Tauberian conditions on some slowly decreasing sequences

Periodica Mathematica Hungarica - We present some Tauberian conditions to recover Cesàro summability of a sequence out of the product methods of Abel and Cesàro summability of the...     

Uniform preconditioners for the time dependent Stokes problem

Summary. Implicit time stepping procedures for the time dependent Stokes problem lead to stationary singular perturbation problems at each time step. These singular perturbation problems are systems of saddle point type, which formally approach a mixed formulation of the Poisson equation as the time step tends to zero. Preconditioners for discrete analogous of these systems are discussed. The preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization. The construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system. Mathematics Subject Classification: 65M55, 65 Revised version received September 18, 2003 An erratum to this article is availabel at .     

A time minimising transportation problem with quality dependent time

The usual assumption made in time minimising transportation problem is that the time for transporting a positive amount in a route is independent of the actual amount transported in that route. In this paper we make a more general and natural assumption that the time depends on the actual amount transported. We assume that the time function for each route is an increasing piecewise constant function. Four algorithms - (1) a threshold algorithm, (2) an upper bounding technique, (3) a primal dual approach, and (4) a branch and bound algorithm - are presented to solve the given problem. A method is also given to compute the minimum bottle-neck shipment corresponding to the optimal time. A numerical example is solved illustrating the algorithms presented in this paper.     

Distribution of eigenvalues of the Sturm-Liouville problem with slowly increasing potential

We establish an asymptotic representation of the function $\tilde n(R) = \int\limits_0^R {\frac{{n(r) - n(0)}}{r}dr, R \in \Re } \subseteq [0, \infty ), R \to \infty ,$ where n(r) is the number of eigenvalues of the Sturm-Liouville problem on [0,∞) in (λ:¦λ¦≤r) (counting multiplicities). This result is obtained under assumption that q(x) slowly (not faster than In x) increases to infinity as x→∞ and satisfies additional requirements on some intervals $[x_ - (R), x_ + (R)],R \in \Re $ .     

Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation: (1) for (t,x) ∈ ?_t × ?. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ∣V(t,x)∣?C(1+t)(1+∣x∣) for a.e. (t,x) ∈ ?_t × ? where C, σ₀>0 are real constants, then for any real number λ that satisfies there exists a real number ρ(f,g,λ)>0 such that the equation has a global solution provided that 0<ρ?ρ(f,g,λ). Copyright © 2004 John Wiley & Sons, Ltd.     

Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials

Partially supported by Bulgarian Committee of Science, Contract No. 52/1988     

Instability results for slowly time varying linear dynamic systems on time scales

We develop eigenvalue criteria under which the solutions of a “slowly” time varying linear dynamic system of the form xΔ(t)=A(t)x(t) are unstable.     

Well-posedness for a cauchy problem associated to time dependent free boundaries with nonlocal leading terms

We study a class of free boundary problems, where the normal velocity of the interface is proportional to the derivative of the solution of an elliptic PDE; we give a simple, explicit criterion for the well-posedness of the linearized Cauchy problem. The method is then applied to two classical problems; the stefan problem and the Muskat problem.     

The moments problem for rapidly decreasing functions

Function classes are described in which the moments problem $$\int_0^{ + \infty } {t^{\alpha _n } f(t)dt = c_n } $$ is solvable for any right-hand sides from a sequence {c _n }. Constructive solutions are given. The results obtained generalize and supplement some theorems proved earlier by other mathematicians.     

On the number of eigenvalues for parameter‐dependent diffusion problem on time scales

In this study, we consider parameter‐dependent diffusion eigenvalue problem on time scales. An upper bound on the number of eigenvalues for this problem on a finite time scale is given.     

Analytical reduction for a concentration dependent diffusion problem

Previous analytical simplifications of Boltzmann's similarity solution are unified and generalized by examining symmetry and transformation properties of the equation. It is shown how to reduce boundary value problems to initial value problems for certain diffusivities. A series solution is derived for this case, and shown to be accurate for a broad range of parameter values.

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Résumé Les simplifications analytiques connues de la solution similaire de Boltzmann sont unifiées et généralisées par l'étude des propriétés de symétrie et des transformations de l'équation de diffusion. On montre comment, pour certaines diffusivités, un problème de conditions aux limites se réduit à un problème de valeurs initiales. Une solution en séries obtenue pour ce cas est très précise pour un domaine étendu des paramètres.