APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE '12): Proceedings of the 38th International Conference Applications of Mathematics in Engineering and Economics
 MATHEMATICAL MODELING


Preface: Applications of Mathematics in Engineering and Economics (AMEE '12)
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Estimation problem for impulsive control systems under ellipsoidal state bounds and with cone constraint on the control
View Description Hide DescriptionThe paper deals with the state estimation problem for the linear control system containing impulsive control terms (or measures). The problem is studied here under uncertainty conditions when the initial system state is unknown but bounded, with given bound. It is assumed also that the system states should belong to the given ellipsoid in the state space. So the main problem of estimating the reachable set of the control system is studied here under more complicated assumption related to the case of state constraints. It is assumed additionally that impulsive controls in the dynamical system must belong to the intersection of a special cone with a generalized ellipsoid both taken in the space of functions of bounded variation. The last constraint is motivated by problems of impulsive control theory and by models from applied areas when not every direction of control impulses is acceptable in the system. We present here the state estimation algorithms that use the special structure of the control system and take into account additional restrictions on states and controls. The algorithms are based on ellipsoidal techniques for estimating the trajectory tubes of uncertain dynamical systems. Numerical simulation results related to proposed procedures are also given.

On the uniqueness of solutions to dynamic and pseudo oscillation problems of linear theory of twocomponent elastic mixture
View Description Hide DescriptionIn the presented work is studied system of partials differential equations of diffusion model of linear theory of twocomponent elastic mixtures. The generalized green's formulas and energy identity are derived for non stationary and pseudo oscillation systems of this theory. Uniqueness of inner and external problems of dynamics and pseudo oscillation of diffusion models of a linear theory of twocomponent elastic mixture is investigated.

Friction induced rail vibrations
View Description Hide DescriptionA model of rail, considered as multiple supported beam, subjected on friction induced vibration is studied in this work using FEM. The model is presented as continuous system and the mass and elastic properties of a real object are taken into account. The friction forces are nonlinear functions of the relative velocity during slipping. The problem is solved using Matlab Simulink.

Balanced truncation of nonlinear systems with error bounds
View Description Hide DescriptionThis paper considers the problem of balanced model reduction for a class of smooth nonlinear systems. The nonlinear system is approximated by local linearizations model around a continuum of constant operating points. An upper bound on the error of approximation is presented. The obtained linearization model can be considered as a linear timevarying (LTV) system model. Three different problems of balanced truncation for LTV systems are considered and the corresponding upper bound on the error between the full order and reduced order models is shown.

A quantum mechanical model for the relationship between stock price and stock ownership
View Description Hide DescriptionThe trade of a fixed stock can be regarded as the basic process that measures its momentary price. The stock price is exactly known only at the time of sale when the stock is between traders, that is, only in the case when the owner is unknown. We show that the stock price can be better described by a function indicating at any moment of time the probabilities for the possible values of price if a transaction takes place. This more general description contains partial information on the stock price, but it also contains partial information on the stock owner. By following the analogy with quantum mechanics, we assume that the time evolution of the function describing the stock price can be described by a Schrödinger type equation.

Shape optimization in flowing medium cooling
View Description Hide DescriptionIn the paper we assume the body which is subjected to influence of given heat source on the part of boundary Γ_{1} and is cooled by flowing cooling medium on the part of boundary Γ_{2}. The problem of shape optimization which focuses to control distribution of temperature on the boundary Γ_{1} of the body with heat source by the change of shape of boundary Γ_{2} is formulated. The energy equation for cooling by flowing medium, which is assumed as a potential flow, is used as the state problem. The cost functional is taken as the second power of the norm in space L ^{2} of difference of trace temperature from the given constant evaluated on the boundary Γ_{1}. Existence and uniqueness of the state problem solution and existence of a solution of shape optimization problem are proved.

Scattering of timeharmonic antiplane shear waves in magnitoelectroelastic materials
View Description Hide DescriptionIn this paper the dynamic behavior of cracked magnetoelectroelastic materials under antiplane mechanical and inplane electric and magnetic load is investigated. The boundary value problem for the coupled system of governing equations is reduced to a nonhypersingular traction boundary integral equation using generalization of the wellknown Jintegrals in elastostatics. Software based on the Boundary Integral Equation Method (BIEM) is developed using FORTRAN. Validation with the results for piezoelectric materials obtained by the dual integral equation method is given. The numerical examples show the dependence of the Stress Intensity Factor (SIF) on the normalized frequency of the incident wave for different materials and different boundary conditions.

Evaluation of indoor air composition time variation in airtight occupied spaces during night periods
View Description Hide DescriptionThis paper presents an easytounderstand procedure for prediction of indoor air composition time variation in airtight occupied spaces during the night periods. The mathematical model is based on the assumptions for homogeneity and perfect mixing of the indoor air, the ideal gas model for nonreacting gas mixtures, mass conservation equations for the entire system and for each species, a model for prediction of basal metabolic rate of humans as well as a model for prediction of consumption rate and both and generation rates by breathing. Time variation of indoor air composition is predicted at constant indoor air temperature for three scenarios based on the analytical solution of the mathematical model. The results achieved reveal both the most probable scenario for indoor air time variation in airtight occupied spaces as well as the cause for morning tiredness after having a sleep in a modern energy efficient space.

Fixed point theorems and existence of equilibrium in discontinuous games
View Description Hide DescriptionIn this paper, we generalize the existence of Berge's strong equilibrium in Deghdak (see [7]) to discontinuous games in infinite dimensional space of srategy. Moreover, we prove that most of Berge's strong games are essential.

Finite volume method for the BlackScholes equation transformed on finite interval
View Description Hide DescriptionIn this paper, we present a fitted FVM for the degenerate at the two ends parabolic equation, derived from the BlackScholes equation after a transformation to a finite interval. For the case of European options we describe a fully discretization of the vertical method of lines, where the spatial discretization is formulated as a PetrovGalerkin FEM. We show that the method is O(h) convergent and monotone. Numerical experiments are presented to verify the theoretical results. Experiments on a powergraded mesh demonstrate higher accuracy.

Analysis of a model of fuel cell  gas turbine hybrid power system for enhanced energy efficiency
View Description Hide DescriptionA simple mathematical model to evaluate the performance of FCGT hybrid system is presented in this paper. The model is used to analyse the influence of various parameters on the performance of a typical hybrid system, where excess heat rejected from the solidoxide fuel cell stack is utilised to generate additional power through a gas turbine system and to provide heat energy for space heating. The model is based on thermodynamic analysis of various components of the plant and can be adapted for various configurations of the plant components. Because there are many parameters defining the efficiency and work output of the hybrid system, the technique is based on mathematical and graphical optimisation of various parameters; to obtain the maximum efficiency for a given plant configuration.

Approximation of nonhomogeneous linear system ODEs with constant coefficients by a homogeneous first order one
View Description Hide DescriptionThe general method to solve a nonhomogeneous system ODEs with constant coefficients is the method of variation of the constants developed by Lagrange. Sometimes this method leads to integrals that cannot be solved exactly. The techniques we propose to approximate such a system allows to find the approximate solution using eigenvalues and eigenvectors [1]. The results are closer to the exact solution than the results after the integration by series (using MAPLE or calculated by hand). Some examples are given, to compare with these from [2].

 NUMERICAL METHODS


Monotoneiterative method for a mixed nonlinear boundary value problem for differential equations with "maxima"
View Description Hide DescriptionThe main object of investigation of the paper is so called differential equation with "maxima". These equations, for example, find a wide application in the theory of automatic regulation. The main characteristic of this type of equations is the presence of the maximum of the unknown function over a past time interval. A nonlinear boundary value problem mixed with an initial condition for a nonlinear equation is studied. An algorithm for constructing two sequences of successive approximations of the solution is given. This algorithm is based on the monotone iterative technique combined by the method of lower/upper solutions. Each term of the constructed sequences is a solution of an initial value problem for linear differential equations with "maxima". Also the terms of the sequences are lower/upper solutions of the given problem. It is proved both sequences are monotonically convergent. The suggested procedure is illustrated on some examples.

Effective implementation of wavelet Galerkin method
View Description Hide DescriptionIt was proved by W. Dahmen et al. that an adaptive wavelet scheme is asymptotically optimal for a wide class of elliptic equations. This scheme approximates the solution u by a linear combination of N wavelets and a benchmark for its performance is the best Nterm approximation, which is obtained by retaining the N largest wavelet coefficients of the unknown solution. Moreover, the number of arithmetic operations needed to compute the approximate solution is proportional to N. The most time consuming part of this scheme is the approximate matrixvector multiplication. In this contribution, we will introduce our implementation of wavelet Galerkin method for Poisson equation on hypercube with homogeneous Dirichlet boundary conditions. In our implementation, we identified nonzero elements of stiffness matrix corresponding to the above problem and we perform matrixvector multiplication only with these nonzero elements.

Quadratic wavelets with short support on the interval
View Description Hide DescriptionIt is wellknown that a Bspline of order m has the shortest support among all compactly supported spline functions with respect to a given smoothness. And recently, B. Han and Z. Shen constructed a Riesz wavelet bases of the space L _{2}(R) with the shortest support and with m vanishing moments based on Bspline of order m. Such wavelets are important for example in signal processing and in numerical solution of differential equations because of their excellent approximation properties and fast algorithms which provide. In our contribution, we present an adaptation of quadratic wavelets to the interval [0,1] which preserves vanishing moments. The proposed adaptation is a modification of the approach proposed by D. Černá et al. and leads to a better conditioned basis.

Discontinuous Galerkin method for numerical solution of the regularized long wave equation
View Description Hide DescriptionIn this paper we are concerned with the development of sufficiently robust, accurate and efficient numerical method for the solution of the regularized long wave (RLW) equation, an important nonlinear equation describing a large class of physical phenomena. The main idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semidiscretization and the backward difference formula for the time discretization. This proposed scheme seems to be a promising technique due to the highorder piecewise polynomial discontinuous approximation and avoiding the time step restriction. The appended numerical experiments investigate the conservative properties of the RLW equation related to mass, momentum and energy, and illustrate the potency of the scheme, consequently.

Wavelet bases on the interval with short support and vanishing moments
View Description Hide DescriptionJia and Zhao have recently proposed a construction of a cubic spline wavelet basis on the interval which satisfies homogeneous Dirichlet boundary conditions of the second order. They used the basis for solving fourth order problems and they showed that Galerkin method with this basis has superb convergence. The stiffness matrices for the biharmonic equation defined on a unit square have very small and uniformly bounded condition numbers. In our contribution, we design wavelet bases with the same scaling functions and different wavelets. We show that our basis has the same quantitative properties as the wavelet basis constructed by Jia and Zhao and additionally the wavelets have vanishing moments. It enables to use this wavelet basis in adaptive wavelet methods and nonadaptive sparse grid methods. Furthermore, we even improve the condition numbers of the stiffness matrices by including lower levels.

Adaptive wavelet scheme for singularly pertubed boundary value problems
View Description Hide DescriptionThe paper is concerned with theoretical and computational issues of a numerical resolution of a singularly pertubed boundary value problem. We use a modification of an adaptive wavelet method from [3] with a cubic splinewavelet basis [2] and efficient matrixvector multiplication [1]. Theoretical advantages of an adaptive wavelet scheme as well as a numerical example will be presented.
