2, 129--145 (1988; Zbl 0637.65016)] is constructed and used. It is proved that the scheme is stable in \(L^{2}\)-norm and the convergence order is determined. Two numerical examples with known exact solutions are given and the results of numerical experiments are presented.