The z-transform is introduced to analyze a full discretization method fora partial integro-differential equation (PIDE) with a weakly singular kernel. In thismethod, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule isconsidered. The global stability and convergence properties of complete discretizationare derived and numerical experiments are reported.
This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular kernel.The integral term is treated by means of the second order convolution quadrature suggested by Lubich.The stability and convergence are proved by the energy method.A numerical experiment is reported to verify the theoretical predictions.
We study discretization in classes of integro-differential equations 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.
