Showing papers in "Annali Dell'universita' Di Ferrara in 2007"

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

Abstract: An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.

On the Navier–Stokes system with pressure boundary condition

Abstract: In this paper we prove the local existence and uniqueness of the weak solution to the non-stationary 2D Navier–Stokes system with pressure boundary condition in a bounded domain.

Rigorous upscaling of the infinite adsorption rate reactive flow under dominant Peclet number through a pore

Abstract: In this paper we present a rigorous derivation of the effective model for enhanced diffusion through a narrow and long 2D pore. The analysis uses the anisotropic singular perturbation technique. Starting point is a local pore scale model describing the transport by convection and diffusion of a reactive solute. The solute particles undergo an adsorption process at the lateral tube boundary, with high adsorption rate. The transport and reaction parameters are such that we have large, dominant Peclet number with respect to the ratio of characteristic transversal and longitudinal lengths (the small parameter $$\varepsilon$$ ). We give a formal derivation of the model using the anisotropic multiscale expansion with respect to $$\varepsilon$$ . Error estimates for the approximation of the physical solution, by the upscaled one, are presented in the energy norm as well as in L ∞ and L 1 norms with respect to the space variable. They give the approximation error as a power of $$\varepsilon$$ and guarantee the validity of the upscaled model through the rigorous mathematical justification of the effective behavior for small $$\varepsilon$$ .

Numerically stable algorithm for cycloidal splines

Abstract: We propose a knot insertion algorithm for splines that are piecewisely in L{1, x, sin x, cos x}. Since an ECC-system on [0, 2π] in this case does not exist, we construct a CCC-system by choosing the appropriate measures in the canonical representation. In this way, a B-basis can be constructed in much the same way as for weighted and tension splines. Thus we develop a corner cutting algorithm for lower order cycloidal curves , though a straightforward generalization to higher order curves, where ECC-systems exist, is more complex. The important feature of the algorithm is high numerical stability and simple implementation.

Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary

Abstract: We consider a viscous incompressible flow in an infinite horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities which size depends on a small parameter, and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using a boundary layer corrector we derive and analyze a first order asymptotic approximation of the flow.

Reiterated homogenization of monotone parabolic problems

Abstract: Reiterated homogenization is studied for divergence structure parabolic problems of the form \({\frac{\partial u_{\varepsilon}}{\partial t}} - \mathrm{div}\left(a\left({\frac{x}{\varepsilon}},{\frac{x}{\varepsilon^2}} ,t, D u_{\varepsilon}\right)\right)=f\) . It is shown that under standard assumptions on the function a(y1,y2,t,ξ) the sequence \(\{u_{\varepsilon}\}\) of solutions converges weakly in \( L^p(0,T;W^{1,p}_0(\Omega))\) to the solution u of the homogenized problem \({\frac{\partial u}{\partial t}} - \mathrm{div}\left( b \left( t,D u \right)\right) = f\) .

Derivation of the model of elastic curved rods from three-dimensional micropolar elasticity

Abstract: In this paper we derive a model of curved elastic rods from the threedimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the rod. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain one-dimensional problem. Appropriate convergence results are proved.

Gradient methods for multiple state optimal design problems

Abstract: We optimise a distribution of two isotropic materials α I and β I (α < β) occupying the given body in R d . The optimality is described by an integral functional (cost) depending on temperatures u 1, . . . , u m of the body obtained for different source terms f 1, . . . ,f m with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by the homogenisation theory. The necessary conditions of optimality are derived via the Gâteaux derivative of the cost functional. Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal design. In the case m < d we show that there exists an optimal design which is a rank-m sequential laminate with matrix material α I almost everywhere on Ω. Contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems (although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient methods for minimising the cost functional.

Numerical multiphase modeling of bubbly flows

Abstract: Bubbly flows appear in a large variety of engineering applications from the petroleum to the nuclear industry. A common model used in these contexts is the so-called drift–flux model where the slip velocity (the difference between the velocities of the gas and of the liquid) is expressed on the basis of empirical correlations. However, depending on these empirical correlations, these models are not always hyperbolic and this induces severe mathematical and numerical difficulties. Using asymptotic analysis in the limit of large drag terms, we propose an Eulerian mixture model where the slip velocity is expressed under the form of a Darcy-like law. We study the mathematical properties of this model and describe a Godunov type scheme for its approximation. Some numerical relevant test-cases are presented.

The Heisenberg magnet equation and the Birkhoff factorization

Abstract: A geometrical description of the Heisenberg magnet (HM) with classical spins is given in terms of flows on the homogeneous space G/H + where G is a Banach Lie group and G + is a subgroup of G. The flows are induced by an action of the abelian group $${\mathbb{R}}^2$$ on G/H +, and the solutions of the HM equation can be found by solving a Birkhoff factorization problem for G. The gauge transformation between the HM and nonlinear Schrodigner (NLS) equations is interpreted as a transformation between a canonical pair of Birkhoff factorizations for G. It is shown that for the HM flows which are Laurent polynomials in the spectral variable this transformation gives rise to a map between the HM and NLS solutions.

Approximation of circular arcs by parametric polynomial curves

Abstract: In this paper the approximation of circular arcs by parametric polynomial curves is studied. If the angular length of the circular arc is h, a parametric polynomial curve of arbitrary degree \(n \in {\mathbb{N}}\) , which interpolates given arc at a particular point, can be constructed with radial distance bounded by h 2n . This is a generalization of the result obtained by Lyche and Morken for odd n.

Homogenization results for climatization problems

Abstract: This paper deals with the homogenization of a nonlinear model for heat conduction through the exterior of a domain containing periodically distributed conductive grains. We assume that on the walls of the grains we have climatizators governing the heat flux through the boundary. The effective behavior of this nonlinear flow is described by a new elliptic boundary-value problem, containing an extra zero-order term which captures the effect of the boundary climatization.

Efficient implementation of WENO schemes to nonuniform meshes

Abstract: Most of the standard papers about the WENO schemes consider their implementation to uniform meshes only. In that case the WENO reconstruction is performed efficiently by using the algebraic expressions for evaluating the reconstruction values and the smoothness indicators from cell averages. The coefficients appearing in these expressions are constant, dependent just on the scheme order, not on the mesh size or the reconstruction function values, and can be found, for example, in Jiang and Shu (J Comp Phys 126:202–228, 1996). In problems where the geometrical properties must be taken into account or the solution has localized fine scale structure that must be resolved, it is computationally efficient to do local grid refinement. Therefore, it is also desirable to have numerical schemes, which can be applied to nonuniform meshes. Finite volume WENO schemes extend naturally to nonuniform meshes although the reconstruction becomes quite complicated, depending on the complexity of the grid structure. In this paper we propose an efficient implementation of finite volume WENO schemes to nonuniform meshes. In order to save the computational cost in the nonuniform case, we suggest the way for precomputing the coefficients and linear weights for different orders of WENO schemes. Furthermore, for the smoothness indicators that are defined in an integral form we present the corresponding algebraic expressions in which the coefficients obtained as a linear combination of divided differences arise. In order to validate the new implementation, resulting schemes are applied in different test examples.

Nice bases for primary Abelian groups

Abstract: Suppose that A is an Abelian p-group. It is proved that if p ω A is bounded, then A has a bounded nice basis and if p ω A is a direct sum of cyclic groups, then A has a nice basis. In particular, all Abelian p-groups of length < ω.2 along with all simply presented Abelian p-groups are equipped with bounded nice bases. It is also shown that if length(A)≤ ω.2 and A/p ω A is countable, then A possesses a bounded nice basis as well as if length(A)≤ ω.2 and p ω A is countable, then A possesses a nice basis. Moreover, contrasting with these claims, we demonstrate that if length(A)=ω.2 and A/p ω A is torsion-complete with finite Ulm-Kaplansky invariants, then A does not have a bounded nice basis. If in addition p ω A is torsion-complete, then A does not have a nice basis, respectively. Finally, we construct a summable $p^{\omega+2}$ -projective group (thus a summable group with a nice basis) which is not a direct sum of countable groups. This answers in negative our question posed in (Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 2005). Keywords: Bounded nice basis, Nice basis, Bounded groups, Direct sums of cyclic groups, Summable groups, $p^{\omega+n}$ -projective groups, Simply presented groups, Σ-groups, Torsion-complete groups, Large subgroups, Countable extensions, Bounded extensions Mathematics Subject Classification: 20K10, 20K15

Deflation in Krylov subspace methods and distance to uncontrollability

Abstract: The task of extracting from a Krylov decomposition the approximation to an eigenpair that yields the smallest backward error can be phrased as finding the smallest perturbation which makes an associated matrix pair uncontrollable. Exploiting this relationship, we propose a new deflation criterion, which potentially admits earlier deflations than standard deflation criteria. Along these lines, a new deflation procedure for shift-and-invert Krylov methods is developed. Numerical experiments demonstrate the merits and limitations of this approach.

On a further condition in the macroscopic extended model for ultrarelativistic gases

Abstract: An exact macroscopic extended model, with many moments, for ultrarelativistic gas has been recently proposed in literature. However, a further condition has not been imposed, even if it is evident in the case of a charged gas and when the electromagnetic field acts as an external force; in the present paper we exploit it and prove that it results in many identities and in residual conditions which allow to determine the arbitrary single variable functions present in the general theory. The result is that they are polynomials determined except for a corresponding number of constants. These are arbitrary constants, so that the macroscopic model remains still more general than the kinetic model. Keywords: Extended Thermodynamics, Fluid Models, Ultra-Relativistic Gas, Entropy Principle Mathematics Subject Classification (2000): 74A15, 74A20

Constructing convex solutions via Perron’s method

Abstract: In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J., 55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces. The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions. Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05

On the justification of the Reynolds equation, describing isentropic compressible flows through a tiny pore

Abstract: We consider the isentropic compressible flow through a tiny pore. Our approach is to adapt the recent results by N. Masmoudi on the homogenization of compressible flows through porous media to our situation. The major difference is in the a priori estimates for the pressure field. We derive the appropriate ones and then Masmoudi’s results allow to conclude the convergence. In this way the compressible Reynolds equation in the lubrication theory is rigorously justified. Keywords: Compressible Navier-Stokes equations, Lubrication, Pressure estimates Mathematics Subject Classification (2000): 35B27, 76M50, 35D05

Evolution model of linear micropolar plate

Abstract: In this paper we derive and mathematically justify the two-dimensional evolution model of linear micropolar plates. We start from the three-dimensional evolution equation of micropolar elasticity for thin plate-like bodies. Using the variational techniques we consider the behavior of the solution of the three-dimensional problem when the thickness tends to zero. The limit function satisfies a certain two-dimensional problem then called the evolution micropolar plate model.

Quadratic convergence of a special quasi-cyclic Jacobi method

Abstract: This paper considers the ultimate asymptotic convergence of a block- oriented, quasi-cyclic Jacobi method for symmetric matrices. The conclusion applies to the new one-sided Jacobi method for computing the singular value decomposition, recently proposed by Drmac and Veselic. Using a simple qualitative analysis, the discussion indicates that a quadratic off-norm reduction per quasi-sweep is to be expected in all perceivable cases.

Random field forward interest rate models, market price of risk and their statistics

Abstract: In this paper we consider discrete time forward interest rate models. In our approach, unlike in the classical Heath–Jarrow–Morton framework, the forward rate curves are driven by a random field. Hence we get a general interest rate structure. Our aim is to give an overview of our results in such a model on the following questions: no-arbitrage conditions, maximum likelihood estimation of the volatility, as well as the joint estimation of the parameters and the asymptotic behaviour of the estimators, relationship with continuous models. Finally we give discussion on the practical problems of the estimation and we show several numerical results on the statistics of such models.

Conditions of matrices in discrete tension spline approximations of DMBVP

Abstract: Some splines can be defined as solutions of differential multi-point boundary value problems (DMBVP). In the numerical treatment of DMBVP, the differential operator is discretized by finite differences. We consider one dimensional discrete hyperbolic tension spline introduced in (Costantini et al. in Adv Comput Math 11:331–354, 1999), and the associated specially structured pentadiagonal linear system. Error in direct methods for the solution of this linear system depends on condition numbers of corresponding matrices. If the chosen mesh is uniform, the system matrix is symmetric and positive definite, and it is easy to compute both, lower and upper bound, for its condition. In the more interesting non-uniform case, matrix is not symmetric, but in some circumstances we can nevertheless find an upper bound on its condition number.

A review of existence criteria for parameter estimation of the Michaelis–Menten regression model

Abstract: In this paper we consider the least squares (LS) and total least squares (TLS) problems for a Michaelis–Menten enzyme kinetic model f(x; a, b) = ax/(b + x), a, b > 0. In various applied research such as biochemistry, pharmacology, biology and medicine there are lots of different applications of this model. We will systematize some of our results pertaining to the existence of the LS and TLS estimate, which were proved in Hadeler et al. (Math Method Appl Sci 30:1231–1241, 2007) and Jukic et al. (J Comput Appl Math 201:230–246, 2007). Finally, we suggest a choice of good initial approximation and give one numerical example.

Tribalj dam-break and flood wave propagation

Abstract: In this paper we present numerical simulations for the dam-break flood wave propagation from Tribalj accumulation to the town of Crikvenica (Croatia). The mathematical models we used were the one-dimensional open channel flow and the two-dimensional shallow water equations. They were solved with the well-balanced finite volume numerical schemes which additionally include special numerical treatment of the wetting/drying front boundary. These schemes were tested on CADAM test problems. The aim of this study was to assess potential damage in the village of Tribalj and the town of Crikvenica. Results of these simulations were used as the basis for urban planning and micro-zoning of the flood-risk areas. Several different dam-break scenarios were considered, ranging from sudden dam disappearance to partial and dynamic breach formation.

The meaning of computer search in the study of some classes of IM-quasigroups

Abstract: The IM-quasigroup C(q) for \(q=\frac{1}{2}(1+\sqrt{5})\) or \(q=\frac{1}{2}(1-\sqrt{5})\) is a GS-quasigroup. Some interesting geometric concepts can be introduced in a general GS-quasigroup and their nice geometric representations can be given in the mentioned GS-quasigroup which justifies the research of this quasigroup. The concept of a parallelogram can be defined by means of several equivalent formulae. Some of them can be obtained by means of a computer. We shall choose a suitable formula for the definition of a parallelogram which allows the characterization of GS-quasigroups by means of commutative groups.

Identification of a time-dependent UV-photon source in an interstellar cloud

Abstract: Consider an interstellar cloud that occupies the region $V\subset{\mathbb R}^3$ , bounded by the known surface $\partial V$ , and assume that the scattering cross section σ s and the total cross section σ are also known. Then, we prove that it is possible to identify the source q=q(x,t) that produces UV-photons inside the cloud, provided that the UV-photon distribution function arriving at a location $\widehat{{\mathbf x}}$ , far from the cloud, is measured at times $\widehat{t}_0$ , $\widehat{t}_1 = \widehat{t}_0+\tau$ , ..., $\widehat{t}_J = \widehat{t}_0+J\tau$ . Keywords: Photon transport, Semigroups and linear evolution equations, Inverse problems Mathematics Subject Classification (2000): 82A25, 82C70, 34K29, 65M32

Time-periodic Poiseuille flow in a pipe for some classes of fluids

Abstract: We consider a fully developed time-periodic pipe flow (Poiseuille flow) for some classes of fluids (micropolar fluids, mixtures of fluids). Such physical cases lead to a parabolic system in which the pressure gradient Γ is a time-periodic function with either only one non vanishing component or the components proportional to a single time-periodic function Γ. For such situations we generalize the results of [7] concerning the Newtonian case. Keywords: Flow in a pipe, Time-periodic Poiseuille flow, Micropolar fluids, Mixtures of fluids Mathematics Subject Classification (2000): 76D03, 76A05, 76T05, 35Q30

Graph representation for asymptotic expansion in homogenisation of nonlinear first-order equations

Abstract: Homogenisation of a linear transport equation leads to an integro-differential equation with the differential part of the same type as the starting equation. The (non-periodic) homogenisation of semilinear transport equations is open. In order to pinpoint technical difficulties, as a first step in that direction, following the approach of Tartar we consider an ordinary differential equation with an oscillating coefficient a $$\left\{\begin{array}{ll} u' +au^2 &=f \\ u(0) &=v \end{array}\right. $$ instead, and expand the solution in terms of a small parameter (the amplitude of oscillations in a). The crucial observation we made is a correspondence between multiple integrals representing the terms in asymptotic expansion of the solution and certain graphs, which allows easy manipulation of otherwise highly complicated expressions, and leads to efficient computation of the terms in expansion.

Cryptosystems based on semi-distributive algebras

Abstract: We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems defined in the paper—semialgebras (Sect. 2). The main examples are semialgebras of polynomial mappings over a finite field K, and their factor-semialgebras. Given such a semialgebra R, one chooses an invertible element a ∈ R* of finite order r, and a random integer s. One chooses also a finite dimensional K-submodule V of R. The 4-tuple (R, V, a, b) where b = as forms the public key for the cryptosystem, while r and s form the secret key. A plain text can be viewed as a sequence of elements of the field K. That sequence is divided into blocks of length dim(V) which, in turn, correspond to uniquely determined elements Xi of V. We propose three different methods (A, B, and C, see Definition 1.1) of encoding/decoding the sequence of Xi. The complexity of cracking the proposed cryptosystem is based on the Discrete Logarithm Problem for polynomial mappings (see Sect. 1.1). No methods of cracking the problem, except for the “brute force” (see Sect. 1.1) with Ω(r) time, are known so far.

Existence of \(g_{g-2}^{1}\) ’s on a double covering of a hyperelliptic curve

Abstract: Let X be a non–hyperelliptic curve of genus g which is a double covering of a hyperelliptic curve C of genus h. In this paper, we prove that, if h≥ 3 and g≥ 4h+5, then X admits a complete, base point free g1g–2. Moreover, if h=3, this result holds under the mild condition g≥ 4h+3=15. Keywords: Double covering of hyperelliptic curves, Pencil of degree g–2 Mathematics Subject Classification (2000:) 14H30, 14H45