Abstract. We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

Laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered. A spectral order, parallelizable method for their numerical inversion is proposed. The method takes into account the available information about the errors arising in the evaluations. Several numerical illustrations are provided.

No abstract

A class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed. It is shown that the k-step method achieves order k, for appropriate starting values, which can be computed by auxiliary routines or by one strategy proposed in the paper. Together with some implementation issues, numerical illustrations are also provided. Mathematics Subject Classifications (2000)65J15 · 65M12 · 65L05 · 65M20

It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) (1+ √ 2)-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of holomorphic functions.2000 Mathematical subject 3assifications : 47A25 ; 47A30