Operator splitting for nonautonomous delay equations

We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated on some typical numerical examples.Comment: to appear in "Computers & Mathematics with Applications (CAMWA)

On exponential stability for linear discrete-time systems in Banach spaces

In this paper we investigate four concepts of exponential stability for difference equations in Banach spaces. Characterizations of these concepts are given. They can be considered as variants for the discrete-time case of the classical results due to E.A. Barbashin [2] and R. Datko [5]. An illustrative example clarifies the relations between these concepts.Comment: Computers & Mathematics with Applications, accepted for publicatio

Distributed order equations as boundary value problems

This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in Computers and mathematics with applications and is available at www.elsevier.comThis preprint discusses the existence and uniqueness of solutions and proposes a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin

A Two-Level Finite Element Discretization of the Streamfunction Formulation of the Stationary Quasi-Geostrophic Equations of the Ocean

In this paper we proposed a two-level finite element discretization of the nonlinear stationary quasi-geostrophic equations, which model the wind driven large scale ocean circulation. Optimal error estimates for the two-level finite element discretization were derived. Numerical experiments for the two-level algorithm with the Argyris finite element were also carried out. The numerical results verified the theoretical error estimates and showed that, for the appropriate scaling between the coarse and fine mesh sizes, the two-level algorithm significantly decreases the computational time of the standard one-level algorithm.Comment: Computers and Mathematics with Applications 66 201

Powersums representing residues mod p^k, from Fermat to Waring

The ring Z_k(+,.) mod p^k with prime power modulus (prime p>2) is analysed. Its cyclic group G_k of units has order (p-1)p^{k-1}, and all p-th power n^p residues form a subgroup F_k with |F_k|=|G_k|/p. The subgroup of order p-1, the core A_k of G_k, extends Fermat's Small Theorem (FST) to mod p^{k>1}, consisting of p-1 residues with n^p = n mod p^k. The concept of "carry", e.g. n' in FST extension n^{p-1} = n'p+1 mod p^2, is crucial in expanding residue arithmetic to integers, and to allow analysis of divisors of 0 mod p^k. . . . . For large enough k \geq K_p (critical precison K_p < p depends on p), all nonzero pairsums of core residues are shown to be distinct, upto commutation. The known FLT case_1 is related to this, and the set F_k + F_k mod p^k of p-th power pairsums is shown to cover half of units group G_k. -- Yielding main result: each residue mod p^k is the sum of at most four p-th power residues. Moreover, some results on the generative power (mod p^{k>2}) of divisors of p^2-1 are derived. -- [Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261]Comment: (9 pgs) Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261. See http://www.iae.nl/users/benschop/nw-abstr.htm - Intro at http://www.iae.nl/users/benschop/fewago.htm -- See also http://arXiv.org/abs/math.GM/0103067 (on primitive roots