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Riemann solution for hyperbolic equations with discontinuous coefficients

Remaki, L. — 2013

Applications of Mathematics 2013

This paper deals with a Riemann solution for scalar hyperbolic equations with discontinuous coefficients. In many numerical schemes of Godunov type in fluid dynamics, electromagnetic and so on, usually hyperbolic problems are solved to estimate fluxes. The exact solution is generally difficult to obtain, but good approximations are provided in many situations like Roe and HLLC Riemann solvers in fluid. However all these solvers assumes that the acoustic waves speeds are continuous which is not...

Differential algebraic equations of Filippov type

Biák, MartinJanovská, Drahoslava — 2015

Application of Mathematics 2015

We will study discontinuous dynamical systems of Filippov-type. Mathematically, Filippov-type systems are defined as a set of first-order differential equations with discontinuous right-hand side. These systems arise in various applications, e.g. in control theory (so called relay feedback systems), in chemical engineering (an ideal gas--liquid system), or in biology (predator-prey models). We will show the way how to extend these models by a set of algebraic equations and then study the resulting...

Numerical approximation of density dependent diffusion in age-structured population dynamics

Gerardo-Giorda, Luca — 2013

Applications of Mathematics 2013

We study a numerical method for the diffusion of an age-structured population in a spatial environment. We extend the method proposed in [2] for linear diffusion problem, to the nonlinear case, where the diffusion coefficients depend on the total population. We integrate separately the age and time variables by finite differences and we discretize the space variable by finite elements. We provide stability and convergence results and we illustrate our approach with some numerical result.

Numerical analysis of a lumped parameter friction model

Janovský, Vladimír — 2015

Application of Mathematics 2015

We consider a contact problem of planar elastic bodies. We adopt Coulomb friction as (an implicitly defined) constitutive law. We will investigate highly simplified lumped parameter models where the contact boundary consists of just one point. In particular, we consider the relevant static and dynamic problems. We are interested in numerical solution of both problems. Even though the static and dynamic problems are qualitatively different, they can be solved by similar piecewise-smooth continuation...

On the interpolation constants over triangular elements

Kobayashi, Kenta — 2015

Application of Mathematics 2015

We propose a simple method to obtain sharp upper bounds for the interpolation error constants over the given triangular elements. These constants are important for analysis of interpolation error and especially for the error analysis in the Finite Element Method. In our method, interpolation constants are bounded by the product of the solution of corresponding finite dimensional eigenvalue problems and constant which is slightly larger than one. Guaranteed upper bounds for these constants are obtained...

Fuzzy sets and small systems

Považan, JaroslavRiečan, Beloslav — 2013

Applications of Mathematics 2013

Independently with [7] a corresponding fuzzy approach has been developed in [3-5] with applications in measure theory. One of the results the Egoroff theorem has been proved in an abstract form. In [1] a necessary and sufficient condition for holding the Egoroff theorem was presented in the case of a space with a monotone measure. By the help of [2] and [6] we prove a variant of the Egoroff theorem stated in [4].

Message doubling and error detection in the binary symmetrical channel

Kárná, LucieKlapka, Štěpán — 2015

Application of Mathematics 2015

The error correcting codes are a common tool to ensure safety in various safety-related systems. The usual technique, employed in the past, is to use two independent transmission systems and to send the safety relevant message two times. This article focuses on analysis of the detection properties of this strategy in the binary symmetrical channel (BSC) model. Besides, various modifications of the mentioned technique can be used. Their impact on the detection properties can be significant, positively...

h p -anisotropic mesh adaptation technique based on interpolation error estimates

Dolejší, Vít — 2013

Applications of Mathematics 2013

We present a completely new h p -anisotropic mesh adaptation technique for the numerical solution of partial differential equations with the aid of a discontinuous piecewise polynomial approximation. This approach generates general anisotropic triangular grids and the corresponding degrees of polynomial approximation based on the minimization of the interpolation error. We develop the theoretical background of this approach and present a numerical example demonstrating the efficiency of this anisotropic...

Application of Richardson extrapolation with the Crank-Nicolson scheme for multi-dimensional advection

Zlatev, ZahariDimov, IvanFaragó, IstvánGeorgiev, KrassimirHavasi, ÁgnesOstromsky, Tzvetan — 2013

Applications of Mathematics 2013

Multi-dimensional advection terms are an important part of many large-scale mathematical models which arise in different fields of science and engineering. After applying some kind of splitting, these terms can be handled separately from the remaining part of the mathematical model under consideration. It is important to treat the multi-dimensional advection in a sufficiently accurate manner. It is shown in this paper that high order of accuracy can be achieved when the well-known Crank-Nicolson...

On the computation of moments of the partial non-central χ -square distribution function

Gil, AmparoSegura, JavierTemme, Nico C. — 2013

Applications of Mathematics 2013

Properties satisfied by the moments of the partial non-central χ -square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.

A method to rigorously enclose eigenpairs of complex interval matrices

Castelli, RobertoLessard, Jean-Philippe — 2013

Applications of Mathematics 2013

In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair x = ( λ , ) is found by solving a nonlinear equation of the form f ( x ) = 0 via a contraction argument. The set-up of the method relies on the notion of r a d i i p o l y n o m i a l s , which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

Wildland fire propagation modelling: A novel approach reconciling models based on moving interface methods and on reaction-diffusion equations

Kaur, InderpreetMentrelli, AndreaBosseur, FredericFilippi, Jean BaptistePagnini, Gianni — 2015

Application of Mathematics 2015

A novel approach to study the propagation of fronts with random motion is presented. This approach is based on the idea to consider the motion of the front, split into a drifting part and a fluctuating part; the front position is also split correspondingly. In particular, the drifting part can be related to existing methods for moving interfaces, for example, the Eulerian level set method and the Lagrangian discrete event system specification. The fluctuating part is the result of a comprehensive...

Numerical method for the mixed Volterra-Fredholm integral equations using hybrid Legendre functions

Nemati, S.Lima, P.Ordokhani, Y. — 2015

Application of Mathematics 2015

A new method is proposed for the numerical solution of linear mixed Volterra-Fredholm integral equations in one space variable. The proposed numerical algorithm combines the trapezoidal rule, for the integration in time, with piecewise polynomial approximation, for the space discretization. We extend the method to nonlinear mixed Volterra-Fredholm integral equations. Finally, the method is tested on a number of problems and numerical results are given.

On the number of stationary patterns in reaction-diffusion systems

Rybář, VojtěchVejchodský, Tomáš — 2015

Application of Mathematics 2015

We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called Turing patterns. These solutions are typically non-unique and it is not clear how many of them exists. Since there are no analytical results available, we look for the number of distinct stationary solutions numerically. As a typical example, we investigate the reaction-diffusion system designed to model...

Finite element modelling of flow and temperature regime in shallow lakes

Podsechin, VictorSchernewski, Gerald — 2013

Applications of Mathematics 2013

A two-dimensional depth-averaged flow and temperature model was applied to study the circulation patterns in the Oder (Szczecin) Lagoon located on the border between Germany and Poland. The system of shallow water and temperature evolution equations is discretized with the modified Utnes scheme [4], which is characterized by a semi-decoupling algorithm. The continuity equation is rearranged to Helmholtz equation form. The upwinding Tabata method [3] is used to approximate convective terms. Averaged...

Why quintic polynomial equations are not solvable in radicals

Křížek, MichalSomer, Lawrence — 2015

Application of Mathematics 2015

We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed by radicals, i.e., by the operations + , - , · , : , and · n . Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.

On the computational identification of temperature-variable characteristics of heat transfer

Vala, Jiří — 2013

Applications of Mathematics 2013

The mathematical analysis of a heat equation and its solutions is a standard part of most textbook of applied mathematics and computational mechanics. However, serious problems from engineering practice do not respect formal simplifications of such analysis, namely at high temperatures, for phase-change materials, etc. This paper, motivated by the material design and testing of a high-temperature thermal accumulator, as a substantial part of the Czech-Swedish project of an original equipment for...

Spherically symmetric solutions to a model for interface motion by interface diffusion

Zhu, Peicheng — 2013

Applications of Mathematics 2013

The existence of spherically symmetric solutions is proved for a new phase-field model that describes the motion of an interface in an elastically deformable solid, here the motion is driven by configurational forces. The model is an elliptic-parabolic coupled system which consists of a linear elasticity system and a non-linear evolution equation of the order parameter. The non-linear equation is non-uniformly parabolic and is of fourth order. One typical application is sintering.

Dynamic contact problems in bone neoplasm analyses and the primal-dual active set (PDAS) method

Nedoma, Jiří — 2015

Application of Mathematics 2015

In the contribution growths of the neoplasms (benign and malignant tumors and cysts), located in a system of loaded bones, will be simulated. The main goal of the contribution is to present the useful methods and efficient algorithms for their solutions. Because the geometry of the system of loaded and possible fractured bones with enlarged neoplasms changes in time, the corresponding mathematical models of tumor's and cyst's evolutions lead to the coupled free boundary problems and the dynamic...

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