Initial-condition problem for a chiral Gross-Neveu system

A time-dependent projection technique is used to treat the initial-value problem for self-interacting fermionic fields. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type for the set of one-body dynamical variables. A nonperturbative mean-field expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Gaussian mean-field approximation, for a uniform system described by the chiral Gross-Neveu Hamiltonian. Standard stationary features of the model, such as dynamical mass generation due to chiral symmetry breaking and a phenomenon analogous to dimensional transmutation, are reobtained in this context. The mean-field time evolution of nonequilibrium initial states is discussed.

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

Stability in Terms of Two Measures for Differential Equations with “Maxima”

Атанаска Георгиева, Стела Глухчева, Снежана Христова - Изследвана е устойчивостта на нелинейни диференциални уравнения с “максимуми” по отношение на две мерки. Приложени са две различни мерки за началните условия и за решението. Използван е методът на Разумихин, а също така и методът на сравнението на обикновени скаларни диференциални уравнения. Приложението на получените резултати и достатъчни условия за устойчивост е илюстрирано с пример.

Coordinate-Independent Computations on Differential Equations

This project investigates the computational representation of differentiable manifolds, with the primary goal of solving partial differential equations using multiple coordinate systems on general n- dimensional spaces. In the process, this abstraction is used to perform accurate integrations of ordinary differential equations using multiple coordinate systems. In the case of linear partial differential equations, however, unexpected difficulties arise even with the simplest equations.

STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs.

On exponential stability of functional differential equations with variable impulse perturbations

We consider a class of functional differential equations subject to perturbations, which vary in time, and we study the exponential stability of solutions of these equations using the theory of generalized ordinary differential equations and Lyapunov functionals. We introduce the concept of variational exponential stability for generalized ordinary differential equations and we develop the theory in this direction by establishing conditions for the trivial solutions of generalized ordinary differential equations to be exponentially stable. Then, we apply the results to get corresponding ones for impulsive functional differential equations. We also present an example of a delay differential equation with Perron integrable right-hand side where we apply our result.

Student Difficulties with units in differential equations in modelling contexts

First-year undergraduate engineering students' understanding of the units of factors and terms in first-order ordinary differential equations used in modelling contexts was investigated using diagnostic quiz questions. Few students appeared to realize that the units of each term in such equations must be the same, or if they did, nevertheless failed to apply that knowledge when needed. In addition, few students were able to determine the units of a proportionality factor in a simple equation. These results indicate that lecturers of modelling courses cannot take this foundational knowledge for granted and should explicitly include it in instruction.

Practical Stability and Vector-Lyapunov Functions for Impulsive Differential Equations with "Supremum"

Stability of nonlinear impulsive differential equations with "supremum" is studied. A special type of stability, combining two different measures and a dot product on a cone, is defined. Perturbing cone-valued piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential equations have been employed.

Modeling some real phenomena by fractional differential equations

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters.

Effect of boundary condition description on convergence of solution in a boundary-value problem

It is well known that the numerical accuracy of a series solution to a boundary-value problem by the direct method depends on the technique of approximate satisfaction of the boundary conditions and on the stage of truncation of the series. On the other hand, it does not appear to be generally recognized that, when the boundary conditions can be described in alternative equivalent forms, the convergence of the solution is significantly affected by the actual form in which they are stated. The importance of the last aspect is studied for three different techniques of computing the deflections of simply supported regular polygonal plates under uniform pressure. It is also shown that it is sometimes possible to modify the technique of analysis to make the accuracy independent of the description of the boundary conditions.

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Exercise sheet 8

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