Showing papers in "Applications of Mathematics in 2010"

A unified approach to singular problems arising in the membrane theory

Abstract: We consider the singular boundary value problem $$({t^n}u't))' + {t^n}f(t,u(t)) = 0,{\rm{ }}\mathop {\lim }\limits_{t \to 0 + } {t^n}u'(t) = 0,{\rm{ }}{a_0}u(1) + {a_1}u'(1 - ) = A,$$ where f(t, x) is a given continuous function defined on the set (0, 1]×(0,∞) which can have a time singularity at t = 0 and a space singularity at x = 0. Moreover, n ∈ ℕ, n ⩾ >2, and a 0, a 1, A are real constants such that a 0 ∈ (0,1), whereas a 1,A ∈ [0,∞). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation.

Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary $p$-summable right-hand side

Abstract: We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the L p norm of the heat sources for exponents p close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell’s equations or to the Navier-Stokes equations (dissipative heating), with many applications such as crystal growth.

Shape and topological sensitivity analysis in domains with cracks

Abstract: The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

Abstract: In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.

An active set strategy based on the multiplier function or the gradient

Abstract: We employ the active set strategy which was proposed by Facchinei for solving large scale bound constrained optimization problems. As the special structure of the bound constrained problem, a simple rule is used for updating the multipliers. Numerical results show that the active set identification strategy is practical and efficient.

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

Abstract: The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation error in the approximation of the nonlinear convective terms. The estimate of this error allows to analyse the error estimate of the method. The results obtained represent the completion and extension of the analysis from V. Dolejsi, M. Feistauer, Numer. Funct. Anal. Optim. 26 (2005), 349–383, where the truncation error in the approximation of the nonlinear convection terms was proved only in the case when the Dirichlet boundary condition on the whole boundary of the computational domain was considered.

Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

Abstract: The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as t → ∞ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these two cases.

The generalized FGM distribution and its application to stereology of extremes

Abstract: The generalized FGM distribution and related copulas are used as bivariate models for the distribution of spheroidal characteristics. It is shown that this model is suitable for the study of extremes of the 3D spheroidal particles observed in terms of their random planar sections.

Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence

Abstract: We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

Modeling the role of constant and time varying recycling delay on an ecological food chain

Abstract: We consider a mathematical model of nutrient-autotroph-herbivore interaction with nutrient recycling from both autotroph and herbivore. Local and global stability criteria of the model are studied in terms of system parameters. Next we incorporate the time required for recycling of nutrient from herbivore as a constant discrete time delay. The resulting DDE model is analyzed regarding stability and bifurcation aspects. Finally, we assume the recycling delay in the oscillatory form to model the daily variation in nutrient recycling and deduce the stability criteria of the variable delay model. A comparison of the variable delay model with the constant delay one is performed to unearth the biological relevance of oscillating delay in some real world ecological situations. Numerical simulations are done in support of analytical results.

Infinitely many solutions of a second-order $p$-Laplacian problem with impulsive condition

Abstract: Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a p-Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the p-Laplacian impulsive problem.

Numerical modelling of semi-coercive beam problem with unilateral elastic subsoil of Winkler’s type

Abstract: A non-linear semi-coercive beam problem is solved in this article. Suitable numerical methods are presented and their uniform convergence properties with respect to the finite element discretization parameter are proved here. The methods are based on the minimization of the total energy functional, where the descent directions of the functional are searched by solving the linear problems with a beam on bilateral elastic “springs”. The influence of external loads on the convergence properties is also investigated. The effectiveness of the algorithms is illustrated on numerical examples.

Weak interaction limit for nuclear matter and the time-dependent Hartree-Fock equation

Abstract: We consider an effective model of nuclear matter including spin and isospin degrees of freedom, described by an N-body Hamiltonian with suitably renormalized two-body and three-body interaction potentials. We show that the corresponding mean-field theory (the time-dependent Hartree-Fock approximation) is “exact” as N tends to infinity.

On the separation of parametric convex polyhedral sets with application in MOLP

Abstract: We investigate diverse separation properties of two convex polyhedral sets for the case when there are parameters in one row of the constraint matrix. In particular, we deal with the existence, description and stability properties of the separating hyperplanes of such convex polyhedral sets. We present several examples carried out on PC. We are also interested in supporting separation (separating hyperplanes support both the convex polyhedral sets at given faces) and permanent separation (a hyperplane separates the convex polyhedral sets for all feasible parameters). Finally, we show how the developed theory is applicable in multiobjective linear programming.

Locally Lipschitz vector optimization with inequality and equality constraints

Abstract: The present paper studies the following constrained vector optimization problem: \(\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0\), where f: ℝ n → ℝ m , g: ℝ n → ℝ p are locally Lipschitz functions, h: ℝ n → ℝ q is C 1 function, and C ⊂ ℝ m and K ⊂ ℝ p are closed convex cones. Two types of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x 0 to be a w-minimizer and first-order sufficient conditions for x 0 to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.

On periodic solutions of non-autonomous second order hamiltonian systems*

Abstract: The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system $$ \left\{ \begin{gathered} \ddot u(t) = abla F(t,u(t)),a.e.t \in [0,T], \hfill \\ u(0) - u(T) = \dot u(0) - \dot u(T) = 0 \hfill \\ \end{gathered} \right. $$ Some new existence theorems are obtained by the least action principle.

A frictional contact problem with wear and damage for electro-viscoelastic materials

Abstract: We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.

Stochastic homogenization of a class of monotone eigenvalue problems

Abstract: Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form $$ - div\left( {a\left( {T_1 \left( {\frac{x} {{\varepsilon _1 }}} \right)\omega _1 ,T_2 \left( {\frac{x} {{\varepsilon _2 }}} \right)\omega _2 , abla u_\varepsilon ^\omega } \right)} \right) = \lambda _\varepsilon ^\omega \mathcal{C}\left( {u_\varepsilon ^\omega } \right) $$ . It is shown, under certain structure assumptions on the random map a(ω 1, ω 2, ξ), that the sequence {λ , u } of kth eigenpairs converges to the kth eigenpair {λ k , u k } of the homogenized eigenvalue problem . For the case of p-Laplacian type maps we characterize b explicitly.

Relation between algebraic and geometric view on nurbs tensor product surfaces

Abstract: NURBS (Non-Uniform Rational B-Splines) belong to special approximation curves and surfaces which are described by control points with weights and B-spline basis functions. They are often used in modern areas of computer graphics as free-form modelling, modelling of processes. In literature, NURBS surfaces are often called tensor product surfaces. In this article we try to explain the relationship between the classic algebraic point of view and the practical geometrical application on NURBS.

On existence of positive periodic solutions of a kind of Rayleigh equation with a deviating argument

Abstract: The existence of positive periodic solutions for a kind of Rayleigh equation with a deviating argument $$ x''(t) + f(x'(t)) + g(t,x(t - \tau (t))) = p(t) $$ is studied. Using the coincidence degree theory, some sufficient conditions on the existence of positive periodic solutions are obtained.

A note on the existence of positive solutions of one-dimensional p-Laplacian boundary value problems

Abstract: This paper is concerned with the existence of positive solutions of a multipoint boundary value problem for higher-order differential equation with one-dimensional p-Laplacian. Examples are presented to illustrate the main results. The result in this paper generalizes those in existing papers.

Global solution to a generalized nonisothermal Ginzburg-Landau system

Abstract: The article deals with a nonlinear generalized Ginzburg-Landau (Allen-Cahn) system of PDEs accounting for nonisothermal phase transition phenomena which was recently derived by A. Miranville and G. Schimperna: Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst., Ser. B, 5 (2005), 753–768. The existence of solutions to a related Neumann-Robin problem is established in an N ⩽ 3- dimensional space setting. A fixed point procedure guarantees the existence of solutions locally in time. Next, Sobolev embeddings, interpolation inequalities, Moser iterations estimates and results on renormalized solutions for a parabolic equation with L 1 data are used to handle a suitable a priori estimate which allows to extend our local solutions to the whole time interval. The uniqueness result is justified by proper contracting estimates.

On a characterization of orthogonality with respect to particular sequences of random variables in L2

Abstract: This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space L2(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of L2(Ω, F, ℙ) to be orthogonal to some other sequence in L2(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is also provided.