The numerical solution of boundary value problems for linear systems of first order equations with a regular singular point at one endpoint is considered. The standard procedure of expanding about the singularity to get a nonsingular problem over a reduced interval is justified in some detail. Quite general boundary conditions are included which permit unbounded solutions. Error estimates are g...

In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R and C, under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the...

We study the ergodic theory of a one-parameter family of interval maps Tα arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of Tα to be Hölder-continuous in the parameter α. Moreover, we prove a central limit theorem for possibly unbounded observables whose bounded variation grows moderately. This class of functions is large enou...

Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the pr...

We study the solvability of a nonlinear quadratic integral equation of Hammerstein type. Using the technique of measures of noncompactness we prove that this equation has solutions on an unbounded interval. Moreover, we also obtain an asymptotic characterization of these solutions. Several special cases of this integral equation are discussed and applications to real world problems are indicate...

This paper provides a numerical approximation theory of algebraic Riccati operator equations with unbounded coefficient operators A and B , such as arise in the study of optimal quadratic cost problems over the time interval [0, oo] for the abstract dynamics y = Ay + Bu . Here, A is the generator of a strongly continuous analytic semigroup, and B is an unbounded operator with any degree of unbo...

We study a class of elliptic operators A with unbounded coefficients defined in I × R for some unbounded interval I ⊂ R. We prove that, for any s ∈ I, the Cauchy problem u(s, ·) = f ∈ Cb(R ) for the parabolic equation Dtu = Au admits a unique bounded classical solution u. This allows to associate an evolution family {G(t, s)} with A, in a natural way. We study the main properties of this evolut...

Interval methods have been shown to be efficient, robust and reliable to solve difficult set-membership localization problems. However they are unsuitable in a probabilistic context, where the approximation of an unbounded probability density function by a set cannot be accepted. This paper proposes a new probabilistic approach which makes possible to use classical set-membership localization m...

Timoshenko beam equations with external damping and internal damping terms and forcing terms are investigated, and boundary conditions (end conditions) to be considered are hinged ends (pinned ends), hinged-sliding ends, and sliding ends. Unboundedness of solutions of boundary value problems for Timoshenko beam equations is studied, and it is shown that the magnitude of the displacement of the ...

We consider the Langevin equation describing a non-viscous Burgers fluid stochastically perturbed by uniform noise. We introduce a deterministic function that corresponds to the mean of the velocity when we keep fixed the value of the position. We study interrelations between this function and the solution of the non-perturbed Burgers equation. Especially we are interested in the property of th...