APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS' 33: 33rd InternationalConference

Control of Double Pendulum with Uncertain Parameters
View Description Hide DescriptionThe subject of the paper is a problem of designing control algorithms for mechanical systems with uncertainty. We consider a generic Lagrangian mechanical system under the assumption that its mass‐inertia parameters are not known exactly and that the system is subjected to unknown bounded disturbances. We assume also that the disturbances are smaller than the control forces which in turn do not exceed the force of gravity. A bounded control is proposed which, under certain conditions, steers the system from an arbitrary initial state to a prescribed terminal state in finite time. To this end, first, for some reference dynamical system close to the original one we design a trajectory connecting the initial and the terminal states. Then, using the trajectory tracking technique we steer the system under consideration to the prescribed terminal state.
The proposed approach is illustrated by the results of the computer simulation of steering a plain two‐link pendulum.

On a Higher‐Gradient Generalization of Fourier's Law of Heat Conduction
View Description Hide DescriptionThis paper deals with a possible generalization of Fourier's law that incorporates spatial memory into the constitutive relation. The integral and differential versions of the memory terms in the constitutive relation are discussed. It is shown that the asymptotically correct model contains the biharmonic operator as the vehicle for the higher‐order heat diffusion that also accounts for the spatial memory of the processes. Different solutions in 1D, 2D and 3D are presented to show the applicability of the new model.

Effective Hydrodynamic Flow of Suspensions in Presence of Apparent Slip at Boundaries
View Description Hide DescriptionCreeping flows may be submitted to an apparent slip at boundaries, which modifies their general behavior. Two cases are considered here: (i) a very dilute suspension of independent solid spheres; a nearby rough wall is equivalent to a smooth wall on which a slip boundary condition applies; (ii) a suspension of solid spheres moving along a smooth wall, for which an apparent slip on the wall arises from the non zero volume fraction and depends much on the distribution of the suspension. Case (i) is linear since the slip may be obtained from the solution of Stokes equations. In case (ii) Stokes equations are solved but the problem is in general non‐linear since the slip boundary condition is obtained in terms of the particle distribution which may depend on the flow field.

Trust Management in an Agent‐Based Grid Resource Brokering System‐Preliminary Considerations
View Description Hide DescriptionIt has been suggested that utilization of autonomous software agents in computational Grids may deliver the needed functionality to speed‐up Grid adoption. I our recent work we have outlined an approach in which agent teams facilitate Grid resource brokering and management. One of the interesting questions is how to manage trust in such a system. The aim of this paper is to outline our proposed solution.

Busy Period of a Delayed‐Service Single‐Server Poisson Queue
View Description Hide DescriptionIn this paper we investigate busy period of a single‐server Poisson queue with delayed‐service. We will analyze this model by considering M/G/1 approximating a non‐Markovian system. We obtain the distribution of the length of a busy period. Additionally, steady‐state mean and distribution of the queue length will be obtained.

Mathematical Aspects of Electoral Systems
View Description Hide DescriptionIn this paper we consider some mathematical aspects of electoral systems. Sometimes the results from elections seem paradoxical although they are mathematically correct. These cases are known as electoral paradoxes. A number of paradoxes of proportional and majoritarian electoral systems are considered.

K–Minimax Stochastic Programming Problems
View Description Hide DescriptionThe purpose of this paper is a discussion of a numerical procedure based on the simplex method for stochastic optimization problems with partially known distribution functions. The convergence of this procedure is proved by the condition on dual problems.

Explicit Formulae for Ellipsoids Approximating Reachable Sets
View Description Hide DescriptionWe obtain explicit formulas for ellipsoids bounding reachable sets for linear dynamic systems with geometric bounds on control. We study both locally and globally optimal ellipsoidal estimates with regard to different optimality criteria. In particular, we solve some essentially nonlinear boundary problems related to the search for globally optimal ellipsoids with regard to the volume criterion. It is shown that by using the explicit formulas one can efficiently pass to limits in several asymptotic problems, including passing to the limit when the phase space dimension goes to infinity.

Forecasting Electricity Demand by Time Series Models
View Description Hide DescriptionElectricity demand is one of the most important variables required for estimating the amount of additional capacity required to ensure a sufficient supply of energy. Demand and technological losses forecasts can be used to control the generation and distribution of electricity more efficiently. The aim of this paper is to utilize time series model, which provides forecasts of both magnitude and timing for lead times of one year.

Christov‐Galerkin Expansion for Localized Solutions in Model Equations with Higher Order Dispersion
View Description Hide DescriptionWe develop a Galerkin spectral technique for computing localized solutions of equations with higher order dispersion. To this end, the complete orthonormal system of functions in proposed by Christov [1] is used.
As a featuring example, the Sixth‐Order Generalized Boussinesq Equation (6GBE) is investigated whose solutions comprise monotone shapes (sech‐es) and damped oscillatory shapes (Kawahara solitons). Localized solutions are obtained here numerically for the case of the moving frame which are used as initial conditions for the time dependent problem.

Stabilization of Chaotic Behavior in the Restricted Three‐Body Problem
View Description Hide DescriptionA new type of orbit in the restricted three‐body problem is constructed. It is analytically shown that along with the well known chaotic and regular orbits in the three‐body problem there also exists a qualitatively different type of orbit which we call “stabilized.” The stabilized orbits are a result of additional orbiting bodies that are placed in the triangular Lagrange points. The results are well confirmed by numerical orbit calculations.

Gravitational Lensing by Rotating Naked Singularities in the Equatorial Plane
View Description Hide DescriptionWe model massive compact objects in galactic nuclei as stationary, axially‐symmetric naked singularities in the massless Einstein scalar field theory and study the resulting gravitational lensing in the strong deflection limit. In this approximation we compute the position of the relativistic images and the flux ratio of the outermost and the innermost images for the weakly naked singularities (those contained within at least one photon sphere) and the results are compared with the case of strongly naked singularities (those not contained within any photon sphere). All of the lensing quantities are compared to particular cases as Schwarzschild and Kerr black holes as well as Janis–Newman–Winicour naked singularities.

On the Beam Functions Spectral Expansions for Fourth‐Order Boundary Value Problems
View Description Hide DescriptionIn this paper we develop further the Galerkin technique based on the so‐called beam functions with application to nonlinear problems. We make use of the formulas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth‐order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth‐order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam‐Galerkin solution and the analytical solution for the linear problem is better than for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than The beam‐Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev‐tau method is attributed to the smaller condition number of the matrices involved in the former.

Phases of 4D Black Holes in Scalar‐Tensor Theories of Gravity Coupled to Non‐Linear Electrodynamics
View Description Hide DescriptionIn the present, work we find new non‐unique, numerical solutions describing charged black holes coupled to non‐linear electrodynamics in a special type of scalar‐tensor theory. The solutions describe three phases of black holes, and each of them is characterized by the value of the asymptotic scalar‐field charge. The presence of multiple phases implies that gravitational phase transition might occur, which is a non‐trivial and poorly studied effect.

Initial Stages of Viscous Drop Impact on Solid Surface
View Description Hide DescriptionDrop impact has various applications like supercooled drops freezing when hitting air‐crafts or electrical power lines, spray cooling, soldering, ink jet printing. When a drop hits a solid surface, a circular wall jet appears, which may spread and/or splash afterwards. The modeling of this jet is still an open problem. In the present paper we propose an axisymmetric dynamic model of the jet appearance, as an inviscid fluid, and its evolution in time towards a thin viscous film, as a boundary layer develops from the wall. Typical values of the drop diameter considered here are and typical values of the impact velocity are Numerical results for the jet thickness and lateral velocity are shown for some typical process parameters and are compared with experimental data obtained with a rapid shutter video camera.

An Elastic Beam Mounted to a Spring‐Mass Dynamic System
View Description Hide DescriptionThe dynamic response of a flexible beam mounted to an elastically supported mass is the subject of study in the paper. The Finite Element Method is used for transforming the partial differential equation describing the transverse vibrations of the beam into an ODE system to which the equations concerning the mass‐spring system are added. The matrix exponent method is applied for the solution of the equations. A numerical example is proposed.

A Double Layer Model of the Electromagnetic and Thermal Processes in Induction Heating of Ferromagnetic Material
View Description Hide DescriptionThis paper presents the modeling of electromagnetic and heating processes in an inductor, where cylindrical ferromagnetic material has been placed. In the first part the electromagnetic mathematical problem is solved, as a result the power density is obtained. The power density takes part in the heat conduction equation. In the second part the thermal mathematical problem is solved, as a result the alteration of the temperature of the ferromagnetic material during the heating process is obtained. The parameters in both mathematical problems depend on the temperature. Because of that the stitching method is used for their finding. In [3, 4] the same mathematical problems are solved by the finite elements method. Comparing our results to those from [3] shows that they are similar. In contrast to [3, 4] our method allows the continuation of the analysis with the finding of the load power during the heating process. Thus result permits the determination of the load power alteration in the supplying inverter [1]. It is well‐known that during the induction hardening it is necessary to maintain constant current amplitude in the load circuit of the inverter. So the next aim of this research is to build up a controller, based on the developed model, which will procure the necessary mode.

Finite Element Solution of 1D Boundary Value Linear and Nonlinear Problems with Nonlocal Jump Conditions
View Description Hide DescriptionWe consider stationary linear and nonlinear problems on non‐connected layers with distinct material properties. A version of the finite element method (FEM) is used for discretization of the continuous problems. We formulate sufficient conditions under which we prove the discrete maximum principle and convergence of the numerical higher‐order finite elements solution. Efficient algorithm for solution of the FEM algebraic equations is proposed. Numerical experiments are also discussed.

Efficient Numerical Solution Schemes Combined with Spatial Analysis Simulation Models—Diffusion and Heat Transfer Problem
View Description Hide DescriptionA numerical analysis was carried out to model and simulate diffusion physical phenomena. The aim of the numerical analysis was to present several useful numerical solutions to the problem under investigation. We focused on the numerical solution of the problem under examination by using the Galerkin finite element method and the Crank–Nicolson numerical discretization method in time and space. Useful numerical analysis conclusions were made focusing on numerical manipulations to achieve an accelerated solution of the problem under investigation. Moreover, efficient combinations are proposed for the utilization of our numerical modeling using other risk assessment spatial tools that can be used in any related spatial analysis monitoring or bioremediation projects for the safety of public health and environmental protection.